From f103fdecc9ff8965cc96872919f91c4c04ef85a7 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Mon, 30 Jan 2023 03:55:13 +0800 Subject: [PATCH] 1. Update extra.css for overwrite the sections title color. 2. Add figures to hash collision. 3. Add inorder traversal in binary search tree. --- .../hash_collision_chaining.png | Bin 0 -> 91385 bytes .../hash_collision_linear_probing.png | Bin 0 -> 60779 bytes docs/chapter_hashing/hash_collision.md | 57 +++++---- docs/chapter_hashing/hash_map.md | 114 +++++++++--------- .../bst_inorder_traversal.png | Bin 0 -> 106267 bytes docs/chapter_tree/binary_search_tree.md | 32 +++-- docs/stylesheets/extra.css | 5 + 7 files changed, 115 insertions(+), 93 deletions(-) create mode 100644 docs/chapter_hashing/hash_collision.assets/hash_collision_chaining.png create mode 100644 docs/chapter_hashing/hash_collision.assets/hash_collision_linear_probing.png create mode 100644 docs/chapter_tree/binary_search_tree.assets/bst_inorder_traversal.png diff --git a/docs/chapter_hashing/hash_collision.assets/hash_collision_chaining.png b/docs/chapter_hashing/hash_collision.assets/hash_collision_chaining.png new file mode 100644 index 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z-YkIa?E?Ul3$@+^vxr!R*>vKhdHr$Hw4&F6lLIziL**OlJJ+X(c>lfUR@)G2!l4U|j%J92AVnlrGRse}lwO_9)R$>CI7%bX3#fw>5Uslvu2cGqfzY73ur`C;3X$dvZJ;OyfH@R~cO5BJa!5 zuC|$OTG)6INrN&g(hCJ0s6vHRFK2(Tqzf&9+0b_zhN`yvqkV9q6Wh;kz|gxk`HUtaq02 z|A%BePkFR69joT?JuCiVpf1zh)_MjRk=9a`(){cCa z<-)8EkrWxD@NL;6y(XmjJW19!~UCy0R$US6PGUl0mkbKUzMRpB(z h3o}NAwBFkDfIruQd!^EfqJV$mPo+h&g>~NiKLFcJ)D!>! literal 0 HcmV?d00001 diff --git a/docs/chapter_hashing/hash_collision.md b/docs/chapter_hashing/hash_collision.md index 45a529970..9c54f22b1 100644 --- a/docs/chapter_hashing/hash_collision.md +++ b/docs/chapter_hashing/hash_collision.md @@ -2,45 +2,46 @@ comments: true --- -# 哈希冲突处理 +# 哈希冲突 -理想情况下,哈希函数应该为每个输入产生唯一的输出,使得 key 和 value 一一对应。而实际上,往往存在向哈希函数输入不同的 key 而产生相同输出的情况,这种情况被称为「哈希冲突 Hash Collision」。 +理想情况下,哈希函数应该为每个输入产生唯一的输出,使得 key 和 value 一一对应。而实际上,往往存在向哈希函数输入不同的 key 而产生相同输出的情况,这种情况被称为「哈希冲突 Hash Collision」。哈希冲突会导致查询结果错误,从而严重影响哈希表的可用性。 -**哈希冲突会严重影响哈希表的实用性**。试想一下,如果在哈希表中总是查找到错误的结果,那么我们肯定不会继续使用这样的数据结构了。 +那么,为什么会出现哈希冲突呢?本质上看,**由于哈希函数的输入空间往往远大于输出空间**,因此不可避免地会出现多个输入产生相同输出的情况,即为哈希冲突。比如,输入空间是全体整数,输出空间是一个固定大小的桶(数组)的索引范围,那么必定会有多个整数同时映射到一个桶索引。 -!!! question "为什么会出现哈希冲突?" +为了缓解哈希冲突,一方面,我们可以通过「哈希表扩容」来减小冲突概率。极端情况下,当输入空间和输出空间大小相等时,哈希表就等价于数组了,可谓“大力出奇迹”。 - 因为 **哈希函数的输入空间往往远大于输出空间**,所以不可避免地会出现多个输入产生相同输出的情况。比如,输入空间是全体整数,输出空间是一个固定大小的桶(数组)的索引范围,那么必定会有多个整数同时映射到一个桶索引。 +另一方面,**考虑通过优化数据结构以缓解哈希冲突**,常见的方法有「链式地址」和「开放寻址」。 -虽然理论上哈希冲突难以避免,**但我们仍然可以在数据结构层面上缓解哈希冲突所带来的负面影响**,尽量保证哈希表的增删查改操作效率。 +## 哈希表扩容 -常见的哈希冲突的解决方案有「链式地址」和「开放寻址」。 +「负载因子 Load Factor」定义为 **哈希表中元素数量除以桶槽数量(即数组大小)**,代表哈希冲突的严重程度。 + +**负载因子常用作哈希表扩容的触发条件**。比如在 Java 中,当负载因子 $> 0.75$ 时则触发扩容,将 HashMap 大小扩充至原先 $2$ 倍。 + +与数组扩容类似,**哈希表扩容操作的开销很大**,因为需要将所有键值对从原哈希表依次移动至新哈希表。 ## 链式地址 -「链式地址」通过引入链表来解决哈希冲突问题,代价是占用空间变大,因为链表或二叉树包含结点指针,相比于数组更加耗费内存空间。 +在原始哈希表中,桶内的每个地址只能存储一个元素(即键值对)。**考虑将单个元素转化成一个链表,将所有冲突元素都存储在一个链表中**。 -### 链表引入 +![hash_collision_chaining](hash_collision.assets/hash_collision_chaining.png) -在原始哈希表中,一个桶地址只能存储一个元素(即键值对)。**考虑将桶地址内的单个元素转变成一个链表,将所有冲突元素都存储在一个链表中**,此时哈希表操作方法为: +链式地址下,哈希表操作方法为: -- **查询元素**:先将 key 输入到哈希函数得到桶地址(即访问链表头部),再遍历链表来确定对应的 value 。 -- **添加元素**:先通过哈希函数访问链表头部,再将元素直接添加到链表头部即可。 -- **删除元素**:同样先访问链表头部,再遍历链表查找对应元素,删除之即可。 +- **查询元素**:先将 key 输入到哈希函数得到桶内索引,即可访问链表头结点,再通过遍历链表查找对应 value 。 +- **添加元素**:先通过哈希函数访问链表头部,再将结点(即键值对)添加到链表头部即可。 +- **删除元素**:同样先根据哈希函数结果访问链表头部,再遍历链表查找对应结点,删除之即可。 -(图) +链式地址虽然解决了哈希冲突问题,但仍存在局限性,包括: -### 二叉树引入 +- **占用空间变大**,因为链表或二叉树包含结点指针,相比于数组更加耗费内存空间; +- **查询效率降低**,因为需要线性遍历链表来查找对应元素; -引入链表虽然解决了哈希冲突,但查询效率也随之降低了,因为需要线性查找(即遍历链表)来确认对应元素。为了缓解此问题,**当某个桶地址内的链表太长时,可以把链表转化为「平衡二叉搜索树」**,将时间复杂度降低至 $O(\log n)$ 。 - -!!! note "工业界方案" - - Java 使用了链式地址来解决哈希冲突。在 JDK 1.8 之后, HashMap 内长度大于 8 的链表会被转化为「红黑树」,以提升查找性能。 +为了缓解时间效率问题,**可以把「链表」转化为「AVL 树」或「红黑树」**,将查询操作的时间复杂度优化至 $O(\log n)$ 。 ## 开放寻址 -「开放寻址」不引入额外数据结构,而是通过“向后探测”来解决哈希冲突。根据探测方法的不同,主要分为 **线性探测、平方探测、多次哈希**。 +「开放寻址」不引入额外数据结构,而是通过“多次探测”来解决哈希冲突。根据探测方法的不同,主要分为 **线性探测、平方探测、多次哈希**。 ### 线性探测 @@ -53,16 +54,16 @@ comments: true 1. 找到对应元素,返回 value 即可; 2. 若遇到空位,则说明查找键值对不在哈希表中; -(图) +![hash_collision_linear_probing](hash_collision.assets/hash_collision_linear_probing.png) -线性探测有以下缺陷: +线性探测存在以下缺陷: - **不能直接删除元素**。删除元素会导致桶内出现一个空位,在查找其他元素时,该空位有可能导致程序认为元素不存在(即上述第 `2.` 种情况)。因此需要借助一个标志位来标记删除元素。 - **容易产生聚集**。桶内被占用的连续位置越长,这些连续位置发生哈希冲突的可能性越大,从而进一步促进这一位置的“聚堆生长”,最终导致增删查改操作效率的劣化。 ### 多次哈希 -顾名思义,「多次哈希」的思路是基于多个哈希函数 $f_1(x)$ , $f_2(x)$ , $f_3(x)$ , $\cdots$ 进行探测。 +顾名思义,「多次哈希」的思路是使用多个哈希函数 $f_1(x)$ , $f_2(x)$ , $f_3(x)$ , $\cdots$ 进行探测。 **插入元素**:若哈希函数 $f_1(x)$ 出现冲突,则尝试 $f_2(x)$ ,以此类推……直到找到空位后插入元素。 @@ -72,3 +73,11 @@ comments: true 2. 到空位或已尝试所有哈希函数,说明哈希表中无此元素; 相比于「线性探测」,「多次哈希」方法更不容易产生聚集,代价是多个哈希函数增加了额外计算量。 + +!!! note "工业界方案" + + Java 采用「链式地址」。在 JDK 1.8 之后,HashMap 内数组长度 $> 64$ 时,长度大于 8 的链表会被转化为「红黑树」,以提升查找性能。 + Python 采用「开放寻址」。字典 dict 采用的是随机探测,即使用伪随机数来探测。 + + + diff --git a/docs/chapter_hashing/hash_map.md b/docs/chapter_hashing/hash_map.md index f3afdcd57..be54c7df1 100644 --- a/docs/chapter_hashing/hash_map.md +++ b/docs/chapter_hashing/hash_map.md @@ -12,7 +12,7 @@ comments: true

Fig. 哈希表抽象表示

-## 哈希表优势 +## 哈希表效率 除了哈希表之外,还可以使用以下数据结构来实现上述查询功能: @@ -42,7 +42,7 @@ comments: true ```java title="hash_map.java" /* 初始化哈希表 */ Map map = new HashMap<>(); - + /* 添加操作 */ // 在哈希表中添加键值对 (key, value) map.put(12836, "小哈"); @@ -50,11 +50,11 @@ comments: true map.put(16750, "小算"); map.put(13276, "小法"); map.put(10583, "小鸭"); - + /* 查询操作 */ // 向哈希表输入键 key ,得到值 value String name = map.get(15937); - + /* 删除操作 */ // 在哈希表中删除键值对 (key, value) map.remove(10583); @@ -65,7 +65,7 @@ comments: true ```cpp title="hash_map.cpp" /* 初始化哈希表 */ unordered_map map; - + /* 添加操作 */ // 在哈希表中添加键值对 (key, value) map[12836] = "小哈"; @@ -73,11 +73,11 @@ comments: true map[16750] = "小算"; map[13276] = "小法"; map[10583] = "小鸭"; - + /* 查询操作 */ // 向哈希表输入键 key ,得到值 value string name = map[15937]; - + /* 删除操作 */ // 在哈希表中删除键值对 (key, value) map.erase(10583); @@ -88,7 +88,7 @@ comments: true ```python title="hash_map.py" """ 初始化哈希表 """ mapp = {} - + """ 添加操作 """ # 在哈希表中添加键值对 (key, value) mapp[12836] = "小哈" @@ -96,11 +96,11 @@ comments: true mapp[16750] = "小算" mapp[13276] = "小法" mapp[10583] = "小鸭" - + """ 查询操作 """ # 向哈希表输入键 key ,得到值 value name = mapp[15937] - + """ 删除操作 """ # 在哈希表中删除键值对 (key, value) mapp.pop(10583) @@ -111,7 +111,7 @@ comments: true ```go title="hash_map.go" /* 初始化哈希表 */ mapp := make(map[int]string) - + /* 添加操作 */ // 在哈希表中添加键值对 (key, value) mapp[12836] = "小哈" @@ -119,11 +119,11 @@ comments: true mapp[16750] = "小算" mapp[13276] = "小法" mapp[10583] = "小鸭" - + /* 查询操作 */ // 向哈希表输入键 key ,得到值 value name := mapp[15937] - + /* 删除操作 */ // 在哈希表中删除键值对 (key, value) delete(mapp, 10583) @@ -141,11 +141,11 @@ comments: true map.set(16750, '小算'); map.set(13276, '小法'); map.set(10583, '小鸭'); - + /* 查询操作 */ // 向哈希表输入键 key ,得到值 value let name = map.get(15937); - + /* 删除操作 */ // 在哈希表中删除键值对 (key, value) map.delete(10583); @@ -165,12 +165,12 @@ comments: true