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<h1 id="25">2.5 &nbsp; 小結<a class="headerlink" href="#25" title="Permanent link">&para;</a></h1>
<h3 id="1">1. &nbsp; 重點回顧<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p><strong>演算法效率評估</strong></p>
<ul>
<li>時間效率和空間效率是衡量演算法優劣的兩個主要評價指標。</li>
<li>我們可以透過實際測試來評估演算法效率，但難以消除測試環境的影響，且會耗費大量計算資源。</li>
<li>複雜度分析可以消除實際測試的弊端，分析結果適用於所有執行平臺，並且能夠揭示演算法在不同資料規模下的效率。</li>
</ul>
<p><strong>時間複雜度</strong></p>
<ul>
<li>時間複雜度用於衡量演算法執行時間隨資料量增長的趨勢，可以有效評估演算法效率，但在某些情況下可能失效，如在輸入的資料量較小或時間複雜度相同時，無法精確對比演算法效率的優劣。</li>
<li>最差時間複雜度使用大 <span class="arithmatex">\(O\)</span> 符號表示，對應函式漸近上界，反映當 <span class="arithmatex">\(n\)</span> 趨向正無窮時，操作數量 <span class="arithmatex">\(T(n)\)</span> 的增長級別。</li>
<li>推算時間複雜度分為兩步，首先統計操作數量，然後判斷漸近上界。</li>
<li>常見時間複雜度從低到高排列有 <span class="arithmatex">\(O(1)\)</span>、<span class="arithmatex">\(O(\log n)\)</span>、<span class="arithmatex">\(O(n)\)</span>、<span class="arithmatex">\(O(n \log n)\)</span>、<span class="arithmatex">\(O(n^2)\)</span>、<span class="arithmatex">\(O(2^n)\)</span> 和 <span class="arithmatex">\(O(n!)\)</span> 等。</li>
<li>某些演算法的時間複雜度非固定，而是與輸入資料的分佈有關。時間複雜度分為最差、最佳、平均時間複雜度，最佳時間複雜度幾乎不用，因為輸入資料一般需要滿足嚴格條件才能達到最佳情況。</li>
<li>平均時間複雜度反映演算法在隨機資料輸入下的執行效率，最接近實際應用中的演算法效能。計算平均時間複雜度需要統計輸入資料分佈以及綜合後的數學期望。</li>
</ul>
<p><strong>空間複雜度</strong></p>
<ul>
<li>空間複雜度的作用類似於時間複雜度，用於衡量演算法佔用記憶體空間隨資料量增長的趨勢。</li>
<li>演算法執行過程中的相關記憶體空間可分為輸入空間、暫存空間、輸出空間。通常情況下，輸入空間不納入空間複雜度計算。暫存空間可分為暫存資料、堆疊幀空間和指令空間，其中堆疊幀空間通常僅在遞迴函式中影響空間複雜度。</li>
<li>我們通常只關注最差空間複雜度，即統計演算法在最差輸入資料和最差執行時刻下的空間複雜度。</li>
<li>常見空間複雜度從低到高排列有 <span class="arithmatex">\(O(1)\)</span>、<span class="arithmatex">\(O(\log n)\)</span>、<span class="arithmatex">\(O(n)\)</span>、<span class="arithmatex">\(O(n^2)\)</span> 和 <span class="arithmatex">\(O(2^n)\)</span> 等。</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>：尾遞迴的空間複雜度是 <span class="arithmatex">\(O(1)\)</span> 嗎？</p>
<p>理論上，尾遞迴函式的空間複雜度可以最佳化至 <span class="arithmatex">\(O(1)\)</span> 。不過絕大多數程式語言（例如 Java、Python、C++、Go、C# 等）不支持自動最佳化尾遞迴，因此通常認為空間複雜度是 <span class="arithmatex">\(O(n)\)</span> 。</p>
<p><strong>Q</strong>：函式和方法這兩個術語的區別是什麼？</p>
<p><u>函式（function）</u>可以被獨立執行，所有參數都以顯式傳遞。<u>方法（method）</u>與一個物件關聯，被隱式傳遞給呼叫它的物件，能夠對類別的例項中包含的資料進行操作。</p>
<p>下面以幾種常見的程式語言為例來說明。</p>
<ul>
<li>C 語言是程序式程式設計語言，沒有物件導向的概念，所以只有函式。但我們可以透過建立結構體（struct）來模擬物件導向程式設計，與結構體相關聯的函式就相當於其他程式語言中的方法。</li>
<li>Java 和 C# 是物件導向的程式語言，程式碼塊（方法）通常作為某個類別的一部分。靜態方法的行為類似於函式，因為它被繫結在類別上，不能訪問特定的例項變數。</li>
<li>C++ 和 Python 既支持程序式程式設計（函式），也支持物件導向程式設計（方法）。</li>
</ul>
<p><strong>Q</strong>：圖解“常見的空間複雜度型別”反映的是否是佔用空間的絕對大小？</p>
<p>不是，該圖展示的是空間複雜度，其反映的是增長趨勢，而不是佔用空間的絕對大小。</p>
<p>假設取 <span class="arithmatex">\(n = 8\)</span> ，你可能會發現每條曲線的值與函式對應不上。這是因為每條曲線都包含一個常數項，用於將取值範圍壓縮到一個視覺舒適的範圍內。</p>
<p>在實際中，因為我們通常不知道每個方法的“常數項”複雜度是多少，所以一般無法僅憑複雜度來選擇 <span class="arithmatex">\(n = 8\)</span> 之下的最優解法。但對於 <span class="arithmatex">\(n = 8^5\)</span> 就很好選了，這時增長趨勢已經佔主導了。</p>

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