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<h1 id="sorting">Sorting</h1>
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<h2 id="sorting-1">Sorting</h2>
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<ul>
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<li>re-arranging elements of a sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mtext> s.t. </mtext><msub><mi>S</mi><mn>0</mn></msub><mo>≤</mo><msub><mi>S</mi><mn>1</mn></msub><mo>≤</mo><msub><mi>S</mi><mn>2</mn></msub><mo>≤</mo><mo>⋯</mo><mo>≤</mo><msub><mi>S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">S \text{ s.t. } S_0 \leq S_1 \leq S_2 \leq \cdots \leq S_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord text"><span class="mord"> s.t. </span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></li>
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<li>We will look at 5 sorting algorithms:
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<ul>
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<li>3 iterative</li>
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<li>2 recursive</li>
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</ul>
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</li>
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</ul>
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<h2 id="the-iterative-algorithms">The iterative algorithms</h2>
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<ul>
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<li>maintain a partition: “unsorted part” & “srtoed part”</li>
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<li>sort a sequence of n elements in n-1 stages</li>
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<li>at each stage, move 1 element from the unsorted part to the sorted part:
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<ul>
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<li>(Diagram of a generic array with unseen “sorted” items on the left and “unsorted” elements on the right. Caption: “1 stage moves 1 element”)</li>
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</ul>
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</li>
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</ul>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>sort(A){
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* initialize
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* repeat n-1 times
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* move 1 element from unsorted and sorted part
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}
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</code></pre></div></div>
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<ul>
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<li>The algorithms differ in how they:
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<ul>
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<li>select an element to remove from the unsorted part</li>
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<li>insert it into the sorted part</li>
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</ul>
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</li>
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</ul>
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<h2 id="insertion-sort">Insertion Sort</h2>
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<ul>
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<li>Initially sorted part is just A[0]</li>
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<li>Repeat n-1 times
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<ul>
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<li>remove the first element from the unsorted part</li>
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<li>insert it into the sorted part (shifting elements to the right as needed)</li>
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</ul>
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</li>
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</ul>
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<p>Diagram of array as it gets sorted in three stages:</p>
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<ul>
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<li>Stage 1: sorted is leftmost (0th) element; n-1 elements are unsorted on the right.</li>
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<li>Stage 2: approximately half of the array is sorted; an arrow points from the leftmost value inside the unsorted side to an arbitrary position inside the sorted side.</li>
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<li>Stage 3: just over half of the array is sorted now.</li>
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</ul>
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<p>Code:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>insertion sort(A){
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for(i=1 to n-1){
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pivot = A[i] // first element in unsorted part
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j=i-1
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// The following loop shifts all elements in sorted parts that are larger than pivot 1 "to the right"
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while(j>=0 AND A[i] > pivot){
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A[j+i] = A[j] // shift jth
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j = j-1
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}
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A[j+i] = pivot // move pivot into position.
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}
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}
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</code></pre></div></div>
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<h2 id="insertion-sort-example">Insertion Sort Example</h2>
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<p>Stages:</p>
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<ul>
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<li>Stage 0: Original
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>5</td>
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<td>4</td>
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<td>2</td>
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<td>6</td>
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<td>1</td>
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<td>3</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Stage 1: (label: 4)
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>4</td>
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<td>5</td>
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<td>2</td>
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<td>6</td>
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<td>1</td>
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<td>3</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Stage 2: (label: 2)
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>2</td>
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<td>4</td>
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<td>5</td>
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<td>6</td>
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<td>1</td>
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<td>3</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Stage 3: (label: 6)
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>2</td>
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<td>4</td>
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<td>5</td>
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<td>6</td>
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<td>1</td>
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<td>3</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Stage 4: (label: 1)
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>1</td>
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<td>2</td>
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<td>4</td>
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<td>5</td>
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<td>6</td>
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<td>3</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Stage 5: (label: 3)
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>1</td>
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<td>2</td>
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<td>3</td>
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<td>4</td>
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<td>5</td>
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<td>6</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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</ul>
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<h2 id="selection-sort">Selection Sort</h2>
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<ul>
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<li>initially sorted part is empty</li>
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<li>repeat n-1 times
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<ul>
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<li>find the smallest element in the unsorted part</li>
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<li>make it the first position which becomes the now last position of sorted part.</li>
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</ul>
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</li>
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</ul>
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<p>Diagram of parts:</p>
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<ul>
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<li>Initially, the entire array is all unsorted.</li>
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<li>Over time the sorted elements stack up on the left.</li>
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<li>Every time an element is moved, it is moved from the unsorted part (lowest element) and swapped with the element just after the end of the sorted part, making the sorted part one element bigger.</li>
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<li>Eventually all elements are sorted in descending order.</li>
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</ul>
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<p>Code:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>selection_sort(A){
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for(i=1 to n-1){
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// find min element of unsorted
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j=i-1 // j is index of min found so far.
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k=i
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while(k<n){
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if(A[k]<A[j]) j=k;
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k=k+1
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}
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swap A[i-1] and A[j]
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}
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}
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</code></pre></div></div>
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<p>Process of Selection Sort:</p>
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<ul>
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<li>Original: all unsorted
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>5</td>
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<td>4</td>
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<td>2</td>
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<td>6</td>
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<td>1</td>
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<td>3</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Stage 1: [0] is sorted; 1 and 5 swap
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>1</td>
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<td>4</td>
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<td>2</td>
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<td>6</td>
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<td>5</td>
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<td>3</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Stage 2: [0..1] is sorted; 2 and 4 swap
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>1</td>
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<td>2</td>
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<td>4</td>
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<td>6</td>
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<td>5</td>
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<td>3</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Stage 3: [0..2] is sorted; 3 and 4 swap
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>1</td>
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<td>2</td>
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<td>3</td>
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<td>6</td>
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<td>5</td>
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<td>4</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Stage 4: [0..3] is sorted; 4 and 6 swap
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>1</td>
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<td>2</td>
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<td>3</td>
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<td>4</td>
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<td>5</td>
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<td>6</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Stage 5: [0..4] is sorted; annotation: s.t. s (final stage)
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>1</td>
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<td>2</td>
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<td>3</td>
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<td>4</td>
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<td>5</td>
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<td>6</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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</ul>
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<h2 id="heapsort-selection-sort-is-crossed-out">Heapsort (Selection Sort is crossed out)</h2>
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<ul>
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<li>Initially sorted part empty</li>
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<li>(highlighted) make unsorted part into a heap</li>
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<li>repeat n-1 times
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<ul>
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<li>find the smallest element in the unsorted part (Note: heap extract takes log(n) time vs. Θ(n) for the scan in selection sort)</li>
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<li>move it to the first position which becomes the new last position of the started part.</li>
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</ul>
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</li>
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</ul>
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<p>Consider the organization of array contents:</p>
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<ol>
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<li>(Diagram of array with sorted half on the right and the unsorted half on the left.) A purple arrow points to the leftmost element in the unsorted portion. The note reads: “if this is the root of the heap, then it is also the smallest element in the unsorted part, so is in its correct final position.
