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<li class="md-nav__item">
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2.4. &nbsp; 权衡时间与空间
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2.5. &nbsp; 小结
2.4. &nbsp; 小结
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6.1. &nbsp; 哈希表
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6.3. &nbsp; 小结
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6.1. &nbsp; 二分查找
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7.1. &nbsp; 二叉树
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7.1. &nbsp; 哈希表
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7.2. &nbsp; 哈希冲突处理
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7.3. &nbsp; 二叉搜索树
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7.4. &nbsp; AVL 树 *
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7.5. &nbsp; 小结
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7.3. &nbsp; 小结
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8.2. &nbsp; 建堆操作 *
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8.2. &nbsp; 二叉树遍历
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8.3. &nbsp; 小结
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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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8.5. &nbsp; 小结
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9. &nbsp; &nbsp; 图
9. &nbsp; &nbsp; 堆
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9. &nbsp; &nbsp; 图
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9.1. &nbsp; 图
<a href="../../chapter_heap/heap/" class="md-nav__link">
9.1. &nbsp; 堆
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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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9.4. &nbsp; 小结
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9.3. &nbsp; 小结
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10. &nbsp; &nbsp; 查找算法
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10.1. &nbsp; 线性查找
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10.1. &nbsp; 图
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10.2. &nbsp; 二分查找
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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
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<a href="../../chapter_searching/summary/" class="md-nav__link">
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10.4. &nbsp; 小结
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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
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</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 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
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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
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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）
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
12.1. &nbsp; 搜索算法（New）
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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
13. &nbsp; &nbsp; 回溯算法
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13.1. &nbsp; 编程环境安装
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<a href="../../chapter_appendix/contribution/" class="md-nav__link">
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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参考文献
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<link rel="prev" href="../time_complexity/">
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2.3.4. &nbsp; 权衡时间与空间
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2.4. &nbsp; 权衡时间与空间
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2.5. &nbsp; 小结
2.4. &nbsp; 小结
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
6.1. &nbsp; 哈希表
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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
<a href="../../chapter_binary_search/binary_search/" class="md-nav__link">
6.1. &nbsp; 二分查找
</a>
</li>
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. &nbsp; &nbsp; 树
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7. &nbsp; &nbsp; 树
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
7.1. &nbsp; 二叉树
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
7.1. &nbsp; 哈希表
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</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
7.2. &nbsp; 哈希冲突处理
</a>
</li>
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
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 *
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</li>
<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
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</li>
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
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8.1. &nbsp; 二叉树
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<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
8.2. &nbsp; 建堆操作 *
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8.2. &nbsp; 二叉树遍历
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</li>
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<a href="../../chapter_heap/summary/" class="md-nav__link">
8.3. &nbsp; 小结
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8.3. &nbsp; 二叉搜索树
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
8.4. &nbsp; AVL 树 *
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</li>
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<a href="../../chapter_tree/summary/" class="md-nav__link">
8.5. &nbsp; 小结
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. &nbsp; &nbsp; 图
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9. &nbsp; &nbsp; 图
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9.1. &nbsp; 堆
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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
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9.3. &nbsp; 小结
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10. &nbsp; &nbsp; 查找算法
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10.1. &nbsp; 线性查找
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@ -1381,8 +1324,8 @@
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10.2. &nbsp; 二分查找
