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<!DOCTYPE html>
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<meta charset="UTF-8">
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<title> | tait.tech</title>
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<script src="/assets/js/"></script>
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</head>
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<body>
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<main>
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<div id="wrapper">
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<h1 id="stacks--queues---1">Stacks & Queues - 1</h1>
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<p>CMPT 225, Fall 2021, Lecture 2</p>
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<h2 id="stack">Stack</h2>
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<ul>
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<li>ADT that stores a collection of objects/values</li>
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<li>Insertions and removals follow last-in-first-out pattern</li>
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<li>Fundamental Operations:
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<ul>
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<li>push: inserts an element on the “top”</li>
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<li>pop: removes + returns the top element</li>
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</ul>
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</li>
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</ul>
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<p>e.g.</p>
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<table>
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<tbody>
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<tr>
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<td>Stack index</td>
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<td>Value</td>
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</tr>
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</tbody>
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</table>
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<p>push a</p>
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<table>
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<tbody>
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<tr>
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<td>Stack index</td>
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<td>Value</td>
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</tr>
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<tr>
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<td>0</td>
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<td>a</td>
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</tr>
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</tbody>
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</table>
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<p>push b</p>
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<table>
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<tbody>
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<tr>
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<td>Stack index</td>
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<td>Value</td>
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</tr>
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<tr>
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<td>0</td>
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<td>b</td>
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</tr>
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<tr>
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<td>1</td>
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<td>a</td>
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</tr>
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</tbody>
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</table>
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<p>push c</p>
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<table>
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<tbody>
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<tr>
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<td>Stack index</td>
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<td>Value</td>
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</tr>
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<tr>
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<td>0</td>
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<td>c</td>
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</tr>
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<tr>
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<td>1</td>
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<td>b</td>
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</tr>
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<tr>
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<td>2</td>
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<td>a</td>
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</tr>
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</tbody>
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</table>
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<p>pop</p>
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<table>
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<tbody>
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<tr>
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<td>Stack index</td>
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<td>Value</td>
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</tr>
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<tr>
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<td>0</td>
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<td>b</td>
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</tr>
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<tr>
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<td>1</td>
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<td>a</td>
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</tr>
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</tbody>
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</table>
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<ul>
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<li>conventient optional operations
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<ul>
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<li>size: return # of elements on stack // encapsulation</li>
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<li>empty: check for emptiness // better than “size=0?”</li>
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<li>top: return top element, but don’t remove it // better than x = pop(); push(x) then use x</li>
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</ul>
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</li>
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</ul>
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<h2 id="algorithm-applications">Algorithm Applications</h2>
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<ul>
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<li>parasing/evaluation/transformation of expressions</li>
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<li>speech recognition coding/decoding</li>
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<li>shared network access control</li>
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<li>programming languages & execution
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<ul>
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<li>Postscript, Forth, the call stack</li>
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</ul>
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</li>
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<li>things we do daily with computers</li>
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</ul>
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<h2 id="parenthasies-checking">Parenthasies Checking</h2>
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<p>Parenthasies: <code class="language-plaintext highlighter-rouge">(()(((()())()()(()))</code></p>
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<p>A diagram showing:
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For every left parenthasies, add one to a counter.
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For every right parthenthasies, remove one from the counter.
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If the counter is not zero by the end of the sequence, there are too many on one side of the parenthasies.</p>
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<p>Can be done by counting unmached left parenthasies:</p>
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<p>Successful example:</p>
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<table>
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<tbody>
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<tr>
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<td>Character</td>
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<td>Counter</td>
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</tr>
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<tr>
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<td> </td>
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<td>0</td>
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</tr>
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<tr>
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<td>(</td>
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<td>1</td>
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</tr>
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<tr>
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<td>(</td>
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<td>2</td>
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</tr>
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<tr>
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<td>)</td>
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<td>1</td>
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</tr>
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<tr>
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<td>(</td>
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<td>2</td>
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</tr>
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<tr>
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<td>(</td>
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<td>3</td>
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</tr>
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<tr>
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<td>)</td>
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<td>2</td>
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</tr>
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<tr>
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<td>)</td>
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<td>1</td>
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</tr>
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<tr>
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<td>)</td>
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<td>0</td>
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</tr>
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<tr>
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<td>Yay!</td>
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<td>0</td>
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</tr>
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</tbody>
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</table>
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<p>Failed example:</p>
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<table>
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<tbody>
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<tr>
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<td>Character</td>
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<td>Country</td>
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</tr>
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<tr>
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<td> </td>
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<td>0</td>
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</tr>
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<tr>
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<td>(</td>
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<td>1</td>
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</tr>
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<tr>
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<td>(</td>
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<td>2</td>
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</tr>
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<tr>
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<td>)</td>
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<td>1</td>
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</tr>
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<tr>
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<td>(</td>
