Fig. 在数组中插入与删除元素
+ === "Java" ```java title="array.java" [class]{array}-[func]{insert} - - [class]{array}-[func]{remove} ``` === "C++" ```cpp title="array.cpp" [class]{}-[func]{insert} - - [class]{}-[func]{remove} ``` === "Python" ```python title="array.py" [class]{}-[func]{insert} - - [class]{}-[func]{remove} ``` === "Go" ```go title="array.go" [class]{}-[func]{insert} - - [class]{}-[func]{remove} ``` === "JavaScript" ```javascript title="array.js" [class]{}-[func]{insert} - - [class]{}-[func]{remove} ``` === "TypeScript" ```typescript title="array.ts" [class]{}-[func]{insert} - - [class]{}-[func]{remove} ``` === "C" ```c title="array.c" [class]{}-[func]{insert} - - [class]{}-[func]{removeItem} ``` === "C#" ```csharp title="array.cs" [class]{array}-[func]{insert} - - [class]{array}-[func]{remove} ``` === "Swift" ```swift title="array.swift" [class]{}-[func]{insert} + ``` +删除元素也是类似，如果我们想要删除索引 $i$ 处的元素，则需要把索引 $i$ 之后的元素都向前移动一位。值得注意的是，删除元素后，原先末尾的元素变得“无意义”了，我们无需特意去修改它。 + + + +=== "Java" + + ```java title="array.java" + [class]{array}-[func]{remove} + ``` + +=== "C++" + + ```cpp title="array.cpp" + [class]{}-[func]{remove} + ``` + +=== "Python" + + ```python title="array.py" + [class]{}-[func]{remove} + ``` + +=== "Go" + + ```go title="array.go" + [class]{}-[func]{remove} + ``` + +=== "JavaScript" + + ```javascript title="array.js" + [class]{}-[func]{remove} + ``` + +=== "TypeScript" + + ```typescript title="array.ts" + [class]{}-[func]{remove} + ``` + +=== "C" + + ```c title="array.c" + [class]{}-[func]{removeItem} + ``` + +=== "C#" + + ```csharp title="array.cs" + [class]{array}-[func]{remove} + ``` + +=== "Swift" + + ```swift title="array.swift" [class]{}-[func]{remove} ``` === "Zig" ```zig title="array.zig" - [class]{}-[func]{insert} - [class]{}-[func]{remove} ``` +总结来看，数组的插入与删除操作有以下缺点： + +- **时间复杂度高**：数组的插入和删除的平均时间复杂度均为 $O(N)$ ，其中 $N$ 为数组长度。 +- **丢失元素**：由于数组的长度不可变，因此在插入元素后，超出数组长度范围的元素会被丢失。 +- **内存浪费**：我们一般会初始化一个比较长的数组，只用前面一部分，这样在插入数据时，丢失的末尾元素都是我们不关心的，但这样做同时也会造成内存空间的浪费。 + ## 数组常用操作 **数组遍历**。以下介绍两种常用的遍历方法。 diff --git a/docs/chapter_array_and_linkedlist/linked_list.assets/linkedlist_insert_remove_node.png b/docs/chapter_array_and_linkedlist/linked_list.assets/linkedlist_insert_node.png similarity index 100% rename from docs/chapter_array_and_linkedlist/linked_list.assets/linkedlist_insert_remove_node.png rename to docs/chapter_array_and_linkedlist/linked_list.assets/linkedlist_insert_node.png diff --git a/docs/chapter_array_and_linkedlist/linked_list.assets/linkedlist_remove_node.png b/docs/chapter_array_and_linkedlist/linked_list.assets/linkedlist_remove_node.png new file mode 100644 index 000000000..3ab4da38c Binary files /dev/null and b/docs/chapter_array_and_linkedlist/linked_list.assets/linkedlist_remove_node.png differ diff --git a/docs/chapter_array_and_linkedlist/linked_list.md b/docs/chapter_array_and_linkedlist/linked_list.md index 834be28e1..bf8c3b05e 100755 --- a/docs/chapter_array_and_linkedlist/linked_list.md +++ b/docs/chapter_array_and_linkedlist/linked_list.md @@ -316,89 +316,131 @@ comments: true ## 链表优点 -**在链表中，插入与删除结点的操作效率高**。例如，如果想在链表中间的两个结点 `A` , `B` 之间插入一个新结点 `P` ，我们只需要改变两个结点指针即可，时间复杂度为 $O(1)$ ，相比数组的插入操作高效很多。在链表中删除某个结点也很方便，只需要改变一个结点指针即可。 +**在链表中，插入与删除结点的操作效率高**。比如，如果我们想在链表中间的两个结点 `A` , `B` 之间插入一个新结点 `P` ，我们只需要改变两个结点指针即可，时间复杂度为 $O(1)$ ，相比数组的插入操作高效很多。 - - -Fig. 在链表中插入与删除结点