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|
|
To use this arrangement, the root of the heap keeps moving, so we have lots of shifting to do.”</li>
|
|
|
<li>(A diagram showing the same array with sorted and unsorted halves.) A purple arrow points to the last element in the array; it points to a purple circle. A purple square is at the leftmost element of the unsorted half (the one discussed in the last item). The note reads: “If this is the root of the, then everything works:
|
|
|
<ul>
|
|
|
<li>We extract the final element (purple circle); move the last leaf (purple square) to the root + do a percolate-down;
|
|
|
store the final element (purple circle) where the last element of the unsorted list (purple square) <em>was</em>,
|
|
|
which is now free, and is the correct final location for the previously final element (purple circle); after which we have:</li>
|
|
|
</ul>
|
|
|
<ul>
|
|
|
<li>(Diagram of array with the “sorted” half extended one cell over to encompass the purple circle)
|
|
|
* <strong>But</strong>: we must re-code our heap implementation s.t. the root is at A[n-1], with the result that the indexing is now less intuitive.</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Instead, we use a max-heap, and this arrangement:
|
|
|
<ul>
|
|
|
<li>(Diagram showcasing, as previously, a sorted half to the right and an unsorted half on the left. An orange circle labeled “root of heap” is the very first element of the list and the unsorted half; an orange square labeled “last leaf” sits at the end (rightmost side) of the unsorted half.)</li>
|
|
|
<li>The heap root is at A[0]</li>
|
|
|
<li>Heap Extraction remove the root of the heap (orange circle), moves the last leaf (orange square) to A[0],
|
|
|
freeing up the spot where the root of the heap (orange circle) belongs.</li>
|
|
|
<li>This leaves us with: (Diagram of the orange circle near the middle of the array, at the leftmost portion of the sorted half. The orange square is in the center of the unsorted half.)</li>
|
|
|
<li>Re-coding a min heap into a max heap is just replacing < with > and vice versa.</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ol>
|
|
|
|
|
|
<h2 id="heapsort-selectioon-sort-is-crossed-out">Heapsort (Selectioon Sort is crossed out)</h2>
|
|
|
|
|
|
<ul>
|
|
|
<li>initially sorted part empty</li>
|
|
|
<li>(highlighted) make unsorted part into a max heap</li>
|
|
|
<li>repeat n-1 times:
|
|
|
<ul>
|
|
|
<li>find the largest (smallest is crossed out) element in the unsorted part</li>
|
|
|
<li>move it to the last (first is crossed out) position which becomes the new first (last is crossed out) position of the sorted part.</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
|
|
|
<p>Code:</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>heapsort(A){
|
|
|
buildMaxHeap(A)
|
|
|
for(i=1 to n-1){
|
|
|
A[n-1] extractMax()
|
|
|
}
|
|
|
}
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<p>Stages of sorting:</p>
|
|
|
|
|
|
<ul>
|
|
|
<li>(Diagram of unsorted array with first element labeled as “heap with max here”.)</li>
|
|
|
<li>(Diagram of a half-sorted array showing the swap between the first and last elements of the unsorted portion of the array. Labeled as “take max element from root…” and “take last leaf from end of heap” with arrows pointing to one another.)</li>
|
|
|
<li>(Diagram of a half+1 sorted array, displaying the new sorted element that has been swapped from the root element of the heap. Labeled as “newest element of sorted part” and “this is the <em>final</em> location” (the new element just swapped), and “y new root of heap (which then gets percolated down)” (what is now the first element of the array, which was also just swapped).)</li>
|
|
|
</ul>
|
|
|
|
|
|
<p>Unsorted heap of size 1 has smallest element.</p>
|
|
|
|
|
|
<h2 id="heapsort-with-in-line-percolate-down">Heapsort with in-line percolate-down</h2>
|
|
|
|
|
|
<p>Code:</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>heapsort(A){
|
|
|
makeMaxHeap(A)
|
|
|
for(i=1 to n-1){
|
|
|
swap A[0] and A[n-1] // move last leaf to root and old root to where last leaf was
|
|
|
size <- n-i+1 // size of heap = size of unsorted part
|
|
|
// start of percolate down
|
|
|
j <- 0
|
|
|
while(2j+1 < size){
|
|
|
child <- 2j+1
|
|
|
if(2j+2 < size AND A[2j+2] < A[2j+1]){
|
|
|
child <- 2j+2
|
|
|
}
|
|
|
if(A[child]<A[j]){
|
|
|
swap A[child] and A[j]
|
|
|
j <- child
|
|
|
} else {
|
|
|
j <- size // termite the while
|
|
|
}
|
|
|
} // end of percolate down
|
|
|
}
|
|
|
}
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<h2 id="heapsort-example">Heapsort Example</h2>
|
|
|
|
|
|
<ul>
|
|
|
<li>Original:
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>5</td>
|
|
|
<td>4</td>
|
|
|
<td>2</td>
|
|
|
<td>6</td>
|
|
|
<td>1</td>
|
|
|
<td>3</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Turn into heap:
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>6</td>
|
|
|
<td>5</td>
|
|
|
<td>3</td>
|
|
|
<td>4</td>
|
|
|
<td>1</td>
|
|
|
<td>2</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Swap root (6) and last unsorted element (2):
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>2</td>
|
|
|
<td>5</td>
|
|
|
<td>3</td>
|
|
|
<td>4</td>
|
|
|
<td>1</td>
|
|
|
<td>6</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Re-heap the unsorted portion: [0..4]
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>5</td>
|
|
|
<td>4</td>
|
|
|
<td>3</td>
|
|
|
<td>2</td>
|
|
|
<td>1</td>
|
|
|
<td>6</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Swap root (5) and the last unsorted element (2):
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>1</td>
|
|
|
<td>4</td>
|
|
|
<td>3</td>
|
|
|
<td>2</td>
|
|
|
<td>5</td>
|
|
|
<td>6</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Re-heap the unsorted portion: [0..3]
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>4</td>
|
|
|
<td>2</td>
|
|
|
<td>3</td>
|
|
|
<td>1</td>
|
|
|
<td>5</td>
|
|
|
<td>6</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Swap root (4) and the last unsorted element (1):
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>1</td>
|
|
|
<td>2</td>
|
|
|
<td>3</td>
|
|
|
<td>4</td>
|
|
|
<td>5</td>
|
|
|
<td>6</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Re-heap unsorted portion: [0..2]
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>3</td>
|
|
|
<td>2</td>
|
|
|
<td>1</td>
|
|
|
<td>4</td>
|
|
|
<td>5</td>
|
|
|
<td>6</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Swap root (3) and last unsorted element (1):
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>1</td>
|
|
|
<td>2</td>
|
|
|
<td>3</td>
|
|
|
<td>4</td>
|
|
|
<td>5</td>
|
|
|
<td>6</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Re-heap unsorted portion: [0..1]
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>2</td>
|
|
|
<td>1</td>
|
|
|
<td>3</td>
|
|
|
<td>4</td>
|
|
|
<td>5</td>
|
|
|
<td>6</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Swap root (2) and last unsorted element (1):
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>1</td>
|
|
|
<td>2</td>
|
|
|
<td>3</td>
|
|
|
<td>4</td>
|
|
|
<td>5</td>
|
|
|
<td>6</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Array is sorted because unsorted portion is only 1 element.</li>
|
|
|
</ul>
|
|
|
|
|
|
<p>Tree version of above (heap):</p>
|
|
|
|
|
|
<p>Original:</p>
|
|
|
|
|
|
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
|
|
|
<ul role="tree">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t11_5">
|
|
|
5
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t11_4">
|
|
|
4
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t11_6">
|
|
|
6
|
|
|
</span>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t11_1">
|
|
|
1
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t11_2">
|
|
|
2
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t11_3">
|
|
|
3 (left)
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
<!-- END OF GENERATED CODE -->
|
|
|
|
|
|
<p>After re-heap and one removal:</p>
|
|
|
|
|
|
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
|
|
|
<ul role="tree">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_2">
|
|
|
2
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_5">
|
|
|
5
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_2">
|
|
|
2
|
|
|
</span>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_1">
|
|
|
1
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_3">
|
|
|
3
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
<!-- END OF GENERATED CODE -->
|
|
|
|
|
|
<p>After a second re-heap and removal:</p>
|
|
|
|
|
|
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
|
|
|
<ul role="tree">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_1">
|
|
|
1
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_4">
|
|
|
4
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_2">
|
|
|
2 (left)
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_3">
|
|
|
3
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
<!-- END OF GENERATED CODE -->
|
|
|
|
|
|
<p>After a third:</p>
|
|
|
|
|
|
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
|
|
|
<ul role="tree">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_1">
|
|
|
1
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_2">
|
|
|
2
|
|
|
</span>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_3">
|
|
|
3
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
<!-- END OF GENERATED CODE -->
|
|
|
|
|
|
<p>Examples stop here.</p>
|
|
|
|
|
|
<h2 id="heapsort-example-2">Heapsort Example (2)</h2>
|
|
|
|
|
|
<p>(Repeat same as above, except with different trees.)</p>
|
|
|
|
|
|
<p>Trees (Transcriber’s note: these trees don’t seem relavant to me…. but maybe I’m wrong):</p>
|
|
|
|
|
|
<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
|
|
|
<ul role="tree">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_2">
|
|
|
2 (crossed out with orange 5)
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_5">
|
|
|
5 (crossed out next to orange 2 which is also crossed out; an orange 4 is not crossed out)
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t0_4">
|
|
|
4 (crossed out with orange 2)
|
|
|
</span>
|
|
|
</li>
|
|
|
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1
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</ul>
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</li>
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<li role="treeitem" tabindex="-1">
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<span id="t0_3">
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3
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</span>
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</ul>
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</li>
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</ul>
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<!-- END OF GENERATED CODE -->