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<td>2</td>
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</tr>
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<tr>
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<td>)</td>
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<td>1</td>
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</tr>
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<tr>
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<td> </td>
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<td>Oh no it’s not zero!</td>
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</tr>
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</tbody>
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</table>
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<p>Another fail:</p>
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<table>
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<tbody>
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<tr>
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<td>Character</td>
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<td>Counter</td>
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</tr>
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<tr>
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<td> </td>
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<td>0</td>
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</tr>
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<tr>
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<td>(</td>
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<td>1</td>
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</tr>
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<tr>
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<td>)</td>
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<td>0</td>
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</tr>
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<tr>
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<td>(</td>
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<td>1</td>
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</tr>
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<tr>
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<td>)</td>
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<td>0</td>
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</tr>
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<tr>
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<td>)</td>
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<td>-1</td>
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</tr>
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<tr>
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<td> </td>
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<td>Negative 1 this time!</td>
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</tr>
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</tbody>
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</table>
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<p>Do this one yourself:</p>
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<table>
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<tbody>
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<tr>
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<td>Character</td>
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<td>Counter</td>
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</tr>
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<tr>
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<td>(</td>
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<td> </td>
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</tr>
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<tr>
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<td>)</td>
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<td> </td>
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</tr>
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<tr>
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<td>)</td>
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<td> </td>
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</tr>
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<tr>
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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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<p>But consider multiple grouping symbols: <code class="language-plaintext highlighter-rouge">({{)</code>, <code class="language-plaintext highlighter-rouge">({){</code>, <code class="language-plaintext highlighter-rouge">({[( ... ]</code>.
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Counting is not enough.</p>
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<h2 id="parenthasies-checking-1">Parenthasies Checking</h2>
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<p>Diagram showing failure on the last character of this sequence: <code class="language-plaintext highlighter-rouge">(()((()()))()(())))</code></p>
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<p>Can be done by counting unmatched left parenthasies:</p>
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<table>
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<tbody>
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<tr>
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<td>Character</td>
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<td>Counter</td>
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</tr>
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<tr>
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<td>(</td>
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<td>1</td>
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</tr>
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<tr>
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<td>(</td>
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<td>2</td>
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</tr>
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<tr>
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<td>)</td>
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<td>1</td>
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</tr>
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<tr>
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<td>(</td>
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<td>2</td>
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</tr>
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<tr>
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<td>(</td>
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<td>3</td>
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</tr>
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<tr>
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<td>)</td>
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<td>2</td>
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</tr>
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<tr>
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<td>)</td>
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<td>1</td>
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</tr>
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<tr>
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<td>)</td>
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<td>0</td>
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</tr>
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</tbody>
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</table>
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<p>Example 2:</p>
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<table>
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<tbody>
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<tr>
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<td>Character</td>
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<td>Counter</td>
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</tr>
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<tr>
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<td>(</td>
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<td>1</td>
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</tr>
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<tr>
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<td>(</td>
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<td>2</td>
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</tr>
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<tr>
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<td>)</td>
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<td>1</td>
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</tr>
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<tr>
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<td>(</td>
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<td>2</td>
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</tr>
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<tr>
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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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<p>Example 3:</p>
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<table>
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<tbody>
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<tr>
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<td>Character</td>
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<td>Counter</td>
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</tr>
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<tr>
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<td>(</td>
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<td>1</td>
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</tr>
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<tr>
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<td>)</td>
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<td>0</td>
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</tr>
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<tr>
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<td>(</td>
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<td>1</td>
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</tr>
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<tr>
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<td>)</td>
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<td>0</td>
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</tr>
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<tr>
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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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<p>Example 4:</p>
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<table>
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<tbody>
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<tr>
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<td>Character</td>
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<td>Counter</td>
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</tr>
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<tr>
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<td>(</td>
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<td>1</td>
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</tr>
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<tr>
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<td>)</td>
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<td>0</td>
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</tr>
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<tr>
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<td>)</td>
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<td>-1</td>
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</tr>
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<tr>
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<td>(</td>
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<td>0</td>
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</tr>
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</tbody>
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</table>
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<p>But consider multiple grouping symbols:</p>
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<ul>
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<li>sequence <code class="language-plaintext highlighter-rouge">({})</code> will and should work</li>
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<li>sequence <code class="language-plaintext highlighter-rouge">({)}</code> will work and should not</li>
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<li>sequence <code class="language-plaintext highlighter-rouge">({[( ...</code> ???</li>
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</ul>
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<p>Counting is not enough.</p>
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<h2 id="stack-based-algorithm-for-checking-grouping-symbols">Stack-based Algorithm for Checking Grouping Symbols</h2>
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<p>Pseudo-code:</p>
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<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>I is an input stream of symbols.
