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第 8 章 &nbsp; ヒープ
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第 8 章 &nbsp; ヒープ
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第 9 章 &nbsp; グラフ
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第 10 章 &nbsp; 探索
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10.2 &nbsp; 二分探索で挿入位置を探す
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10.3 &nbsp; 二分探索で境界を探す
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10.4 &nbsp; ハッシュを使った最適化
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10.5 &nbsp; 探索アルゴリズムを改めて考える
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第 11 章 &nbsp; ソート
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11.4 &nbsp; 挿入ソート
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11.5 &nbsp; クイックソート
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11.6 &nbsp; マージソート
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11.8 &nbsp; バケットソート
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11.9 &nbsp; カウントソート
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11.10 &nbsp; 基数ソート
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11.11 &nbsp; まとめ
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第 12 章 &nbsp; 分割統治
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12.1 &nbsp; 分割統治アルゴリズム
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12.2 &nbsp; 分割統治による探索戦略
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12.3 &nbsp; 二分木の構築問題
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12.4 &nbsp; ハノイの塔の問題
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12.5 &nbsp; まとめ
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第 13 章 &nbsp; バックトラッキング
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13.1 &nbsp; バックトラッキングアルゴリズム
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第 14 章 &nbsp; 動的計画法
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14.1 &nbsp; 動的計画法への第一歩
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14.2 &nbsp; 動的計画法の問題の特徴
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14.3 &nbsp; 動的計画法の解き方
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14.4 &nbsp; 0-1 ナップサック問題
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<h1 id="77">7.7 &nbsp; 練習問題<a class="headerlink" href="#77" title="この見出しへのリンク">&para;</a></h1>
<h2 id="771">7.7.1 &nbsp; 知識の確認<a class="headerlink" href="#771" title="この見出しへのリンク">&para;</a></h2>
<h3 id="1">1. &nbsp; 完全二分木・全二分木・パーフェクト二分木<a class="headerlink" href="#1" title="この見出しへのリンク">&para;</a></h3>
<p>次の二つの配列は、二分木をレベル順に表しています。<code>None</code> は空の位置を表します。</p>
<ul>
<li>木 A:<code>[1, 2, 3, 4, 5, 6]</code></li>
<li>木 B:<code>[1, 2, 3, None, None, 6, 7]</code></li>
</ul>
<!-- numbered-subquestions -->
<ol>
<li>どちらが完全二分木ですか。</li>
<li>どちらが全二分木、つまり葉以外のすべてのノードに二つの子がある木ですか。</li>
<li>この二つの中にパーフェクト二分木はありますか。それぞれ理由を説明してください。</li>
</ol>
<details class="success">
<summary>解答例</summary>
<ol>
<li>
<p>木 A は完全二分木です。埋まっていないのは最下層だけで、ノードは左から右へ連続して並んでいます。
木 B は完全二分木ではありません。最下層の左側に空きがあるのに、右側にはノードがあるためです。</p>
</li>
<li>
<p>木 B は全二分木です。ノード 1 と 3 はそれぞれ二つの子を持ち、ほかはすべて葉です。
木 A は全二分木ではありません。ノード 3 にあるのは左の子 6 だけだからです。</p>
</li>
<li>
<p>どちらも最下層が完全には埋まっていないため、パーフェクト二分木ではありません。</p>
</li>
</ol>
</details>
<h3 id="2">2. &nbsp; 同じ木を走査する三つの順序<a class="headerlink" href="#2" title="この見出しへのリンク">&para;</a></h3>
<p>配列 <code>[1, 2, 3, 4, 5, 6, 7]</code> を、レベル順に完全二分木へ格納します。</p>
<!-- numbered-subquestions -->
<ol>
<li>この木を描いてください。</li>
<li>行きがけ順、通りがけ順、帰りがけ順の走査列を書いてください。</li>
<li>通りがけ順の列で、根ノード 1 の左側と右側の列は、それぞれ木のどの部分に対応しますか。</li>
</ol>
<details class="success">
<summary>解答例</summary>
<ol>
<li>
