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第 11 章 &nbsp; ソート
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11.6 &nbsp; マージソート
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<h1 id="1112">11.12 &nbsp; 演習<a class="headerlink" href="#1112" title="Permanent link">&para;</a></h1>
<h2 id="11121">11.12.1 &nbsp; 確認問題<a class="headerlink" href="#11121" title="Permanent link">&para;</a></h2>
<h3 id="1">1. &nbsp; 選択ソートとバブルソートの最初の数回<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p>配列 <code>[4, 2, 5, 1, 3]</code> が与えられます。次の操作では、いずれも小さい順にソートします。</p>
<!-- numbered-subquestions -->
<ol>
<li>選択ソートの最初の 2 回を行い、各回の終了時の配列を書いてください。また、この時点で位置が確定した要素を示してください。</li>
<li>バブルソートの最初の 1 回を行い、配列の状態と交換回数を書いてください。また、どの位置が確定したかを示してください。</li>
</ol>
<details class="success">
<summary>解答</summary>
<ol>
<li>
<p>最初の 2 回は次のとおりです。</p>
<table>
<thead>
<tr>
<th>回</th>
<th>配列の状態</th>
<th>説明</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td><code>[1, 2, 5, 4, 3]</code></td>
<td>最小要素 1 を先頭の要素と交換する</td>
</tr>
<tr>
<td>2</td>
<td><code>[1, 2, 5, 4, 3]</code></td>
<td>値 2 はすでにインデックス 1 にあるため、交換は不要</td>
</tr>
</tbody>
</table>
<p>この時点で、先頭の 2 つの位置が確定しています。以後は <code>[5, 4, 3]</code> から最小要素を選び続ければよいです。</p>
</li>
<li>
<p>隣り合う要素を順に比較します。4 と 2 は交換し、4 と 5 は交換せず、5 と 1、5 と 3 はそれぞれ交換します。
その結果は <code>[2, 4, 1, 3, 5]</code> で、交換回数は 3 回です。最大要素 5 が配列の末尾へ移動したため、最後の位置が確定しています。</p>
</li>
</ol>
</details>
<h3 id="2">2. &nbsp; 等しい要素の順序は変わるか<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<p>配列 <span class="arithmatex">\([2_a, 2_b, 1]\)</span> で、<span class="arithmatex">\(2_a\)</span> と <span class="arithmatex">\(2_b\)</span> の値は等しいものの、添字によって元の順序を区別しています。</p>
<!-- numbered-subquestions -->
<ol>
<li>選択ソートの 1 回目が終わった後の配列を書いてください。<span class="arithmatex">\(2_a\)</span> と <span class="arithmatex">\(2_b\)</span> の相対的な順序は変わりますか?</li>
<li>バブルソートの 1 回目が終わった後の配列を書いてください。<span class="arithmatex">\(2_a\)</span> と <span class="arithmatex">\(2_b\)</span> の相対的な順序は変わりますか?</li>
<li>問い 1、2 の結果から、等しい要素の元の順序を保つという点で、2 つのソートにはどのような違いがあるかを説明してください。</li>
</ol>
<details class="success">
<summary>解答</summary>
<ol>
<li>
<p>選択ソートの 1 回目では最小要素 1 を選び、先頭の <span class="arithmatex">\(2_a\)</span> と交換するため、
<span class="arithmatex">\([1, 2_b, 2_a]\)</span> となります。<span class="arithmatex">\(2_a\)</span> は <span class="arithmatex">\(2_b\)</span> の後ろへ移動し、相対的な順序が変わっています。</p>
</li>
<li>
<p>バブルソートでは、まず <span class="arithmatex">\(2_a\)</span> と <span class="arithmatex">\(2_b\)</span> を比較します。両者は等しいので交換しません。次に <span class="arithmatex">\(2_b\)</span> と 1 を比較して交換すると、