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S is a new empty stack.
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while there are signals to read from I {
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c is next symbol from I
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if c is a left grouping signal {
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push c on S
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} else if c is a right grouping signal{
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if s in empty { report error and stop }
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d is pop S
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if c and d do not match { report error and stop }
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endif // if c is not a grouping symbol, ignore it
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end while
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if S is not empty { report error and exit }
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report ok
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</code></pre></div></div>
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<p>Diagram showing how each set of parenthasies are matched, innermost first torwards the outermost.</p>
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<h2 id="stack-partially-filled-array-implementation">Stack: Partially-Filled Array Implementation</h2>
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<p>Variables:</p>
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<ul>
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<li>A - an array of stack elements</li>
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<li>capacity - size of array</li>
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<li>top - index in array of top stack element
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<ul>
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<li>-1 if stack is empty</li>
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</ul>
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</li>
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</ul>
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<p>A starts out as:</p>
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<table>
|
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<tbody>
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<tr>
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|
<td>Index</td>
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<td>0</td>
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<td>1</td>
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<td>2</td>
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<td>…</td>
|
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<td>capacity-1</td>
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</tr>
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|
<tr>
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|
<td>Value</td>
|
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<td>a</td>
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|
<td>b</td>
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|
<td>c</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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<ul>
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<li>Capacity = <size of A></li>
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<li>top = 2</li>
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</ul>
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<p>push e</p>
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<p>A now equals</p>
|
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<table>
|
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|
<tbody>
|
|
|
<tr>
|
|
|
<td>Index</td>
|
|
|
<td>0</td>
|
|
|
<td>1</td>
|
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|
<td>2</td>
|
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|
<td>3</td>
|
|
|
<td>4</td>
|
|
|
<td>…</td>
|
|
|
<td>capacity-1</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>Value</td>
|
|
|
<td>a</td>
|
|
|
<td>b</td>
|
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|
<td>c</td>
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|
<td>d</td>
|
|
|
<td>e</td>
|
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|
<td>…</td>
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|
<td> </td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<ul>
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|
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<li>capacity = <size of A></li>
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|
<li>top = 3</li>
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</ul>
|
|
|
|
|
|
<h2 id="the-queue-adt">The Queue ADT</h2>
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|
|
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<p>A container that stores a collection of items with insertion and removal in “first-in-first-out” order.</p>
|
|
|
|
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|
<p>Stack: a digram showing circled being placed on top of eachother, pushing circles into the tube, then popping them out from the same size (the top). Like a pringles can of elements.</p>
|
|
|
|
|
|
<p>Queue: a diagram showing circles being unqueued in on the left and dequeued on the right. Like waiting in line for food.</p>
|
|
|
|
|
|
<p>Basic operations:</p>
|
|
|
<ul>
|
|
|
<li>enqueue: add an item to the back of a list</li>
|
|
|
<li>dequeue: remove & return the item at the front of the list</li>
|
|
|
</ul>
|
|
|
|
|
|
<h2 id="example-palindrome-checking">Example: Palindrome Checking</h2>
|
|
|
|
|
|
<p>Sequence <code class="language-plaintext highlighter-rouge">aba</code> should succeed.
|
|
|
Sequence <code class="language-plaintext highlighter-rouge">aab</code> should not.</p>
|
|
|
|
|
|
<p>In an array: a diagram showing the following pattern with the variable <code class="language-plaintext highlighter-rouge">i</code> set to the leftmost element and <code class="language-plaintext highlighter-rouge">j</code> set to the rightmost element.</p>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>a</td>
|
|
|
<td>b</td>
|
|
|
<td>c</td>
|
|
|
<td>…</td>
|
|
|
<td>s</td>
|
|
|
<td>b</td>
|
|
|
<td>a</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>With a stack and queue:</p>
|
|
|
<ol>
|
|
|
<li>read symbols, starting each in a stack and a queue</li>
|
|
|
<li>while the stack, queue are not empty:
|
|
|
<ul>
|
|
|
<li>pop and element from the stack + delete an element from the queue</li>
|
|
|
<li>if different -> not a palindrome</li>
|
|
|
</ul>
|
|
|
</li>
|
|
|
<li>palindrome</li>
|
|
|
</ol>
|
|
|
|
|
|
<h2 id="queue-array-implementations">Queue: Array Implementations</h2>
|
|
|
|
|
|
<p>Enqueue: a,b,c,d.
|
|
|
Front is now a, last is now d.</p>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>a</td>
|
|
|
<td>b</td>
|
|
|
<td>c</td>
|
|
|
<td>d</td>
|
|
|
<td>…</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>Dequeue returns and deletes first element.