<p>木は次のようになります。</p>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a> 1
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a> / \
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a> 2 3
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> / \ / \
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> 4 5 6 7
</code></pre></div>
</li>
<li>
<p>行きがけ順は <code>1, 2, 4, 5, 3, 6, 7</code>、
通りがけ順は <code>4, 2, 5, 1, 6, 3, 7</code>、
帰りがけ順は <code>4, 5, 2, 6, 7, 3, 1</code> です。</p>
</li>
<li>
<p>根ノード 1 の左側の <code>4, 2, 5</code> は、左部分木の通りがけ順の列です。
右側の <code>6, 3, 7</code> は、右部分木の通りがけ順の列です。</p>
</li>
</ol>
</details>
<h3 id="3">3. &nbsp; 二つの二分探索木を比べる<a class="headerlink" href="#3" title="この見出しへのリンク">&para;</a></h3>
<p>次の二つの列を、それぞれ空の二分探索木へ左から右へ順に挿入します。</p>
<ul>
<li>列 A:<code>[4, 2, 6, 1, 3, 5, 7]</code></li>
<li>列 B:<code>[1, 2, 3, 4, 5, 6, 7]</code></li>
</ul>
<!-- numbered-subquestions -->
<ol>
<li>7 を探すときに通るノードを、それぞれ書いてください。</li>
<li>高さを、根から最も遠い葉までに通る辺の数とすると、それぞれの木の高さはいくつですか。</li>
<li>前の二問から、7 を探す効率は二つの木で同じだと思いますか。木の形と探索経路に基づいて理由を説明してください。</li>
</ol>
<details class="success">
<summary>解答例</summary>
<ol>
<li>
<p>列 A で作った木の探索経路は <code>4 → 6 → 7</code>、
列 B で作った木の探索経路は <code>1 → 2 → 3 → 4 → 5 → 6 → 7</code> です。</p>
</li>
<li>
<p>一つ目はすべての層が埋まっており、高さは 2 です。二つ目は右の子しかなく、高さは 6 です。</p>
</li>
<li>
<p>7 を探す効率は異なります。挿入順によって二分探索木の形と高さが変わるためです。7 を探すとき、一つ目の木で通るのは 3 個のノードだけですが、
二つ目では 7 個のノードを順に通る必要があります。木が高いほど、最悪の場合に経路に沿って比較するノード数が増えます。</p>
</li>
</ol>
</details>
<h2 id="772">7.7.2 &nbsp; プログラミング演習<a class="headerlink" href="#772" title="この見出しへのリンク">&para;</a></h2>
<h3 id="1_1">1. &nbsp; 二分木の最大の深さ<a class="headerlink" href="#1_1" title="この見出しへのリンク">&para;</a></h3>
<p>二分木の根ノード <code>root</code> が与えられます。各ノードは整数値と、左右の子への参照を持ちます。</p>
<p>根から最も遠い葉までに通る<strong>ノード数</strong>を、二分木の最大の深さと呼びます。この木の最大の深さを返してください。空の木の最大の深さは 0 とします。
再帰を使って解いてください。</p>
<details class="tip">
<summary>解くためのヒント</summary>
<ol>
<li>この問題ではノード数で深さを数えます。根が一つだけなら、最大の深さは 1 です</li>
<li>再帰関数は、現在のノードを根とする部分木の最大の深さを返すようにします</li>
<li>空のノードでは 0、それ以外では max(depth(left), depth(right)) + 1 を返します</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/maximum-depth-of-binary-tree/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
<h3 id="2_1">2. &nbsp; 二分木を層ごとに走査する<a class="headerlink" href="#2_1" title="この見出しへのリンク">&para;</a></h3>
<p>二分木の根ノード <code>root</code> が与えられます。キューを使って上から下へ層ごとに全ノードを訪問し、同じ層では左から右の順に訪問してください。</p>
<p>二次元配列を返してください。最初の部分配列には根の層のノード値、二番目には次の層のノード値、というように格納します。
木が空なら、空配列を返してください。</p>
<details class="tip">
<summary>解くためのヒント</summary>
<ol>
<li>レベル順走査では、先に入ったノードを先に訪問するため、キューを使います</li>
<li>各反復の開始時にキューにあるノードは、ちょうど同じ層に属しています</li>
<li>まずキューの長さを記録し、その個数だけノードを取り出して、それぞれの子を追加します</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/binary-tree-level-order-traversal/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
<h3 id="3-k">3. &nbsp; 二分探索木の k 番目に小さい要素<a class="headerlink" href="#3-k" title="この見出しへのリンク">&para;</a></h3>
<p>二分探索木に <code>n</code> 個のノードがあり、各ノードの値はすべて異なります。
全ノードの値を昇順に並べ、その位置に 1 から番号を付けます。</p>
<p>根ノード <code>root</code> と、<code>1 &lt;= k &lt;= n</code> を満たす整数 <code>k</code> が与えられます。<code>k</code> 番目のノードの値を返してください。
先に全ノードの値を集めず、通りがけ順の走査中に直接答えを見つけてください。</p>
<details class="tip">
<summary>解くためのヒント</summary>
<ol>
<li>二分探索木の通りがけ順走査では、ノードの値を小さい順に訪問します</li>
<li>通りがけ順では左部分木、現在のノード、右部分木の順に処理します。現在のノードを訪問したら、個数を 1 増やします</li>
<li>個数が初めて k に達したときのノード値が答えです。それ以降は走査する必要がありません</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/kth-smallest-element-in-a-bst/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
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