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<h1 id="36-exercises">3.6 &nbsp; Exercises<a class="headerlink" href="#36-exercises" title="Permanent link">&para;</a></h1>
<h2 id="361-check-your-understanding">3.6.1 &nbsp; Check your understanding<a class="headerlink" href="#361-check-your-understanding" title="Permanent link">&para;</a></h2>
<h3 id="1-relationships-between-data-in-everyday-situations">1. &nbsp; Relationships between data in everyday situations<a class="headerlink" href="#1-relationships-between-data-in-everyday-situations" title="Permanent link">&para;</a></h3>
<p>For each situation below, choose a linear, tree, or network structure based on the relationships between the data, and explain your choice:</p>
<!-- numbered-subquestions -->
<ol>
<li>Students stand in a line, each concerned only with the people immediately in front of and behind them.</li>
<li>A school is organized in a hierarchy: school → grade → class.</li>
<li>City roads connect intersections. An intersection can lead to several others, and the roads may form cycles.</li>
</ol>
<details class="success">
<summary>Reference answer</summary>
<ol>
<li>
<p>A linear structure. Apart from the first and last people, each person has exactly one neighbor in front and one behind, forming a line of relationships.</p>
</li>
<li>
<p>A tree structure. Each class belongs to a grade, and each grade belongs to the school, forming a hierarchy from top to bottom.</p>
</li>
<li>
<p>A network structure. An intersection can connect to several others, and routes can form cycles. These relationships cannot be arranged in a single sequence or a strict hierarchy.</p>
<p>To identify a structure, first examine the relationships between its elements, rather than how much memory it occupies.</p>
</li>
</ol>
</details>
<h3 id="2-storing-a-logical-sequence-in-memory">2. &nbsp; Storing a logical sequence in memory<a class="headerlink" href="#2-storing-a-logical-sequence-in-memory" title="Permanent link">&para;</a></h3>
<p>Consider two simplified memory layouts for storing the logical sequence <code>A → B → C</code>:</p>
<ul>
<li>Layout A: <code>A, B, C</code> are stored in memory cells numbered <code>20, 21, 22</code>, respectively.</li>
<li>Layout B: <code>A, B, C</code> are stored in cells numbered <code>20, 7, 31</code>, respectively. <code>A</code> records the location of <code>B</code>, and <code>B</code> records the location of <code>C</code>.</li>
</ul>
<!-- numbered-subquestions -->
<ol>
<li>Which layout uses contiguous storage, and which uses noncontiguous storage?</li>
<li>Which layout is more like an array, and which is more like a linked list?</li>
<li>The cell numbers in layout B are not in ascending order. Why can it still represent the logical sequence <code>A → B → C</code>?</li>
</ol>
<details class="success">
<summary>Reference answer</summary>
<ol>
<li>
<p>Layout A uses consecutive memory cells, so its storage is contiguous. In layout B, nodes occupy separate locations, so its storage is noncontiguous.</p>
</li>
<li>
<p>Layout A is more like an array; layout B is more like a linked list.</p>
</li>
<li>
<p>The logical order is determined by the links recorded between nodes, not by the numerical order of their memory cell numbers.
The location stored in <code>A</code> leads to <code>B</code>, and the location stored in <code>B</code> leads to <code>C</code>, so we can still visit <code>A, B, C</code> in sequence.</p>
<p>This also shows that logical and physical structure are two ways of looking at the same data.</p>
</li>
</ol>
</details>
<h3 id="3-data-types-and-structures-in-homework-records">3. &nbsp; Data types and structures in homework records<a class="headerlink" href="#3-data-types-and-structures-in-homework-records" title="Permanent link">&para;</a></h3>
<p>A study group records whether four students have submitted their homework, in seating order:</p>
<p><code>[true, false, true, true]</code></p>
<!-- numbered-subquestions -->
<ol>
<li>Which basic data type is suitable for each element?</li>
<li>These four elements form a sequence in seating order. What logical structure does this use?</li>
<li>If the group later records homework scores instead, as <code>[90, 0, 85, 100]</code>, has the type of content changed, or the way it is organized? Explain.</li>
</ol>
<details class="success">
<summary>Reference answer</summary>
<ol>
<li>
<p>Each element represents only yes or no, so the Boolean type <code>bool</code> is appropriate.</p>
</li>
<li>
<p>The elements form a linear structure in seating order and can be stored in an array.</p>
</li>
<li>
<p>The content type has changed: the elements are now integers instead of Boolean values. Their organization has not changed.
The data still forms a sequence in seating order and can still use the linear structure of an array.</p>
<p>A basic data type describes what is stored; a data structure describes how the data is organized.</p>
</li>
</ol>
</details>
<h2 id="362-programming-exercises">3.6.2 &nbsp; Programming exercises<a class="headerlink" href="#362-programming-exercises" title="Permanent link">&para;</a></h2>
<h3 id="1-counting-ones-in-a-binary-representation">1. &nbsp; Counting ones in a binary representation<a class="headerlink" href="#1-counting-ones-in-a-binary-representation" title="Permanent link">&para;</a></h3>
<p>Given a nonnegative integer <code>n</code>, count the ones in its binary representation.</p>
<p>Use bitwise operations. Do not convert the binary representation to a string or use a built-in function that counts the ones directly.</p>
<details class="tip">
<summary>Hints</summary>
<ol>
<li>n &amp; 1 extracts the rightmost bit of n, allowing you to check whether it is 1.</li>
<li>Shifting right by one bit discards the current rightmost bit; most languages use the &gt;&gt; operator.</li>
<li>After implementing the method that checks each bit and shifts right, observe that n &amp; (n - 1) changes the rightmost 1 in n to 0.</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/number-of-1-bits/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
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