+ === "Java" ```java title="linked_list.java" [class]{linked_list}-[func]{insert} - - [class]{linked_list}-[func]{remove} ``` === "C++" ```cpp title="linked_list.cpp" [class]{}-[func]{insert} - - [class]{}-[func]{remove} ``` === "Python" ```python title="linked_list.py" [class]{}-[func]{insert} - - [class]{}-[func]{remove} ``` === "Go" ```go title="linked_list.go" [class]{}-[func]{insertNode} - - [class]{}-[func]{removeNode} ``` === "JavaScript" ```javascript title="linked_list.js" [class]{}-[func]{insert} - - [class]{}-[func]{remove} ``` === "TypeScript" ```typescript title="linked_list.ts" [class]{}-[func]{insert} - - [class]{}-[func]{remove} ``` === "C" ```c title="linked_list.c" [class]{}-[func]{insertNode} - - [class]{}-[func]{removeNode} ``` === "C#" ```csharp title="linked_list.cs" [class]{linked_list}-[func]{insert} - - [class]{linked_list}-[func]{remove} ``` === "Swift" ```swift title="linked_list.swift" [class]{}-[func]{insert} - - [class]{}-[func]{remove} ``` === "Zig" ```zig title="linked_list.zig" [class]{}-[func]{insert} + ``` + +在链表中删除结点也很方便，只需要改变一个结点指针即可。如下图所示，虽然在完成删除后结点 `P` 仍然指向 `n2` ，但实际上 `P` 已经不属于此链表了，因为遍历此链表是无法访问到 `P` 的。 + + + +=== "Java" + + ```java title="linked_list.java" + [class]{linked_list}-[func]{remove} + ``` + +=== "C++" + + ```cpp title="linked_list.cpp" + [class]{}-[func]{remove} + ``` + +=== "Python" + + ```python title="linked_list.py" + [class]{}-[func]{remove} + ``` +=== "Go" + + ```go title="linked_list.go" + [class]{}-[func]{removeNode} + ``` + +=== "JavaScript" + + ```javascript title="linked_list.js" + [class]{}-[func]{remove} + ``` + +=== "TypeScript" + + ```typescript title="linked_list.ts" + [class]{}-[func]{remove} + ``` + +=== "C" + + ```c title="linked_list.c" + [class]{}-[func]{removeNode} + ``` + +=== "C#" + + ```csharp title="linked_list.cs" + [class]{linked_list}-[func]{remove} + ``` + +=== "Swift" + + ```swift title="linked_list.swift" + [class]{}-[func]{remove} + ``` + +=== "Zig" + + ```zig title="linked_list.zig" [class]{}-[func]{remove} ``` diff --git a/docs/chapter_computational_complexity/time_complexity.assets/time_complexity_first_example.png b/docs/chapter_computational_complexity/time_complexity.assets/time_complexity_simple_example.png similarity index 100% rename from docs/chapter_computational_complexity/time_complexity.assets/time_complexity_first_example.png rename to docs/chapter_computational_complexity/time_complexity.assets/time_complexity_simple_example.png diff --git a/docs/chapter_computational_complexity/time_complexity.md b/docs/chapter_computational_complexity/time_complexity.md index f4c410e22..162bcbd57 100755 --- a/docs/chapter_computational_complexity/time_complexity.md +++ b/docs/chapter_computational_complexity/time_complexity.md @@ -369,7 +369,7 @@ $$ ``` - +Fig. 算法 A, B, C 的时间增长趋势
