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<link rel="canonical" href="https://www.hello-algo.com/chapter_computational_complexity/summary/">
<link rel="prev" href="../space_time_tradeoff/">
<link rel="prev" href="../space_complexity/">
<link rel="next" href="../../chapter_data_structure/data_and_memory/">
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<title>2.5.   小结 - Hello 算法</title>
<title>2.4.   小结 - Hello 算法</title>
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<div data-md-component="skip">
<a href="#25" class="md-skip">
<a href="#24" class="md-skip">
跳转至
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<div class="md-header__topic" data-md-component="header-topic">
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2.5. &nbsp; 小结
2.4. &nbsp; 小结
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<li class="md-nav__item">
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2.4. &nbsp; 权衡时间与空间
</a>
</li>
@@ -530,55 +514,10 @@
<label class="md-nav__link md-nav__link--active" for="__toc">
2.5. &nbsp; 小结
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
2.5. &nbsp; 小结
2.4. &nbsp; 小结
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<label class="md-nav__title" for="__toc">
<span class="md-nav__icon md-icon"></span>
目录
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#_1" class="md-nav__link">
算法效率评估
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时间复杂度
</a>
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<li class="md-nav__item">
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空间复杂度
</a>
</li>
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<li class="md-nav__item">
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6.1. &nbsp; 哈希表
</a>
</li>
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
6.2. &nbsp; 哈希冲突处理
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
6.3. &nbsp; 小结
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6.1. &nbsp; 二分查找
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7. &nbsp; &nbsp; 树
7. &nbsp; &nbsp; 散列表
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7.1. &nbsp; 二叉树
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
7.1. &nbsp; 哈希表
</a>
</li>
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7.2. &nbsp; 二叉树遍历
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</a>
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7.3. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.4. &nbsp; AVL 树 *
</a>
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<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.5. &nbsp; 小结
<a href="../../chapter_hashing/summary/" class="md-nav__link">
7.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 堆
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8.3. &nbsp; 二叉搜索树
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8.4. &nbsp; AVL 树 *
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9.1. &nbsp; 图
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9.1. &nbsp; 堆
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<li class="md-nav__item">
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9.2. &nbsp; 图基础操作
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9.2. &nbsp; 建堆操作 *
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9.3. &nbsp; 图的遍历
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<a href="../../chapter_graph/summary/" class="md-nav__link">
9.4. &nbsp; 小结
<a href="../../chapter_heap/summary/" class="md-nav__link">
9.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. &nbsp; &nbsp; 查找算法
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<li class="md-nav__item">
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
10.1. &nbsp; 线性查找
<a href="../../chapter_graph/graph/" class="md-nav__link">
10.1. &nbsp; 图
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</li>
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<li class="md-nav__item">
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10.2. &nbsp; 二分查找
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
10.2. &nbsp; 图基础操作
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<li class="md-nav__item">
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
10.3. &nbsp; 哈希查找
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
10.3. &nbsp; 图的遍历
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
<a href="../../chapter_graph/summary/" class="md-nav__link">
10.4. &nbsp; 小结
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
11.6. &nbsp; 桶排序（New）
11.6. &nbsp; 桶排序
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
11.7. &nbsp; 计数排序（New）
11.7. &nbsp; 计数排序
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</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
11.8. &nbsp; 基数排序（New）
11.8. &nbsp; 基数排序
</a>
</li>
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<label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
12. &nbsp; &nbsp; 回溯算法
12. &nbsp; &nbsp; 搜索算法
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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12. &nbsp; &nbsp; 回溯算法
12. &nbsp; &nbsp; 搜索算法
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12.1. &nbsp; 回溯算法（New）
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12.1. &nbsp; 搜索算法（New）
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12.2. &nbsp; 哈希优化策略
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12.3. &nbsp; 小结
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13. &nbsp; &nbsp; 附录
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<li class="md-nav__item">
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13.1. &nbsp; 编程环境安装
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13.2. &nbsp; 一起参与创作
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13.1. &nbsp; 回溯算法（New）
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</li>
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<label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="0">
14. &nbsp; &nbsp; 附录
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14.1. &nbsp; 编程环境安装
</a>
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14.2. &nbsp; 一起参与创作
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参考文献
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<label class="md-nav__title" for="__toc">
<span class="md-nav__icon md-icon"></span>
目录
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#_1" class="md-nav__link">
算法效率评估
</a>
</li>
<li class="md-nav__item">
<a href="#_2" class="md-nav__link">
时间复杂度
</a>
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<a href="#_3" class="md-nav__link">
空间复杂度
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</li>
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@@ -1808,14 +1757,14 @@
<h1 id="25">2.5. &nbsp; 小结<a class="headerlink" href="#25" title="Permanent link">&para;</a></h1>
<h3 id="_1">算法效率评估<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
<h1 id="24">2.4. &nbsp; 小结<a class="headerlink" href="#24" title="Permanent link">&para;</a></h1>
<p><strong>算法效率评估</strong></p>
<ul>
<li>时间效率和空间效率是评价算法性能的两个关键维度。</li>
<li>我们可以通过实际测试来评估算法效率，但难以消除测试环境的影响，且会耗费大量计算资源。</li>
<li>复杂度分析可以克服实际测试的弊端，分析结果适用于所有运行平台，并且能够揭示算法在不同数据规模下的效率。</li>
</ul>
<h3 id="_2">时间复杂度<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h3>
<p><strong>时间复杂度</strong></p>
<ul>
<li>时间复杂度用于衡量算法运行时间随数据量增长的趋势，可以有效评估算法效率，但在某些情况下可能失效，如在输入数据量较小或时间复杂度相同时，无法精确对比算法效率的优劣。</li>
<li>最差时间复杂度使用大 <span class="arithmatex">\(O\)</span> 符号表示，即函数渐近上界，反映当 <span class="arithmatex">\(n\)</span> 趋向正无穷时，<span class="arithmatex">\(T(n)\)</span> 的增长级别。</li>
@@ -1824,7 +1773,7 @@
<li>某些算法的时间复杂度非固定，而是与输入数据的分布有关。时间复杂度分为最差、最佳、平均时间复杂度，最佳时间复杂度几乎不用，因为输入数据一般需要满足严格条件才能达到最佳情况。</li>
<li>平均时间复杂度反映算法在随机数据输入下的运行效率，最接近实际应用中的算法性能。计算平均时间复杂度需要统计输入数据分布以及综合后的数学期望。</li>
</ul>
<h3 id="_3">空间复杂度<a class="headerlink" href="#_3" title="Permanent link">&para;</a></h3>
<p><strong>空间复杂度</strong></p>
<ul>
<li>类似于时间复杂度，空间复杂度用于衡量算法占用空间随数据量增长的趋势。</li>
<li>算法运行过程中的相关内存空间可分为输入空间、暂存空间、输出空间。通常情况下，输入空间不计入空间复杂度计算。暂存空间可分为指令空间、数据空间、栈帧空间，其中栈帧空间通常仅在递归函数中影响空间复杂度。</li>
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