1 回目の終了時は <span class="arithmatex">\([2_a, 1, 2_b]\)</span> となります。<span class="arithmatex">\(2_a\)</span> は引き続き <span class="arithmatex">\(2_b\)</span> より前にあり、相対的な順序は変わっていません。</p>
</li>
<li>
<p>この例では、選択ソートは等しい要素の元の順序を変えます。バブルソートは、左側の要素が右側の要素より大きい場合にだけ
隣り合う要素を交換します。等しい要素同士は交換しないため、元の順序を保てます。</p>
</li>
</ol>
</details>
<h3 id="3">3. &nbsp; 計数ソートと基数ソートの比較<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
<p>学校で、8 桁に固定された多数の学籍番号をソートします。次の問いに答えてください。</p>
<!-- numbered-subquestions -->
<ol>
<li>最下位桁から始める基数ソートでは、何回の処理が必要ですか?</li>
<li>学籍番号をそのまま整数として計数ソートを行うと、使われないカウント位置を大量に用意することになるのはなぜですか?</li>
<li>問い 1、2 の結果から、8 桁に固定された多数の学籍番号を計数ソートと基数ソートのどちらかで処理するなら、どちらを選びますか?その理由も説明してください。</li>
</ol>
<details class="success">
<summary>解答</summary>
<ol>
<li>
<p>学籍番号には 8 個の桁があるため、最下位桁から最上位桁まで、合計 8 回処理します。各回では 0~9 だけを使って分類します。</p>
</li>
<li>
<p>そのまま数えるには、取り得るすべての 8 桁の値に対して位置を用意する必要があります。しかし、実際に生徒が使うのはそのうちのごく一部なので、
ほとんどのカウント位置は常に 0 のままです。</p>
</li>
<li>
<p>基数ソートを選びます。「桁数が固定され、各桁が 10 種類の値だけを取る」という特徴を利用し、安定した分類を 8 回繰り返すだけで済むためです。
8 桁の学籍番号全体を整数のインデックスとして計数ソートを行うと、実際には現れない多数の値にもカウント位置を用意する必要があります。</p>
</li>
</ol>
</details>
<h2 id="11122">11.12.2 &nbsp; プログラミング演習<a class="headerlink" href="#11122" title="Permanent link">&para;</a></h2>
<h3 id="1_1">1. &nbsp; マージソートで配列を並べる<a class="headerlink" href="#1_1" title="Permanent link">&para;</a></h3>
<p>整数配列 <code>nums</code> が与えられます。マージソートを自分で実装し、要素を非減少順に並べて返してください。言語に組み込まれたソート関数を呼び出してはいけません。</p>
<details class="tip">
<summary>解法のヒント</summary>
<ol>
<li>区間の長さが 1 以下なら、すでにソートされています</li>
<li>中点で区間を 2 つに分け、それぞれを再帰的にソートします</li>
<li>2 つのポインタでソート済みの左右の区間をマージし、結果を元の配列へ書き戻します</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/sort-an-array/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
<h3 id="2_1">2. &nbsp; 計数ソートで整数配列を並べる<a class="headerlink" href="#2_1" title="Permanent link">&para;</a></h3>
<p>整数配列 <code>nums</code> と 0 以上の整数 <span class="arithmatex">\(K\)</span> が与えられ、配列の各要素は <span class="arithmatex">\(0\)</span> 以上 <span class="arithmatex">\(K\)</span> 以下です。</p>
<p>計数ソートを実装し、要素を非減少順に <code>nums</code> へ書き戻して、<code>nums</code> を返してください。
要素同士の大小比較によって順序を決めたり、言語に組み込まれたソート関数を呼び出したりしてはいけません。</p>
<details class="tip">
<summary>解法のヒント</summary>
<ol>
<li>各要素は 0 以上 K 以下なので、その値をカウント配列のインデックスとして直接使えます</li>
<li>1 回目に nums を走査し、対応する位置のカウントに 1 を足します</li>
<li>次にカウント配列を 0 から K まで走査します。値 x が現れた回数だけ、nums に x を続けて書き込みます</li>
</ol>
</details>
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