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<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
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<ul role="tree">
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<li role="treeitem" tabindex="-1">
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<span id="t0_1">
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1 (crossed out with an orange 4)
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<li role="treeitem" tabindex="-1">
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4 (crossed out with orange 1, which is also crossed out; an orange 2, not crossed out is next to it)
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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2 (left; crossed out with orange 1)
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3
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<!-- END OF GENERATED CODE -->
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<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
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<ul role="tree">
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<li role="treeitem" tabindex="-1">
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<span id="t0_1">
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1 (orange 2)
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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<span id="t0_2">
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2 (left; orange 1)
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<h2 id="time-complexity-of-iterative-sorting-algorithms">Time Complexity of Iterative Sorting Algorithms</h2>
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<ul>
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<li>each algorithm does exactly n-1 stages</li>
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<li>the work done at the i<sup>th</sup> stage varies with the algorithm (& input).</li>
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<li>we take # from item comparisons as a measure of work/time*.</li>
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</ul>
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<dl>
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<dt>Selection Sort</dt>
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|
<dd>exactly n-i comparisons to find num element in unsorted part</dd>
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<dt>Insertion Sort</dt>
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<dd>between 1 and i comparisons to find location for pivot</dd>
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<dt>HeapSort</dt>
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<dd>between 1 and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><msub><mo><mi>log</mi><mo></mo></mo><mn>2</mn></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2\log_{2} (n-i-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.74285em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> comparisons for percolate-down</dd>
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</dl>
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<h2 id="-number-of-comparisons">* Number of comparisons</h2>
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<ul>
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<li>We must verify # comparisons (or some constant times # comparisons) is an upper bound on work done by each algorithm.</li>
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<li>
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<h1 id="of-assignments--swaps-also-matters-in-actual-run-time">of assignments (& swaps) also matters in actual run time.</h1>
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</li>
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</ul>
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<h2 id="selection-sort-1">Selection Sort</h2>
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<p>On input of size n, # of comparisons is always (regardless of input):</p>
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>i</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mi>n</mi></mrow><mn>2</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">
|
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|
\begin{aligned}
|
|
|
\sum_{i=1}^{n-1} (n-i) & = \sum_{i=1}^{n-1} i\\
|
|
|
& = S(n-i)\\
|
|
|
& = \frac{(n-1)(n)}{2}\\
|
|
|
& = \frac{n^2 -n}{2}\\
|
|
|
& = \Theta(n^2)
|
|
|
\end{aligned}
|
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.292998em;vertical-align:-5.396499000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.8964989999999995em;"><span style="top:-7.896499em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011130000000004em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">i</span><span class="mclose">)</span></span></span><span style="top:-5.47883em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"></span></span><span style="top:-3.3918299999999997em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"></span></span><span style="top:-0.9147220000000001em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"></span></span><span style="top:0.9353860000000007em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:5.396499000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.8964989999999995em;"><span style="top:-7.896499em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011130000000004em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span></span></span><span style="top:-5.47883em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">i</span><span class="mclose">)</span></span></span><span style="top:-3.3918299999999997em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.9147220000000001em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:0.9353860000000007em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:5.396499000000001em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
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<h2 id="insertion-sort--worst-case">Insertion Sort – Worst Case</h2>
|
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<p>Upper Bound: <span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext># comparisons</mtext><mo>≤</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>i</mi><mo>=</mo><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mi>n</mi></mrow><mn>2</mn></mfrac><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{\# comparisons} \leq \sum_{i=1}^{n-1} i = \frac{n^{2} -n}{2} = O(n^{2})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord"># comparisons</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0787820000000004em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011130000000004em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.177108em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
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<p>Lower Bound:</p>
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<ul>
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<li>Worst case initial sequence is in reverse order. e.g.:
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>n</td>
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<td>n-1</td>
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<td>n-2</td>
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<td>…</td>
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<td>1</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>In the i<sup>th</sup> stage we have:
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>n-i+1</td>
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<td>n-1+2</td>
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<td>…</td>
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<td>n</td>
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<td>n-1</td>
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<td>n-1-1</td>
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<td>…</td>
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<td>2</td>
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<td>1</td>
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</tr>
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</tbody>
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</table>
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</li>
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<li>
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<table>
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<tbody>
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<tr>
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<td>n-i</td>
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<td>n-i+1</td>
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<td>…</td>
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<td>n-1</td>
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<td>n</td>
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<td>n-i-1</td>
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<td>…</td>
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<td>2</td>
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<td>1</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>This takes i comparisons, because the sorted part is of size i.</li>
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<li>So, <span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext># comparisons</mtext><mo>≤</mo><mo>∑</mo><msup><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{\# comparisons} \leq \sum{i=1}^{n-1} = \Omega(n^{2})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord"># comparisons</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.864108em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em;"></span><span class="mord">Ω</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></li>
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</ul>
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<p>So, insertion sort worst case is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n^{2})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p>
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<p>(Transcriber’s note: I’m fairly certain you can only use big-O notation when talking about worst case scenario, not Theta. But I’m leaving it as written.)</p>
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<h2 id="insertion-sort-best-case">Insertion Sort Best Case</h2>
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<p>Best case: initial sequence is fully ordered.</p>
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<p>Then: In each stage, exactly 1 comparison is made.</p>
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<p>So, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext># comparisons</mtext><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>=</mo><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{\# comparisons} = n-1 = \Theta(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord"># comparisons</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span>.</p>