map.set(10583, '小鸭'); console.info('\n添加完成后,哈希表为\nKey -> Value'); console.info(map); - + /* 查询操作 */ // 向哈希表输入键 key ,得到值 value let name = map.get(15937); console.info('\n输入学号 15937 ,查询到姓名 ' + name); - + /* 删除操作 */ // 在哈希表中删除键值对 (key, value) map.delete(10583); @@ -181,7 +181,7 @@ comments: true === "C" ```c title="hash_map.c" - + ``` === "C#" @@ -189,7 +189,7 @@ comments: true ```csharp title="hash_map.cs" /* 初始化哈希表 */ Dictionary map = new (); - + /* 添加操作 */ // 在哈希表中添加键值对 (key, value) map.Add(12836, "小哈"); @@ -197,11 +197,11 @@ comments: true map.Add(16750, "小算"); map.Add(13276, "小法"); map.Add(10583, "小鸭"); - + /* 查询操作 */ // 向哈希表输入键 key ,得到值 value String name = map[15937]; - + /* 删除操作 */ // 在哈希表中删除键值对 (key, value) map.Remove(10583); @@ -212,7 +212,7 @@ comments: true ```swift title="hash_map.swift" /* 初始化哈希表 */ var map: [Int: String] = [:] - + /* 添加操作 */ // 在哈希表中添加键值对 (key, value) map[12836] = "小哈" @@ -220,11 +220,11 @@ comments: true map[16750] = "小算" map[13276] = "小法" map[10583] = "小鸭" - + /* 查询操作 */ // 向哈希表输入键 key ,得到值 value let name = map[15937]! - + /* 删除操作 */ // 在哈希表中删除键值对 (key, value) map.removeValue(forKey: 10583) @@ -341,7 +341,7 @@ comments: true === "C" ```c title="hash_map.c" - + ``` === "C#" @@ -413,7 +413,7 @@ $$ this.val = val; } } - + /* 基于数组简易实现的哈希表 */ class ArrayHashMap { private List bucket; @@ -424,13 +424,13 @@ $$ bucket.add(null); } } - + /* 哈希函数 */ private int hashFunc(int key) { int index = key % 100; return index; } - + /* 查询操作 */ public String get(int key) { int index = hashFunc(key); @@ -438,14 +438,14 @@ $$ if (pair == null) return null; return pair.val; } - + /* 添加操作 */ public void put(int key, String val) { Entry pair = new Entry(key, val); int index = hashFunc(key); bucket.set(index, pair); } - + /* 删除操作 */ public void remove(int key) { int index = hashFunc(key); @@ -468,7 +468,7 @@ $$ this->val = val; } }; - + /* 基于数组简易实现的哈希表 */ class ArrayHashMap { private: @@ -478,27 +478,27 @@ $$ // 初始化一个长度为 100 的桶(数组) bucket= vector(100); } - + /* 哈希函数 */ int hashFunc(int key) { int index = key % 100; return index; } - + /* 查询操作 */ string get(int key) { int index = hashFunc(key); Entry* pair = bucket[index]; return pair->val; } - + /* 添加操作 */ void put(int key, string val) { Entry* pair = new Entry(key, val); int index = hashFunc(key); bucket[index] = pair; } - + /* 删除操作 */ void remove(int key) { int