diff --git a/docs/chapter_data_structure/data_and_memory.assets/IEEE-754-float.png b/docs/chapter_data_structure/data_and_memory.assets/ieee_754_float.png similarity index 100% rename from docs/chapter_data_structure/data_and_memory.assets/IEEE-754-float.png rename to docs/chapter_data_structure/data_and_memory.assets/ieee_754_float.png diff --git a/docs/chapter_data_structure/data_and_memory.md b/docs/chapter_data_structure/data_and_memory.md index 4dd649b8c..78f4f7e87 100644 --- a/docs/chapter_data_structure/data_and_memory.md +++ b/docs/chapter_data_structure/data_and_memory.md @@ -80,7 +80,7 @@ $$ \end{aligned} $$ - + 以上图为例，$\mathrm{S} = 0$ ， $\mathrm{E} = 124$ ，$\mathrm{N} = 2^{-2} + 2^{-3} = 0.375$ ，易得 diff --git a/docs/chapter_heap/heap.assets/heapify_count.png b/docs/chapter_heap/heap.assets/heapify_operations_count.png similarity index 100% rename from docs/chapter_heap/heap.assets/heapify_count.png rename to docs/chapter_heap/heap.assets/heapify_operations_count.png diff --git a/docs/chapter_heap/heap.md b/docs/chapter_heap/heap.md index 09f9a15a9..1b016aa7e 100644 --- a/docs/chapter_heap/heap.md +++ b/docs/chapter_heap/heap.md @@ -793,7 +793,7 @@ $$ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{(h-1)}\times1 $$ - + 化简上式需要借助中学的数列知识，先对 $T(h)$ 乘以 $2$ ，易得 diff --git a/docs/chapter_preface/about_the_book.assets/mindmap.png b/docs/chapter_preface/about_the_book.assets/hello_algo_mindmap.png similarity index 100% rename from docs/chapter_preface/about_the_book.assets/mindmap.png rename to docs/chapter_preface/about_the_book.assets/hello_algo_mindmap.png diff --git a/docs/chapter_preface/about_the_book.md b/docs/chapter_preface/about_the_book.md index 62b1eb435..a7c219f92 100644 --- a/docs/chapter_preface/about_the_book.md +++ b/docs/chapter_preface/about_the_book.md @@ -36,7 +36,7 @@ comments: true 本书主要内容分为复杂度分析、数据结构、算法三个部分。 - +Fig. 知识点思维导图
diff --git a/docs/chapter_sorting/bubble_sort.assets/bubble_sort.png b/docs/chapter_sorting/bubble_sort.assets/bubble_sort_overview.png similarity index 100% rename from docs/chapter_sorting/bubble_sort.assets/bubble_sort.png rename to docs/chapter_sorting/bubble_sort.assets/bubble_sort_overview.png diff --git a/docs/chapter_sorting/bubble_sort.md b/docs/chapter_sorting/bubble_sort.md index 79861d62b..3b22c33ba 100755 --- a/docs/chapter_sorting/bubble_sort.md +++ b/docs/chapter_sorting/bubble_sort.md @@ -43,7 +43,7 @@ comments: true 2. 同理，对剩余 $n - 1$ 个元素执行「冒泡」，可将第二大元素交换至正确位置，因而待排序元素只剩 $n - 2$ 个。 3. 以此类推…… **循环 $n - 1$ 轮「冒泡」，即可完成整个数组的排序**。 - +Fig. 冒泡排序流程
diff --git a/docs/chapter_sorting/insertion_sort.assets/insertion_sort.png b/docs/chapter_sorting/insertion_sort.assets/insertion_sort_overview.png similarity index 100% rename from docs/chapter_sorting/insertion_sort.assets/insertion_sort.png rename to docs/chapter_sorting/insertion_sort.assets/insertion_sort_overview.png diff --git a/docs/chapter_sorting/insertion_sort.md b/docs/chapter_sorting/insertion_sort.md index 4e3dd2b6a..e660e3e0b 100755 --- a/docs/chapter_sorting/insertion_sort.md +++ b/docs/chapter_sorting/insertion_sort.md @@ -20,7 +20,7 @@ comments: true 2. 第 2 轮选取 **第 3 个元素** 为 `base` ，执行「插入操作」后，**数组前 3 个元素已完成排序**。 3. 以此类推……最后一轮选取 **数组尾元素** 为 `base` ，执行「插入操作」后，**所有元素已完成排序**。 - +Fig. 插入排序流程