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<h2 id="heapsort-worst-case">Heapsort Worst Case</h2>
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<p>Upper bound:</p>
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext># comparisons</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>≤</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mn>2</mn><msub><mo><mi>log</mi><mo></mo></mo><mn>2</mn></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mo><mi>log</mi><mo></mo></mo><mn>2</mn></msub><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>≤</mo><mn>2</mn><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><msub><mo><mi>log</mi><mo></mo></mo><mn>2</mn></msub><mi>n</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>≤</mo><mn>2</mn><mi>n</mi><msub><mo><mi>log</mi><mo></mo></mo><mn>2</mn></msub><mi>n</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">
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\begin{aligned}
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\text{\# comparisons} & \leq \sum_{i=1}^{n-1} 2\log_{2} (n-i+1)\\
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& = 2\sum_{i=1}^{n-1} \log_{2} (i+1)\\
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& = \leq 2\sum_{i=1}^{n-1} \log_{2} n\\
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& = \leq 2n\log_{2} n\\
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& = O(n \log n)
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\end{aligned}
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:13.136346000000001em;vertical-align:-6.318173000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.818173000000001em;"><span style="top:-8.818173000000002em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord text"><span class="mord"># comparisons</span></span></span></span><span style="top:-5.4393910000000005em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"></span></span><span style="top:-2.0606089999999995em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"></span></span><span style="top:0.35706000000000016em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"></span></span><span style="top:1.8570600000000002em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.318173000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.818173000000001em;"><span style="top:-8.818173000000002em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011130000000004em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-5.4393910000000005em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011130000000004em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-2.0606089999999995em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011130000000004em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span></span></span><span style="top:0.35706000000000016em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">2</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span></span></span><span style="top:1.8570600000000002em;"><span class="pstrut" style="height:3.8011130000000004em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.318173000000001em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
|
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<p>Lower Bound? (empty space)</p>
|
|
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|
|
<p>Base Case? (What input would lead to <strong>no</strong> movement during percolate-down?
|
|
|
What if we exclude this case?)</p>
|
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|
<h2 id="recursive-divide--conquer-sorting">Recursive Divide & Conquer Sorting</h2>
|
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<ul>
|
|
|
<li>Partition the sequence A into two parts <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
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<li>Recursively sort each of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
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<li>Combine the sorted versions of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to obtain a sorted version of A.</li>
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</ul>
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<p id="rec_diag">Diagram showing A: (<a href="#ret_diag">Back to second occurrence of the diagram)</a></p>
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<ol>
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<li>Original
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<ul>
|
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<li>
|
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<table>
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<tbody>
|
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<tr>
|
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<td>20</td>
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<td>…</td>
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<td>30</td>
|
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<td>…</td>
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<td>1</td>
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<td>…</td>
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<td>6</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Partition
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<ul>
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<li>Two separate arrays with no items shown.</li>
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</ul>
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</li>
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<li>Combine
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<ul>
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<li>
|
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<table>
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<tbody>
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<tr>
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<td>1</td>
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<td>…</td>
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<td>6</td>
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<td>…</td>
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<td>20</td>
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<td>…</td>
|
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<td>30</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
|
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</ol>
|
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<p>The algorithms differ in how they choose the partition, and how they combine the sorted parts.</p>
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<h2 id="mergesort">Mergesort</h2>
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<ul>
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<li>Uses the fact that merging two sorted lists is easy.</li>
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<li>Exmaple 1:
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<ul>
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|
<li>Two lists:
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<ul>
|
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<li>
|
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<table>
|
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<tbody>
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<tr>
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<td>2 (crossed out)</td>
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<td>5 (crossed out)</td>
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<td>6 (crossed out)</td>
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<td>9</td>
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|
<td>…</td>
|
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</tr>
|
|
|
</tbody>
|
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</table>
|
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</li>
|
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<li>
|
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|
<table>
|
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<tbody>
|
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<tr>
|
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<td>1 (crossed out)</td>
|
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|
<td>3 (crossed out)</td>
|
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<td>4 (crossed out)</td>
|
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<td>8</td>
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|
<td>…</td>
|
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|
</tr>
|
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|
</tbody>
|
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|
</table>
|
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|
</li>
|
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</ul>
|
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|
</li>
|
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|
<li>Combined:
|
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|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
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<tbody>
|
|
|
<tr>
|
|
|
<td>1</td>
|
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|
<td>2</td>
|
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|
<td>3</td>
|
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<td>4</td>
|
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|
<td>5</td>
|
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|
<td>6</td>
|
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|
<td> </td>
|
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|
<td>…</td>
|
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|
</tr>
|
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|
</tbody>
|
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|
</table>
|
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|
</li>
|
|
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</ul>
|
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|
</li>
|
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</ul>
|
|
|
</li>
|
|
|
<li>Example 2:
|
|
|
<ul>
|
|
|
<li>Two lists:
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>…</td>
|
|
|
<td>90 (crossed out)</td>
|
|
|
<td>91 (crossed out)</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>…</td>
|
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|
<td>80 (crossed out)</td>
|
|
|
<td>100</td>
|
|
|
<td>101</td>
|
|
|
<td>102</td>
|
|
|
<td>103</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>Combined:
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>…</td>
|
|
|
<td>80</td>
|
|
|
<td>90</td>
|
|
|
<td>91</td>
|
|
|
<td>100</td>
|
|
|
<td>101</td>
|
|
|
<td>102</td>
|
|
|