|
|
|
Front is now b, last is now d.</p>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>?</td>
|
|
|
<td>b</td>
|
|
|
<td>c</td>
|
|
|
<td>d</td>
|
|
|
<td>…</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>Many enquques/dequeues later:</p>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>s</td>
|
|
|
<td>…</td>
|
|
|
<td>m</td>
|
|
|
<td>n</td>
|
|
|
<td>o</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>Front element is now <code class="language-plaintext highlighter-rouge">m</code>, last element is now s.</p>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>?</td>
|
|
|
<td>t</td>
|
|
|
<td>u</td>
|
|
|
<td>v</td>
|
|
|
<td>w</td>
|
|
|
<td>…</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>A similar slide is here, but I can’t see the differnece.</p>
|
|
|
|
|
|
<p>Variables:</p>
|
|
|
<ul>
|
|
|
<li>array A is array of size capacity</li>
|
|
|
<li>front is 0</li>
|
|
|
<li>back is 0</li>
|
|
|
<li>size is 0</li>
|
|
|
<li>capacity = <size of array></li>
|
|
|
</ul>
|
|
|
|
|
|
<p>Pseudo-code:</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>array A = array of size capacity // holds queue contents
|
|
|
front = 0 // index in A of front element, if queue not empty
|
|
|
back = 0 // index where next unqueued element will go
|
|
|
size = 0 // number of elements in queue
|
|
|
capacity = <size of array> // ???(can't read) queue size supported
|
|
|
|
|
|
enqueue(x) { // adds x to the back of the queue
|
|
|
// requires size<capacity>
|
|
|
A[back] = x
|
|
|
size = size + 1
|
|
|
back = (back + 1) % capacity
|
|
|
}
|
|
|
|
|
|
dequeue() { // removes front element of queue and returns it
|
|
|
// required size > 0
|
|
|
temp = A[front]
|
|
|
front = (front + 1) % capacity
|
|
|
size = size - 1
|
|
|
return temp
|
|
|
}
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<p>Is <code class="language-plaintext highlighter-rouge">size == front - back</code>?</p>
|
|
|
|
|
|
<p>Diagram showing only true if front and back haven’t been switched around due to wrapping around back to the beginning o the array.</p>
|
|
|
|
|
|
<h2 id="circular-array-queue-example">“Circular Array” Queue Example</h2>
|
|
|
|
|
|
<ul>
|
|
|
<li>size = 0</li>
|
|
|
<li>front = 0</li>
|
|
|
<li>back = 0</li>
|
|
|
</ul>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>enqueue: a,b,c,d</p>
|
|
|
|
|
|
<ul>
|
|
|
<li>front = 0</li>
|
|
|
<li>back = 4</li>
|
|
|
</ul>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>a</td>
|
|
|
<td>b</td>
|
|
|
<td>c</td>
|
|
|
<td>d</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>dequque 3 times</p>
|
|
|
|
|
|
<ul>
|
|
|
<li>front = 3</li>
|
|
|
<li>back = 4</li>
|
|
|
</ul>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>d</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>enqueue e,f,g,h</p>
|
|
|
|
|
|
<ul>
|
|
|
<li>front = 3</li>
|
|
|
<li>back = 2</li>
|
|
|
</ul>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>g</td>
|
|
|
<td>h</td>
|
|
|
<td>empty</td>
|
|
|
<td>d</td>
|
|
|
<td>e</td>
|
|
|
<td>f</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>unqueue j</p>
|
|
|
|
|
|
<ul>
|
|
|
<li>size = 6</li>
|
|
|
<li>font = 3</li>
|
|
|
<li>back = 3</li>
|
|
|
</ul>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>g</td>
|
|
|
<td>h</td>
|
|
|
<td>j</td>
|
|
|
<td>d</td>
|
|
|
<td>e</td>
|
|
|
<td>f</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>dequeue 3 times</p>
|
|
|
|
|
|
<ul>
|
|
|
<li>back = 3</li>
|
|
|
<li>front = 0</li>
|
|
|
</ul>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>g</td>
|
|
|
<td>h</td>
|
|
|
<td>j</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>dequeue 3 times</p>
|
|
|
|
|
|
<ul>
|
|
|
<li>back = 3</li>
|
|
|
<li>front = 3</li>
|
|
|
</ul>
|
|
|
|
|
|
<table>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>Values</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
<td>empty</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<h2 id="end">End</h2>
|
|
|
|
|
|
</div>
|
|
|
</main>
|
|
|
<hr>
|
|
|
<footer>
|
|
|
</footer>
|
|
|
</body>
|
|
|
</html>
|