diff --git a/docs/chapter_sorting/merge_sort.assets/merge_sort_preview.png b/docs/chapter_sorting/merge_sort.assets/merge_sort_overview.png similarity index 100% rename from docs/chapter_sorting/merge_sort.assets/merge_sort_preview.png rename to docs/chapter_sorting/merge_sort.assets/merge_sort_overview.png diff --git a/docs/chapter_sorting/merge_sort.md b/docs/chapter_sorting/merge_sort.md index 9933896f3..57e1c18ec 100755 --- a/docs/chapter_sorting/merge_sort.md +++ b/docs/chapter_sorting/merge_sort.md @@ -9,7 +9,7 @@ comments: true 1. **划分阶段**：通过递归不断 **将数组从中点位置划分开**，将长数组的排序问题转化为短数组的排序问题； 2. **合并阶段**：划分到子数组长度为 1 时，开始向上合并，不断将 **左、右两个短排序数组** 合并为 **一个长排序数组**，直至合并至原数组时完成排序； - +Fig. 归并排序两阶段：划分与合并
diff --git a/docs/chapter_sorting/quick_sort.assets/quick_sort.png b/docs/chapter_sorting/quick_sort.assets/quick_sort_overview.png similarity index 100% rename from docs/chapter_sorting/quick_sort.assets/quick_sort.png rename to docs/chapter_sorting/quick_sort.assets/quick_sort_overview.png diff --git a/docs/chapter_sorting/quick_sort.md b/docs/chapter_sorting/quick_sort.md index 9b1200a0c..ef299d450 100755 --- a/docs/chapter_sorting/quick_sort.md +++ b/docs/chapter_sorting/quick_sort.md @@ -129,7 +129,7 @@ comments: true 观察发现，快速排序和「二分查找」的原理类似，都是以对数阶的时间复杂度来缩小处理区间。 - +Fig. 快速排序流程
diff --git a/docs/chapter_tree/avl_tree.assets/degradation_from_inserting_node.png b/docs/chapter_tree/avl_tree.assets/avltree_degradation_from_inserting_node.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/degradation_from_inserting_node.png rename to docs/chapter_tree/avl_tree.assets/avltree_degradation_from_inserting_node.png diff --git a/docs/chapter_tree/avl_tree.assets/degradation_from_removing_node.png b/docs/chapter_tree/avl_tree.assets/avltree_degradation_from_removing_node.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/degradation_from_removing_node.png rename to docs/chapter_tree/avl_tree.assets/avltree_degradation_from_removing_node.png diff --git a/docs/chapter_tree/avl_tree.assets/left_right_rotate.png b/docs/chapter_tree/avl_tree.assets/avltree_left_right_rotate.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/left_right_rotate.png rename to docs/chapter_tree/avl_tree.assets/avltree_left_right_rotate.png diff --git a/docs/chapter_tree/avl_tree.assets/left_rotate.png b/docs/chapter_tree/avl_tree.assets/avltree_left_rotate.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/left_rotate.png rename to docs/chapter_tree/avl_tree.assets/avltree_left_rotate.png diff --git a/docs/chapter_tree/avl_tree.assets/left_rotate_with_grandchild.png b/docs/chapter_tree/avl_tree.assets/avltree_left_rotate_with_grandchild.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/left_rotate_with_grandchild.png rename to docs/chapter_tree/avl_tree.assets/avltree_left_rotate_with_grandchild.png diff --git a/docs/chapter_tree/avl_tree.assets/right_left_rotate.png b/docs/chapter_tree/avl_tree.assets/avltree_right_left_rotate.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/right_left_rotate.png rename to docs/chapter_tree/avl_tree.assets/avltree_right_left_rotate.png diff --git a/docs/chapter_tree/avl_tree.assets/right_rotate_step1.png b/docs/chapter_tree/avl_tree.assets/avltree_right_rotate_step1.