<td>103</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
|
|
|
<p>Takes O(n) time, where n is the total size.</p>
|
|
|
|
|
|
<h2 id="mergesort-1">Mergesort</h2>
|
|
|
|
|
|
<ul>
|
|
|
<li>Partition: first half & second half.</li>
|
|
|
<li>Combine: merge the parts</li>
|
|
|
</ul>
|
|
|
|
|
|
<p>Diagram showing the sorting of a list:</p>
|
|
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|
|
|
<ul>
|
|
|
<li>Original, length: n
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>5</td>
|
|
|
<td>9</td>
|
|
|
<td>4</td>
|
|
|
<td>3</td>
|
|
|
<td>7</td>
|
|
|
<td>8</td>
|
|
|
<td>2</td>
|
|
|
<td>1</td>
|
|
|
<td>6</td>
|
|
|
<td>10</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
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<li>Two parts, length <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌊</mo><mi>n</mi><mi mathvariant="normal">/</mi><mn>2</mn><mo stretchy="false">⌋</mo></mrow><annotation encoding="application/x-tex">\lfloor n/2 \rfloor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mord mathnormal">n</span><span class="mord">/2</span><span class="mclose">⌋</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo><mi>n</mi><mi mathvariant="normal">/</mi><mn>2</mn><mo stretchy="false">⌉</mo></mrow><annotation encoding="application/x-tex">\lceil n/2 \rceil</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌈</span><span class="mord mathnormal">n</span><span class="mord">/2</span><span class="mclose">⌉</span></span></span></span> respectively. All numbers are crossed out. Annotation: “recursively sort n/2 elements”
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>3</td>
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<td>4</td>
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<td>5</td>
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<td>7</td>
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<td>9</td>
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</tr>
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</tbody>
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</table>
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</li>
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<li>
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<table>
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<tbody>
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<tr>
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<td>1</td>
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<td>2</td>
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<td>6</td>
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<td>8</td>
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<td>10</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Annotation: merge two parts.
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>1</td>
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<td>2</td>
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<td>3</td>
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<td>4</td>
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<td>5</td>
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<td>6</td>
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<td>7</td>
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<td>8</td>
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<td>9</td>
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<td>10</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>works with linked-list or array implementations</li>
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<li>in array implementations, used <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> extra space.</li>
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</ul>
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<h2 id="mergesort-2">Mergesort</h2>
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<p>Code:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>mergesort(A,lo,hi){
|
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if(lo<hi){// there are >=2 items, so work to do
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mid <- floor((lo+hi)/2)
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mergesort(A,lo,mid)
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mergesort(A,mid+1,hi)
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merge(A,low,mid,hi)
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}
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}
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</code></pre></div></div>
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<h2 id="merge-for-merge-sort">Merge for Merge sort</h2>
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<p>Code:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>merge(A,lo,mid,hi){
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l <- lo
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r <- mid+1
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n <- lo
|
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while(l<mid AND r<hi){
|
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|
if(A[l]<A[r]){
|
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|
B[n] <- A[i]
|
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l++
|
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|
} else {
|
|
|
B[n] <- A[r]
|
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|
r++
|
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|
}
|
|
|
}
|
|
|
while(l<mid){
|
|
|
B[n] <- A[l]
|
|
|
l++;n++
|
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|
}
|
|
|
while(r<hi){
|
|
|
B[n] <- A[r]
|
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|
r++;n++
|
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|
}
|
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}// *
|
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|
</code></pre></div></div>
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<p>After *, the sorted sequence is in B[lo]…B[hi].</p>
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<ul>
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<li>Lazy solution:
|
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<ul>
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|
<li>copy B[lo]..B[hi] to A.</li>
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</ul>
|
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</li>
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<li>Fast solution:
|
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<ul>
|
|
|
<li>swap A,B, as in:
|
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|
<ul>
|
|
|
<li>temp <- A</li>
|
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<li>A <- B</li>
|
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<li>B <- temp</li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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<h2 id="time-complexity-of-mergesort-via-tree-of-recursive-calls">Time Complexity of MergeSort: via tree of recursive calls</h2>
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<!-- AUTOGENERATED HTML FROM PYTHON TOOL https://github.com/TTWNO/transcription-tools -->
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<ul role="tree">
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<li role="treeitem" tabindex="-1">
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<span id="t9_n">
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n
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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<span id="t9_n/2">
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n/2
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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<span id="t9_n/4">
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n/4
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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<span id="t9_n/8">
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n/8
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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<span id="t9_...">
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...
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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<span id="t9_2">
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2
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<span id="t9_1">
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1
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<span id="t9_1">
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1
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</span>
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</li>
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</ul>
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</li>
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</ul>
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</li>
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<li role="treeitem" tabindex="-1">
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<span id="t9_n/8">
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n/8
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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<span id="t9_...">
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...
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</span>
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</li>
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</ul>
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</li>
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</ul>
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</li>
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<li role="treeitem" tabindex="-1">
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<span id="t9_n/4">
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n/4
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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<span id="t9_n/8">
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n/8
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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<span id="t9_...">
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...
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</span>
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</li>
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</ul>
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</li>
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<li role="treeitem" tabindex="-1">
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<span id="t9_n/8">
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n/8
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
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<span id="t9_...">
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...