index = hashFunc(key); @@ -557,24 +557,24 @@ $$ key int val string } - + /* 基于数组简易实现的哈希表 */ type arrayHashMap struct { bucket []*entry } - + func newArrayHashMap() *arrayHashMap { // 初始化一个长度为 100 的桶(数组) bucket := make([]*entry, 100) return &arrayHashMap{bucket: bucket} } - + /* 哈希函数 */ func (a *arrayHashMap) hashFunc(key int) int { index := key % 100 return index } - + /* 查询操作 */ func (a *arrayHashMap) get(key int) string { index := a.hashFunc(key) @@ -584,14 +584,14 @@ $$ } return pair.val } - + /* 添加操作 */ func (a *arrayHashMap) put(key int, val string) { pair := &entry{key: key, val: val} index := a.hashFunc(key) a.bucket[index] = pair } - + /* 删除操作 */ func (a *arrayHashMap) remove(key int) { index := a.hashFunc(key) @@ -610,7 +610,7 @@ $$ this.val = val; } } - + /* 基于数组简易实现的哈希表 */ class ArrayHashMap { #bucket; @@ -618,12 +618,12 @@ $$ // 初始化一个长度为 100 的桶(数组) this.#bucket = new Array(100).fill(null); } - + /* 哈希函数 */ #hashFunc(key) { return key % 100; } - + /* 查询操作 */ get(key) { let index = this.#hashFunc(key); @@ -631,13 +631,13 @@ $$ if (entry === null) return null; return entry.val; } - + /* 添加操作 */ set(key, val) { let index = this.#hashFunc(key); this.#bucket[index] = new Entry(key, val); } - + /* 删除操作 */ delete(key) { let index = this.#hashFunc(key); @@ -735,7 +735,7 @@ $$ === "C" ```c title="array_hash_map.c" - + ``` === "C#" @@ -752,7 +752,7 @@ $$ this.val = val; } } - + /* 基于数组简易实现的哈希表 */ class ArrayHashMap { @@ -804,44 +804,44 @@ $$ class Entry { var key: Int var val: String - + init(key: Int, val: String) { self.key = key self.val = val } } - + /* 基于数组简易实现的哈希表 */ class ArrayHashMap { private var bucket: [Entry?] = [] - + init() { // 初始化一个长度为 100 的桶(数组) for _ in 0 ..< 100 { bucket.append(nil) } } - + /* 哈希函数 */ private func hashFunc(key: Int) -> Int { let index = key % 100 return index } - + /* 查询操作 */ func get(key: Int) -> String? { let index = hashFunc(key: key) let pair = bucket[index] return pair?.val } - + /* 添加操作 */ func put(key: Int, val: String) { let pair = Entry(key: key, val: val) let index = hashFunc(key: key) bucket[index] = pair } - + /* 删除操作 */ func remove(key: Int) { let index = hashFunc(key: key) @@ -853,13 +853,13 @@ $$ ## 哈希冲突 -细心的同学可能会发现,**哈希函数 $f(x) = x \% 100$ 会在某些情况下失效**。具体地,当输入的 key 后两位相同时,哈希函数的计算结果也相同,指向同一个 value 。例如,分别查询两个学号 12836 和 20336 ,则有 +细心的同学可能会发现,**哈希函数 $f(x) = x \% 100$ 会在某些情况下失效**。