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@ -1395,8 +1338,8 @@
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10.3. &nbsp; 哈希查找
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10.3. &nbsp; 图的遍历
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@ -1409,7 +1352,7 @@
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@ -1551,7 +1494,7 @@
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@ -1565,7 +1508,7 @@
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@ -1579,7 +1522,7 @@
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@ -3239,6 +3276,10 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
<p>对数阶常见于分治算法和数据类型转换等。</p>
<p>例如“归并排序”算法，输入长度为 <span class="arithmatex">\(n\)</span> 的数组，每轮递归将数组从中点划分为两半，形成高度为 <span class="arithmatex">\(\log n\)</span> 的递归树，使用 <span class="arithmatex">\(O(\log n)\)</span> 栈帧空间。</p>
<p>再例如“数字转化为字符串”，输入任意正整数 <span class="arithmatex">\(n\)</span> ，它的位数为 <span class="arithmatex">\(\log_{10} n\)</span> ，即对应字符串长度为 <span class="arithmatex">\(\log_{10} n\)</span> ，因此空间复杂度为 <span class="arithmatex">\(O(\log_{10} n) = O(\log n)\)</span> 。</p>
<h2 id="234">2.3.4. &nbsp; 权衡时间与空间<a class="headerlink" href="#234" title="Permanent link">&para;</a></h2>
<p>理想情况下，我们希望算法的时间复杂度和空间复杂度都能达到最优。然而在实际情况中，同时优化时间复杂度和空间复杂度通常是非常困难的。</p>
<p><strong>降低时间复杂度通常需要以提升空间复杂度为代价，反之亦然</strong>。我们将牺牲内存空间来提升算法运行速度的思路称为“以空间换时间”；反之，则称为“以时间换空间”。</p>
<p>选择哪种思路取决于我们更看重哪个方面。在大多数情况下，时间比空间更宝贵，因此以空间换时间通常是更常用的策略。当然，在数据量很大的情况下，控制空间复杂度也是非常重要的。</p>
@ -3332,13 +3373,13 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
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6.2. &nbsp; 哈希冲突处理
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@ -995,21 +902,17 @@
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@ -1019,8 +922,8 @@
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
7.1. &nbsp; 二叉树
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
7.1. &nbsp; 哈希表
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@ -1033,8 +936,8 @@
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. &nbsp; 二叉树遍历
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
7.2. &nbsp; 哈希冲突处理
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@ -1047,36 +950,8 @@
<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
7.3. &nbsp; 二叉搜索树
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<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.4. &nbsp; AVL 树 *
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7.5. &nbsp; 小结
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7.3. &nbsp; 小结
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@ -1114,17 +989,21 @@
<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
<span class="md-nav__icon md-icon"></span>
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8. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 树
</label>
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@ -1134,8 +1013,8 @@
<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
8.1. &nbsp; 堆
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
8.1. &nbsp; 二叉树
</a>
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@ -1148,8 +1027,8 @@
<li class="md-nav__item">
<a href="../../chapter_heap/build_heap/" class="md-nav__link">
8.2. &nbsp; 建堆操作 *
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
8.2. &nbsp; 二叉树遍历
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@ -1162,8 +1041,36 @@
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
8.4. &nbsp; AVL 树 *
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8.5. &nbsp; 小结
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@ -1201,19 +1108,17 @@
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9. &nbsp; &nbsp; 图
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@ -1223,8 +1128,8 @@
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9.1. &nbsp; 图
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9.1. &nbsp; 堆
</a>
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@ -1237,8 +1142,8 @@
<li class="md-nav__item">
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<a href="../../chapter_heap/build_heap/" class="md-nav__link">
9.2. &nbsp; 建堆操作 *
</a>
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@ -1251,22 +1156,8 @@
<li class="md-nav__item">
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
9.3. &nbsp; 图的遍历
</a>
</li>
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9.4. &nbsp; 小结
<a href="../../chapter_heap/summary/" class="md-nav__link">
9.3. &nbsp; 小结
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@ -1309,14 +1200,14 @@
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10. &nbsp; &nbsp; 查找算法
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@ -1326,8 +1217,8 @@
<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
</a>
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@ -1340,8 +1231,8 @@
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<a href="../../chapter_searching/binary_search/" class="md-nav__link">
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
</a>
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@ -1354,8 +1245,8 @@
<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>
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@ -1368,7 +1259,7 @@
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@ -1510,7 +1401,7 @@
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11.6. &nbsp; 桶排序（New）
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@ -1524,7 +1415,7 @@
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11.7. &nbsp; 计数排序（New）
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@ -1538,7 +1429,7 @@
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@ -1586,17 +1477,21 @@
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算法效率评估
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时间复杂度
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<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>
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<p><strong>时间复杂度</strong></p>
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<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>
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<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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