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</span>
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</li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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</li>
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<li role="treeitem" tabindex="-1">
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<span id="t9_n/2">
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n/2
|
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</span>
|
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<ul role="group">
|
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<li role="treeitem" tabindex="-1">
|
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<span id="t9_n/4">
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n/4
|
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</span>
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<ul role="group">
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<li role="treeitem" tabindex="-1">
|
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<span id="t9_n/8">
|
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n/8
|
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</span>
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<ul role="group">
|
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<li role="treeitem" tabindex="-1">
|
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<span id="t9_...">
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...
|
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</span>
|
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|
</li>
|
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|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_n/8">
|
|
|
n/8
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_...">
|
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...
|
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</span>
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</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_n/4">
|
|
|
n/4
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_n/8">
|
|
|
n/8
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
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<span id="t9_...">
|
|
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...
|
|
|
</span>
|
|
|
</li>
|
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|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_n/8">
|
|
|
n/8
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_...">
|
|
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...
|
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</span>
|
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</li>
|
|
|
</ul>
|
|
|
</li>
|
|
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</ul>
|
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</li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
|
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|
<!-- END OF AUTOGENERATED HTML FROM PYTHON TOOL https://github.com/TTWNO/transcription-tools -->
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<p>Each level takes O(n), so total time is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo></mo></mrow><mn>2</mn></msub><mi>n</mi></mrow><annotation encoding="application/x-tex">\log_{2} n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.93858em;vertical-align:-0.24414em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span></span></span></span></p>
|
|
|
|
|
|
<ul>
|
|
|
<li>If n is a power of 2, the tree of recursive calls is a perfect binary tree with n leaves and height <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo></mo></mrow><mn>2</mn></msub><mi>n</mi></mrow><annotation encoding="application/x-tex">\log_{2} n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.93858em;vertical-align:-0.24414em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20696799999999996em;"><span style="top:-2.4558600000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span></span></span></span>.</li>
|
|
|
<li>At depth i there are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">2^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.824664em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> calls to merge, each to merge two lists of size <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>n</mi><msup><mn>2</mn><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup></mfrac></mrow><annotation encoding="application/x-tex">\frac{n}{2^{i+1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.040392em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7570857142857143em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> into one of size <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>n</mi><msup><mn>2</mn><mi>i</mi></msup></mfrac></mrow><annotation encoding="application/x-tex">\frac{n}{2^i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.040392em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7570857142857143em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</li>
|
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<li>Total work at depth i is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>i</mi></msup><mo>⋅</mo><mi>O</mi><mo stretchy="false">(</mo><mfrac><mi>n</mi><msup><mn>2</mn><mi>i</mi></msup></mfrac><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mfrac><msup><mn>2</mn><mi>i</mi></msup><msup><mn>2</mn><mi>i</mi></msup></mfrac><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2^{i}\cdot O(\frac{n}{2^i}) = O(n\frac{2^i}{2^i}) = O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.824664em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.095em;vertical-align:-0.345em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7570857142857143em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.3704599999999998em;vertical-align:-0.345em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0254599999999998em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7570857142857143em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9020857142857143em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>.</li>
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<li>Total work is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext># depths</mtext><mo>⋅</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>log</mi><mo></mo><mi>n</mi><mo>⋅</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{\# depths}\cdot O(n) = \log n \cdot O(n) = O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord"># depths</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>.</li>
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</ul>
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<h2 id="recursive-divide--conquer-sorting-1">Recursive Divide & Conquer Sorting</h2>
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<ul>
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<li>Partition the sequence A into two parts <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
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<li>Recursively sort each of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
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<li>Combine the sorted versions of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> to obtain a sorted version of A.</li>
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<li>
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<p id="ret_diag"><a href="#rec_diag">Diagram from earlier</a></p>
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</li>
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<li>The algorithms differ in how they choose the partition, and how they combine the sorted parts.</li>
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</ul>
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<h2 id="quicksort">Quicksort</h2>
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<ul>
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<li>Uses a pivot p to partition sequence into “small” and “large” elements: <em>small elements</em> < p < <em>large elements</em></li>
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<li>Combining sorted versions is trivial</li>
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</ul>
|
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<p>Diagram:</p>
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<ul>
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<li>list of length n</li>
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<li>now split into “partitions”; annotation: “choose a pivot p”
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>values <= p</td>
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<td>p</td>
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<td>values > p</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
|
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<li>recursively sort two parts</li>
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<li>Now:
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>values <= p in order</td>
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<td>p</td>
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<td>values > p in order</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>Choosing pivots is key to performance.</li>
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</ul>
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<h2 id="quicksort-1">Quicksort</h2>
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<p>Code:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>quicksort(A,lo,hi){
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if(lo<hi){// there are >= 2 items
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pivot position <- partition(A,lo,hi)// partition
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quicksort(A,lo,pivot position-1)
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quicksort(A,pivot position+1,hi)
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}
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}
|
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</code></pre></div></div>
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<p>Quicksort is correct as long as every call to partition() returns and leaves the variables satisfying the following:</p>
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<ol>
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<li>lo <= pivot position <= hi</li>
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<li>for every i,j with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>lo</mtext><mo>≤</mo><mi>i</mi><mo>≤</mo><mtext>pivot position</mtext><mo>≤</mo><mi>j</mi><mo>≤</mo><mtext>hi</mtext><mo separator="true">;</mo><mi>A</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>≤</mo><mi>A</mi><mo stretchy="false">[</mo><mtext>pivot position</mtext><mo stretchy="false">]</mo><mo>≤</mo><mi>A</mi><mo stretchy="false">[</mo><mtext>hi</mtext><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text{lo} \leq i \leq \text{pivot position} \leq j \leq \text{hi}; A[i] \leq A[\text{pivot position}] \leq A[\text{hi}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83041em;vertical-align:-0.13597em;"></span><span class="mord text"><span class="mord">lo</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79549em;vertical-align:-0.13597em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8623000000000001em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">pivot position</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">hi</span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord text"><span class="mord">pivot position</span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord text"><span class="mord">hi</span></span><span class="mclose">]</span></span></span></span></li>