具体地,当输入的 key 后两位相同时,哈希函数的计算结果也相同,指向同一个 value 。例如,分别查询两个学号 $12836$ 和 $20336$ ,则有 $$ f(12836) = f(20336) = 36 $$ -两个学号指向了同一个姓名,这明显是不对的,我们将这种现象称为「哈希冲突 Hash Collision」,其会严重影响查询的正确性。如何避免哈希冲突的问题将被留在下章讨论。 +两个学号指向了同一个姓名,这明显是不对的,我们将这种现象称为「哈希冲突 Hash Collision」。如何避免哈希冲突的问题将被留在下章讨论。 ![hash_collision](hash_map.assets/hash_collision.png) diff --git a/docs/chapter_tree/binary_search_tree.assets/bst_inorder_traversal.png 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cur.left - + # 插入结点 val node = TreeNode(num) if pre.val < num: @@ -410,7 +410,7 @@ comments: true === "C" ```c title="binary_search_tree.c" - + ``` === "C#" @@ -433,7 +433,7 @@ comments: true // 插入位置在 cur 的左子树中 else cur = cur.left; } - + // 插入结点 val TreeNode node = new TreeNode(num); if (pre != null) @@ -519,6 +519,14 @@ comments: true 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 $O(\log n)$ 。 +### 排序 + +我们知道,「中序遍历」遵循“左 $\rightarrow$ 根 $\rightarrow$ 右”的遍历优先级,而二叉搜索树遵循“左子结点 $<$ 根结点 $<$ 右子结点”的大小关系。因此,在二叉搜索树中进行中序遍历时,总是会优先遍历下一个最小结点,从而得出一条重要性质:**二叉搜索树的中序遍历序列是升序的**。 + +借助中序遍历升序的性质,我们在二叉搜索树中获取有序数据仅需 $O(n)$ 时间,而无需额外排序,非常高效。 + +![bst_inorder_traversal](binary_search_tree.assets/bst_inorder_traversal.png) + === "Java" ```java title="binary_search_tree.java" @@ -614,10 +622,10 @@ comments: true # 若树为空,直接提前返回 if root is None: return None - + cur = root pre = None - + # 循环查找,越过叶结点后跳出 while cur is not None: # 找到待删除结点,跳出循环 @@ -628,11 +636,11 @@ comments: true cur = cur.right else: # 待删除结点在 cur 的左子树中 cur = cur.left - + # 若无待删除结点,则直接返回 if cur is None: return None - + # 子结点数量 = 0 or 1 if cur.left is None or cur.right is None: # 当子结点数量 = 0 / 1 时, child = null / 该子结点 @@ -811,7 +819,7 @@ comments: true === "C" ```c title="binary_search_tree.c" - + ``` === "C#" @@ -926,7 +934,7 @@ comments: true } ``` -## 二叉搜索树的优势 +## 二叉搜索树的效率 假设给定 $n$ 个数字,最常用的存储方式是「数组」,那么对于这串乱序的数字,常见操作的效率为: diff --git a/docs/stylesheets/extra.css b/docs/stylesheets/extra.css index bbbfe4417..4f153c794 100644 --- a/docs/stylesheets/extra.css +++ b/docs/stylesheets/extra.css @@ -22,6 +22,11 @@ --md-typeset-a-color: #21C8B8; } +/* https://github.com/squidfunk/mkdocs-material/issues/4832#issuecomment-1374891676 */ +.md-nav__link[for] { + color: var(--md-default-fg-color) !important +} + /* Center Markdown Tables (requires md_in_html extension) */ .center-table { text-align: center;