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/right_rotate_step1.png rename to docs/chapter_tree/avl_tree.assets/avltree_right_rotate_step1.png diff --git a/docs/chapter_tree/avl_tree.assets/right_rotate_step2.png b/docs/chapter_tree/avl_tree.assets/avltree_right_rotate_step2.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/right_rotate_step2.png rename to docs/chapter_tree/avl_tree.assets/avltree_right_rotate_step2.png diff --git a/docs/chapter_tree/avl_tree.assets/right_rotate_step3.png b/docs/chapter_tree/avl_tree.assets/avltree_right_rotate_step3.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/right_rotate_step3.png rename to docs/chapter_tree/avl_tree.assets/avltree_right_rotate_step3.png diff --git a/docs/chapter_tree/avl_tree.assets/right_rotate_step4.png b/docs/chapter_tree/avl_tree.assets/avltree_right_rotate_step4.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/right_rotate_step4.png rename to docs/chapter_tree/avl_tree.assets/avltree_right_rotate_step4.png diff --git a/docs/chapter_tree/avl_tree.assets/right_rotate_with_grandchild.png b/docs/chapter_tree/avl_tree.assets/avltree_right_rotate_with_grandchild.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/right_rotate_with_grandchild.png rename to docs/chapter_tree/avl_tree.assets/avltree_right_rotate_with_grandchild.png diff --git a/docs/chapter_tree/avl_tree.assets/rotation_cases.png b/docs/chapter_tree/avl_tree.assets/avltree_rotation_cases.png similarity index 100% rename from docs/chapter_tree/avl_tree.assets/rotation_cases.png rename to docs/chapter_tree/avl_tree.assets/avltree_rotation_cases.png diff --git a/docs/chapter_tree/avl_tree.md b/docs/chapter_tree/avl_tree.md index 34b681762..5d2c7e0d9 100644 --- a/docs/chapter_tree/avl_tree.md +++ b/docs/chapter_tree/avl_tree.md @@ -8,11 +8,11 @@ comments: true 如下图所示，执行两步删除结点后，该二叉搜索树就会退化为链表。 - + 再比如，在以下完美二叉树中插入两个结点后，树严重向左偏斜，查找操作的时间复杂度也随之发生劣化。 - + G. M. Adelson-Velsky 和 E. M. Landis 在其 1962 年发表的论文 "An algorithm for the organization of information" 中提出了「AVL 树」。**论文中描述了一系列操作，使得在不断添加与删除结点后，AVL 树仍然不会发生退化**，进而使得各种操作的时间复杂度均能保持在 $O(\log n)$ 级别。 @@ -314,20 +314,20 @@ AVL 树的独特之处在于「旋转 Rotation」的操作，其可 **在不影 如下图所示（结点下方为「平衡因子」），从底至顶看，二叉树中首个失衡结点是 **结点 3**。我们聚焦在以该失衡结点为根结点的子树上，将该结点记为 `node` ，将其左子结点记为 `child` ，执行「右旋」操作。完成右旋后，该子树已经恢复平衡，并且仍然为二叉搜索树。 === "<1>" -  +  === "<2>" -  +  === "<3>" -  +  === "<4>" -  +  进而，如果结点 `child` 本身有右子结点（记为 `grandChild` ），则需要在「右旋」中添加一步：将 `grandChild` 作为 `node` 的左子结点。 - + “向右旋转”是一种形象化的说法，实际需要通过修改结点指针实现，代码如下所示。 @@ -395,11 +395,11 @@ AVL 树的独特之处在于「旋转 Rotation」的操作，其可 **在不影 类似地，如果将取上述失衡二叉树的“镜像”，那么则需要「左旋」操作。 - + 同理，若结点 `child` 本身有左子结点（记为 `grandChild` ），则需要在「左旋」中添加一步：将 `grandChild` 作为 `node` 的右子结点。 - + 观察发现，**「左旋」和「右旋」操作是镜像对称的，两者对应解决的两种失衡情况也是对称的**。根据对称性，我们可以很方便地从「右旋」推导出「左旋」。具体地，只需将「右旋」代码中的把所有的 `left` 替换为 `right` 、所有的 `right` 替换为 `left` ，即可得到「左旋」代码。 @@ -467,19 +467,19 @@ AVL 树的独特之处在于「旋转 Rotation」的操作，其可 **在不影 对于下图的失衡结点 3 ，**单一使用左旋或右旋都无法使子树恢复平衡**，此时需要「先左旋后右旋」，即先对 `child` 执行「左旋」，再对 `node` 执行「右旋」。 - + ### Case 4 - 先右后左 同理，取以上失衡二叉树的镜像，则需要「先右旋后左旋」，即先对 `child` 执行「右旋」，然后对 `node` 执行「左旋」。 - + ### 旋转的选择 下图描述的四种失衡情况与上述 Cases 逐个对应，分别需采用 **右旋、左旋、先右后左、先左后右** 的旋转操作。 - + 具体地，在代码中使用 **失衡结点的平衡因子、较高一侧子结点的平衡因子** 来确定失衡结点属于上图中的哪种情况。 