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</ol>
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<p>However efficiency relies critically on choice of pivot.</p>
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<h2 id="ex-perfect-pivots">Ex: Perfect Pivots</h2>
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<ul>
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<li>Suppose all elements are distinct, and the pivot is chosen to be the median element in A[lo]…A[hi].</li>
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|
<li>Then, every call to Quicksort on sequence of size <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">k \geq 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83041em;vertical-align:-0.13597em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span> makes two recursive calls on sequences of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>size</mtext><mo>≤</mo><mi>k</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\text{size} \leq k/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.80383em;vertical-align:-0.13597em;"></span><span class="mord text"><span class="mord">size</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord">/2</span></span></span></span>: (Diagram shows partition labels):
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>(length) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mi>k</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\leq k/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord">/2</span></span></span></span></td>
|
|
|
<td>partition</td>
|
|
|
<td>(length) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mi>k</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\leq k/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord">/2</span></span></span></span></td>
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</tr>
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</tbody>
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</table>
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|
|
</li>
|
|
|
</ul>
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|
</li>
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|
<li>By essentially the same augment as used for MergeSort, this gives us running time of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo></mo><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\log(n)\cdot f(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>, where f(n) is the time to run partition on a sequence of size n.</li>
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|
|
<li>Assuming O(n) time for partition, this would give us <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> time for Quicksort.</li>
|
|
|
<li>But: finding medians is too slow in practice</li>
|
|
|
<li>Optional exercise: Can the median be found in O(n) time?</li>
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|
|
</ul>
|
|
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|
|
<h2 id="ex-worst-case-path">Ex: Worst Case Path</h2>
|
|
|
|
|
|
<ul>
|
|
|
<li>Suppose all elements are distinct, and the max is always chosen as pivot.</li>
|
|
|
<li>Then every call to Quicksort on a sequence of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">k\geq 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83041em;vertical-align:-0.13597em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span> elements makes one recursive call on a sequence of size <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>, and one on a sequence of size 0: (diagram of length of array partitions):
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>k-1</td>
|
|
|
<td>max</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>The recursion tree looks like this:</li>
|
|
|
</ul>
|
|
|
|
|
|
<!-- AUTOGENERATED HTML FROM PYTHON TOOL https://github.com/TTWNO/transcription-tools -->
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<ul role="tree">
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<li role="treeitem" tabindex="-1">
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<span id="t9_n">
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|
|
n
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|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_n-1">
|
|
|
n-1
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_n-2">
|
|
|
n-2
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_n-3">
|
|
|
n-3
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_...">
|
|
|
...
|
|
|
</span>
|
|
|
<ul role="group">
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_1">
|
|
|
1
|
|
|
</span>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_0">
|
|
|
0
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_0">
|
|
|
0
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_0">
|
|
|
0
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li role="treeitem" tabindex="-1">
|
|
|
<span id="t9_0">
|
|
|
0
|
|
|
</span>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
</ul>
|
|
|
<!-- END OF AUTOGENERATED HTML FROM PYTHON TOOL https://github.com/TTWNO/transcription-tools -->
|
|
|
|
|
|
<ul>
|
|
|
<li>This tree is of height <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>, giving us a running time of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n)\cdot f(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>, or <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n^{2})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> assuming <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> for partition.</li>
|
|
|
</ul>
|
|
|
|
|
|
<p>Quicksort takes time <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n^{2})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> in the worst case.</p>
|
|
|
|
|
|
<h2 id="partition">Partition</h2>
|
|
|
|
|
|
<p>Partition must choose a pivot p and efficiently re-arrange elements</p>
|
|
|
|
|
|
<p>Code:</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>partition(A,lo,hi){
|
|
|
pivotindex <- choosePivot(A,lo,hi) // choose pivot
|
|
|
swap A[pivotindex] and A[hi] // move pivot out of the way
|
|
|
p <- A[hi] // p is the pivot
|
|
|
i <- lo // known "small" values will be at nodes < i
|
|
|
for(j=lo;j<hi;j++){ // "already inspected" values will be at incicies < j
|
|
|
if(A[j]<=p){ // if we are inspecting a "small"
|
|
|
swap A[i] and A[j] // swap it with first "non small"
|
|
|
i <- i+1 // increase size of "smalls" part
|
|
|
}
|
|
|
}
|
|
|
swap A[i] and A[hi] // move pivot where it belongs
|
|
|
return i // this is pivot position
|
|
|
}
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Name</td>
|
|
|
<td>Description</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>lo</td>
|
|
|
<td>known to be small</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>…</td>
|
|
|
<td>…</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>i</td>
|
|
|
<td>known to be large</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>…</td>
|
|
|
<td>…</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>j</td>
|
|
|
<td>not inspected yet; j itself is currently being inspected</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>…</td>
|
|
|
<td>…</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>hi</td>
|
|
|
<td>pivot</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<h2 id="partition-example">Partition Example</h2>
|
|
|
|
|
|
<p>(Start of a diagram of how a quick sort happens over time.)</p>
|
|
|
|
|
|
<p>Legend:</p>
|
|
|
|
|
|
<dd>
|
|
|
<dt>-</dt>
|
|
|
<dd>known small</dd>
|
|
|
<dt>+</dt>
|
|
|
<dd>known large</dd>
|
|
|
</dd>
|
|
|
|
|
|
<p>The modifier will be displayed <em>after</em> the value to not be confused with negative and positive symbols.</p>
|
|
|
|
|
|
<p>If a variable (i.e. i, j) are set to an index of the array, it will show up in parenthesis after the value.</p>
|
|
|
|
|
|
<p>NOTE: Every table below has unseen elements to the left and right like so:</p>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>…</td>
|
|
|
<td>array values</td>
|
|
|
<td>…</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>However, because all the action is happening inside the visible part of the array, the left and right ellipses will not be shown below.</p>
|
|
|
|
|
|
<ul>
|
|
|
<li>Original:
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>5 (lo)</td>
|
|
|
<td>3</td>
|
|
|
<td>4 (pivot)</td>
|
|
|
<td>8</td>
|
|
|
<td>2</td>
|
|
|
<td>9</td>
|
|
|
<td>1</td>
|
|
|
<td>7 (hi)</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>swap pivot into hi
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>5 (i,j)</td>
|
|
|
<td>3</td>
|
|
|
<td>7</td>
|
|
|
<td>8</td>
|
|
|
<td>2</td>
|
|
|
<td>9</td>
|
|
|
<td>1</td>
|
|
|
<td>4</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>
|
|
|
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo>≮</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">A[j] \nless 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel amsrm">≮</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span></span>
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>5+ (i)</td>
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<td>3 (j)</td>
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<td>7</td>
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<td>8</td>
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<td>2</td>
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<td>9</td>
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<td>1</td>
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<td>4</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo><</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">A[j] < 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span></span>
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>3-</td>
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<td>5+ (i)</td>
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<td>7 (j)</td>
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<td>8</td>
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<td>2</td>
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<td>9</td>
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<td>1</td>
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<td>4</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo>≮</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">A[j] \nless 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel amsrm">≮</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span></span>
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>3-</td>
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<td>5+ (i)</td>
