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<h1 id="55-exercises">5.5 &nbsp; Exercises<a class="headerlink" href="#55-exercises" title="Permanent link">&para;</a></h1>
<h2 id="551-concept-review">5.5.1 &nbsp; Concept Review<a class="headerlink" href="#551-concept-review" title="Permanent link">&para;</a></h2>
<h3 id="1-which-element-leaves-a-stack-or-queue-first">1. &nbsp; Which Element Leaves a Stack or Queue First?<a class="headerlink" href="#1-which-element-leaves-a-stack-or-queue-first" title="Permanent link">&para;</a></h3>
<p>Prepare an empty stack <code>S</code> and an empty queue <code>Q</code>. Perform the same sequence of operations on each one:</p>
<p>Step 1: Add <code>A</code>.
Step 2: Add <code>B</code>.
Step 3: Remove and record one element.
Step 4: Add <code>C</code>.
Step 5: Keep removing and recording elements until the container is empty.</p>
<p>Write the order in which elements are removed from <code>S</code> and from <code>Q</code>. Explain the difference using "last-in-first-out" or "first-in-first-out."</p>
<details class="success">
<summary>Answer</summary>
<p>The removal order for stack <code>S</code> is <code>B, C, A</code>. After the elements <code>A, B</code> are added, the most recently added element, <code>B</code>, is popped first. After <code>C</code> is added,
<code>C, A</code> are popped in that order. This is "last-in-first-out."</p>
<p>The removal order for queue <code>Q</code> is <code>A, B, C</code>. After the elements <code>A, B</code> are added, the earliest added element, <code>A</code>, is removed first.
After <code>C</code> is added, <code>B, C</code> are removed in that order. This is "first-in-first-out."</p>
</details>
<h3 id="2-what-happens-when-the-rear-passes-the-end-of-the-array">2. &nbsp; What Happens When the Rear Passes the End of the Array?<a class="headerlink" href="#2-what-happens-when-the-rear-passes-the-end-of-the-array" title="Permanent link">&para;</a></h3>
<p>A queue is implemented with a circular array of length 5, whose indices are <code>0–4</code>.
Currently, <code>front = 3</code> and <code>size = 2</code>; <code>A, B</code> are stored at indices 3 and 4, respectively.</p>
<!-- numbered-subquestions -->
<ol>
<li>When <code>C</code> is enqueued, at which index should <code>C</code> be stored? What is <code>size</code> after the enqueue?</li>
<li>Next, dequeue once. Which element is removed? What are the new values of <code>front</code> and <code>size</code>?</li>
<li>What is the logical order from the front to the rear now? Does dequeuing require moving the other elements in the array? Why or why not?</li>
</ol>
<details class="success">
<summary>Answer</summary>
<ol>
<li>
<p>The new element's position is
<code>(front + size) % 5 = (3 + 2) % 5 = 0</code>,
so <code>C</code> is stored at index 0. After the enqueue, <code>size = 3</code>.</p>
</li>
<li>
<p>Dequeuing removes the current front element, <code>A</code>. The new front index is
<code>(3 + 1) % 5 = 4</code>, so <code>front = 4</code> and <code>size = 2</code>.</p>
</li>
<li>
<p>The logical order of valid elements is <code>B, C</code>, with <code>B</code> at index 4 and <code>C</code> at index 0.
Dequeuing requires only changing <code>front</code> and <code>size</code>. The circular array uses the remainder operation to wrap the index back to the beginning,
so there is no need to shift all the other elements forward.</p>
</li>
</ol>
</details>
<h3 id="3-operations-at-both-ends-of-a-deque">3. &nbsp; Operations at Both Ends of a Deque<a class="headerlink" href="#3-operations-at-both-ends-of-a-deque" title="Permanent link">&para;</a></h3>
<p>Here, <code>push_first</code> adds an element at the front, <code>push_last</code> adds one at the rear,
<code>pop_first</code> removes one from the front, and <code>pop_last</code> removes one from the rear.</p>
<p>Perform the following operations on an empty deque <code>deq</code>:</p>
<ol>
<li><code>push_last(A)</code></li>
<li><code>push_last(B)</code></li>
<li><code>push_first(C)</code></li>
<li><code>pop_last()</code></li>
<li><code>push_last(D)</code></li>
<li><code>pop_first()</code></li>
</ol>
<!-- numbered-subquestions -->
<ol>
<li>Which elements are returned by the two pop operations?</li>
<li>After all operations are complete, which elements remain from front to rear?</li>
<li>Examine the six operations. Can a queue that allows insertion only at the rear and removal only at the front perform all of them? If not, identify the operations it cannot perform. Then state whether a deque can perform them and explain why.</li>
</ol>
<details class="success">
<summary>Answer</summary>
<p>After the first three steps, the deque from front to rear is <code>[C, A, B]</code>.</p>
<!-- numbered-subquestions -->
<ol>
<li>
<p><code>pop_last()</code> removes <code>B</code>. After <code>D</code> is added, the deque is <code>[C, A, D]</code>,
and <code>pop_first()</code> then removes <code>C</code>.</p>
</li>
<li>
<p><code>[A, D]</code> remains.</p>
</li>
<li>
<p>A queue that permits insertion only at the rear and removal only at the front cannot perform all the operations.
Step 3, <code>push_first(C)</code>, requires insertion at the front, and Step 4, <code>pop_last()</code>, requires removal at the rear. Both are outside the operations supported by such a queue.
A deque permits insertion and removal at both ends, so it can perform all six operations.</p>
</li>
</ol>
</details>
<h2 id="552-programming-exercises">5.5.2 &nbsp; Programming Exercises<a class="headerlink" href="#552-programming-exercises" title="Permanent link">&para;</a></h2>
<h3 id="1-check-a-bracket-sequence">1. &nbsp; Check a Bracket Sequence<a class="headerlink" href="#1-check-a-bracket-sequence" title="Permanent link">&para;</a></h3>
<p>Given a string <code>s</code> containing only the three types of brackets <code>()</code>, <code>[]</code>, and <code>{}</code>, use a stack to determine whether the string is valid.</p>
<p>A valid sequence must satisfy both conditions: every closing bracket matches the type of the most recent unmatched opening bracket,
and no unmatched opening bracket remains after the traversal. Return a Boolean value for the result.</p>
<details class="tip">
<summary>Hints</summary>
<ol>
<li>You can create a mapping from each closing bracket to its matching opening bracket</li>
<li>When you encounter a closing bracket, first check whether the stack is empty, and then check whether the top matches</li>
<li>The stack must also be empty after the traversal</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/valid-parentheses/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
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