diff --git a/docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_1.png b/docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_step1.png similarity index 100% rename from docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_1.png rename to docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_step1.png diff --git a/docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_2.png b/docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_step2.png similarity index 100% rename from docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_2.png rename to docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_step2.png diff --git a/docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_3.png b/docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_step3.png similarity index 100% rename from docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_3.png rename to docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_step3.png diff --git a/docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_4.png b/docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_step4.png similarity index 100% rename from docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_4.png rename to docs/chapter_tree/binary_search_tree.assets/bst_remove_case3_step4.png diff --git a/docs/chapter_tree/binary_search_tree.assets/bst_search_1.png b/docs/chapter_tree/binary_search_tree.assets/bst_search_step1.png similarity index 100% rename from docs/chapter_tree/binary_search_tree.assets/bst_search_1.png rename to docs/chapter_tree/binary_search_tree.assets/bst_search_step1.png diff --git a/docs/chapter_tree/binary_search_tree.assets/bst_search_2.png b/docs/chapter_tree/binary_search_tree.assets/bst_search_step2.png similarity index 100% rename from docs/chapter_tree/binary_search_tree.assets/bst_search_2.png rename to docs/chapter_tree/binary_search_tree.assets/bst_search_step2.png diff --git a/docs/chapter_tree/binary_search_tree.assets/bst_search_3.png b/docs/chapter_tree/binary_search_tree.assets/bst_search_step3.png similarity index 100% rename from docs/chapter_tree/binary_search_tree.assets/bst_search_3.png rename to docs/chapter_tree/binary_search_tree.assets/bst_search_step3.png diff --git a/docs/chapter_tree/binary_search_tree.assets/bst_search_4.png b/docs/chapter_tree/binary_search_tree.assets/bst_search_step4.png similarity index 100% rename from docs/chapter_tree/binary_search_tree.assets/bst_search_4.png rename to docs/chapter_tree/binary_search_tree.assets/bst_search_step4.png diff --git a/docs/chapter_tree/binary_search_tree.md b/docs/chapter_tree/binary_search_tree.md index dcfcbb350..8c6dbb829 100755 --- a/docs/chapter_tree/binary_search_tree.md +++ b/docs/chapter_tree/binary_search_tree.md @@ -22,16 +22,16 @@ comments: true - 若 `cur.val = num` ，说明找到目标结点，跳出循环并返回该结点即可； === "<1>" -  +  === "<2>" -  +  === "<3>" -  +  === "<4>" -  +  二叉搜索树的查找操作和二分查找算法如出一辙，也是在每轮排除一半情况。循环次数最多为二叉树的高度，当二叉树平衡时，使用 $O(\log n)$ 时间。 @@ -189,16 +189,16 @@ comments: true 3. 使用 `nex` 替换待删除结点； === "<1>" -  +  === "<2>" -  +  === "<3>" -  +  === "<4>" -  +  删除结点操作也使用 $O(\log n)$ 时间，其中查找待删除结点 $O(\log n)$ ，获取中序遍历后继结点 $O(\log n)$ 。