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<td>7+</td>
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<td>8 (j)</td>
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<td>2</td>
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<td>9</td>
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<td>1</td>
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<td>4</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo>≮</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">A[j] \nless 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel amsrm">≮</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span></span>
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>3-</td>
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<td>5+ (i)</td>
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<td>7+</td>
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<td>8+</td>
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<td>2 (j)</td>
|
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<td>9</td>
|
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<td>1</td>
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<td>4</td>
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</tr>
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</tbody>
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</table>
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</li>
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</ul>
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</li>
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<li>
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo><</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">A[j] < 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span></span>
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<ul>
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<li>
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<table>
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<tbody>
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<tr>
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<td>3-</td>
|
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<td>2-</td>
|
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<td>7+ (i)</td>
|
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<td>8+</td>
|
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<td>5+</td>
|
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<td>9 (j)</td>
|
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<td>1</td>
|
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<td>4</td>
|
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</tr>
|
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</tbody>
|
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</table>
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</li>
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</ul>
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</li>
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<li>
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo>≮</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">A[j] \nless 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel amsrm">≮</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span></span>
|
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<ul>
|
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<li>
|
|
|
<table>
|
|
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<tbody>
|
|
|
<tr>
|
|
|
<td>3-</td>
|
|
|
<td>2-</td>
|
|
|
<td>7+ (i)</td>
|
|
|
<td>8+</td>
|
|
|
<td>5+</td>
|
|
|
<td>9+</td>
|
|
|
<td>1 (j)</td>
|
|
|
<td>4</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
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</li>
|
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</ul>
|
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</li>
|
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<li>
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|
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo><</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">A[j] < 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span></span>
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>3-</td>
|
|
|
<td>2-</td>
|
|
|
<td>1-</td>
|
|
|
<td>8+ (i)</td>
|
|
|
<td>5+</td>
|
|
|
<td>9+</td>
|
|
|
<td>7+</td>
|
|
|
<td>4 (j)</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>swap A[i] & A[hi] <code class="language-plaintext highlighter-rouge">(small is first 3, pivot is 4, large is last 4)</code>
|
|
|
<ul>
|
|
|
<li>
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>3-</td>
|
|
|
<td>2-</td>
|
|
|
<td>1-</td>
|
|
|
<td>4 (i)</td>
|
|
|
<td>5+</td>
|
|
|
<td>9+</td>
|
|
|
<td>7+</td>
|
|
|
<td>8+ (j)</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
</li>
|
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|
</ul>
|
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|
</li>
|
|
|
</ul>
|
|
|
|
|
|
<p>(End of diagram)</p>
|
|
|
|
|
|
<h2 id="partition-1">Partition</h2>
|
|
|
|
|
|
<ul>
|
|
|
<li>Time complexity of partition is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>+</mo><mi>g</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n)+g(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>, where g(n) is time taken by choosePivot.</li>
|
|
|
<li>Q: How can we choose “good” pivots “fast”?</li>
|
|
|
<li>Quicksort is the most-used sorting algorithm in practice, so there must be a way.</li>
|
|
|
<li>But, what qualifies as “fast”? Probably a very small constant.</li>
|
|
|
<li>What qualifies as “good”? (given that it must be fast)</li>
|
|
|
</ul>
|
|
|
|
|
|
<h2 id="consider">Consider</h2>
|
|
|
|
|
|
<ul>
|
|
|
<li>A <em>small</em> number of bad pivots makes a <em>small</em> difference in height.</li>
|
|
|
<li>(A digram explaining the above. First it shows a perfect binary tree of height 5, next to a binary tree with height 6, where at depth 2, the “right right (RR)” subtree is extended with its own perfect binary tree of height 4, whereas the “right left (RL)” subtree contains nothing.)</li>
|
|
|
<li>Perfect pivot points are not needed for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> time.</li>
|
|
|
<li>(Diagram showing the progressive shortening of the list into smaller and smaller pieces. There are no number, markings or annotations.)</li>
|
|
|
<li>If pivots are all better than <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>⋅</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{3}\cdot \frac{2}{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, the depth is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo></mo><mn>3</mn><mi mathvariant="normal">/</mi><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">\log 3/2 n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">3/2</span><span class="mord mathnormal">n</span></span></span></span>, so we still get <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>.</li>
|
|
|
</ul>
|
|
|
|
|
|
<h2 id="some-simple-choosepivot-options">Some “simple” choosePivot options</h2>
|
|
|
|
|
|
<ul>
|
|
|
<li>A[hi] - fast, but performs badly on many inputs.</li>
|
|
|
<li>median - perfect pivots, but too slow to compute.</li>
|
|
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<li>random - If pivots are chosen uniformly at random, then Quicksort runs in time <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> with probability <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></mrow><annotation encoding="application/x-tex">1-\frac{1}{2^{n}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5935428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> – i.e. almost always. But: good random numbers are not fast to make.</li>
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<li><code class="language-plaintext highlighter-rouge">median{A[lo],A[hi],A[(hi+lo)/2]}</code>
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<ul>
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<li>fast</li>
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<li>not very easy to come up with a “very bad” input.</li>
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</ul>
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</li>
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</ul>
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<h2 id="complexity-of-quicksort">Complexity of Quicksort</h2>
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<ul>
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<li>Depends critically on how pivots are chosen.</li>
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<li>Choosing “perfect” pivots is too slow for a practical sorting algorithm.</li>
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<li>Fortunately, choosing pivots that are “good enough” for <em>most inputs</em> can be done fast.</li>
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<li>Quicksort – with practical pivot choice strategies – is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n^{2})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, but is often observed as “like <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> in practice.</li>
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</ul>
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<h2 id="in-practice">In practice</h2>
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<ul>
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<li>There are settings where Merge sort & Insertion sort are preferred.</li>
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<li>In most settings, the preferred algorithm is Quicksort.</li>
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<li>For small sets, SelectionSort is faster.</li>
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<li>Often, this variant (or similar) is faster:</li>
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</ul>
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<p>Code:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>quicksort(A,lo,hi){
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if(lo<hi){// there are >=2 items
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if(lo+15>hi){// less than 15 items
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selectionSort(A,lo,hi)
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}
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else {
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pivotposition <- partition(A,lo,hi) // partition
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quicksort(A,lo,pivotposition-1)
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quicksort(A,pivotposition+1,hi)
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}
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}
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}
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</code></pre></div></div>
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<h2 id="end">End</h2>
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</div>
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</main>
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