From d1efac36ca14bcd06b80feb14269d340b28e7536 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Tue, 6 Dec 2022 01:00:21 +0800 Subject: [PATCH] Update the chapter of hashing. --- .../java/chapter_hashing/array_hash_map.java | 24 ++--- codes/java/chapter_hashing/hash_map.java | 18 ++-- codes/python/chapter_stack_and_queue/queue.py | 2 +- .../hash_map.assets/hash_collision.png | Bin 0 -> 84373 bytes .../hash_map.assets/hash_function.png | Bin 0 -> 89110 bytes .../hash_map.assets/hash_map.png | Bin 0 -> 64231 bytes docs/chapter_hashing/hash_map.md | 93 ++++++++++-------- 7 files changed, 76 insertions(+), 61 deletions(-) create mode 100644 docs/chapter_hashing/hash_map.assets/hash_collision.png create mode 100644 docs/chapter_hashing/hash_map.assets/hash_function.png create mode 100644 docs/chapter_hashing/hash_map.assets/hash_map.png diff --git a/codes/java/chapter_hashing/array_hash_map.java b/codes/java/chapter_hashing/array_hash_map.java index 5fad6f3ea..89082c2f9 100644 --- a/codes/java/chapter_hashing/array_hash_map.java +++ b/codes/java/chapter_hashing/array_hash_map.java @@ -22,15 +22,15 @@ class ArrayHashMap { private List bucket; public ArrayHashMap() { // 初始化一个长度为 10 的桶(数组) - bucket = new ArrayList<>(10); - for (int i = 0; i < 10; i++) { + bucket = new ArrayList<>(); + for (int i = 0; i < 100; i++) { bucket.add(null); } } /* 哈希函数 */ private int hashFunc(int key) { - int index = key % 10000; + int index = key % 100; return index; } @@ -102,23 +102,23 @@ public class array_hash_map { /* 添加操作 */ // 在哈希表中添加键值对 (key, value) - map.put(10001, "小哈"); - map.put(10002, "小啰"); - map.put(10003, "小算"); - map.put(10004, "小法"); - map.put(10005, "小哇"); + map.put(12836, "小哈"); + map.put(15937, "小啰"); + map.put(16750, "小算"); + map.put(13276, "小法"); + map.put(10583, "小鸭"); System.out.println("\n添加完成后,哈希表为\nKey -> Value"); map.print(); /* 查询操作 */ // 向哈希表输入键 key ,得到值 value - String name = map.get(10002); - System.out.println("\n输入学号 10002 ,查询到姓名 " + name); + String name = map.get(15937); + System.out.println("\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.remove(10005); - System.out.println("\n删除 10005 后,哈希表为\nKey -> Value"); + map.remove(10583); + System.out.println("\n删除 10583 后,哈希表为\nKey -> Value"); map.print(); /* 遍历哈希表 */ diff --git a/codes/java/chapter_hashing/hash_map.java b/codes/java/chapter_hashing/hash_map.java index bd145b76e..bb72a4b0f 100644 --- a/codes/java/chapter_hashing/hash_map.java +++ b/codes/java/chapter_hashing/hash_map.java @@ -15,23 +15,23 @@ public class hash_map { /* 添加操作 */ // 在哈希表中添加键值对 (key, value) - map.put(10001, "小哈"); - map.put(10002, "小啰"); - map.put(10003, "小算"); - map.put(10004, "小法"); - map.put(10005, "小哇"); + map.put(12836, "小哈"); + map.put(15937, "小啰"); + map.put(16750, "小算"); + map.put(13276, "小法"); + map.put(10583, "小鸭"); System.out.println("\n添加完成后,哈希表为\nKey -> Value"); PrintUtil.printHashMap(map); /* 查询操作 */ // 向哈希表输入键 key ,得到值 value - String name = map.get(10002); - System.out.println("\n输入学号 10002 ,查询到姓名 " + name); + String name = map.get(15937); + System.out.println("\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.remove(10005); - System.out.println("\n删除 10005 后,哈希表为\nKey -> Value"); + map.remove(10583); + System.out.println("\n删除 10583 后,哈希表为\nKey -> Value"); PrintUtil.printHashMap(map); /* 遍历哈希表 */ diff --git a/codes/python/chapter_stack_and_queue/queue.py b/codes/python/chapter_stack_and_queue/queue.py index 93729a868..740d9b1e1 100644 --- a/codes/python/chapter_stack_and_queue/queue.py +++ b/codes/python/chapter_stack_and_queue/queue.py @@ -1,5 +1,5 @@ ''' -File: que.py +File: queue.py Created Time: 2022-11-29 Author: Peng Chen (pengchzn@gmail.com) ''' diff --git a/docs/chapter_hashing/hash_map.assets/hash_collision.png b/docs/chapter_hashing/hash_map.assets/hash_collision.png new file mode 100644 index 0000000000000000000000000000000000000000..c2d869bd2b2363d5af0ba76d2ef7626cc98e7661 GIT binary patch literal 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zK2%?m8+FMGtH(Xjjk24pbPLP9m*k>UZ`$fI@2|a(nPotN{qoaV^vR>sRt=FAPS_Cf SA(@u?I_2*DJ^6U>l|KP`b9XoZ literal 0 HcmV?d00001 diff --git a/docs/chapter_hashing/hash_map.md b/docs/chapter_hashing/hash_map.md index 62725faae..77bc82321 100644 --- a/docs/chapter_hashing/hash_map.md +++ b/docs/chapter_hashing/hash_map.md @@ -4,9 +4,34 @@ comments: true # 哈希表 -哈希表通过建立「键 Key」和「值 Value」之间的映射,实现高效的元素查找。具体地,查询操作(给定一个 Key 查询得到 Value)的时间复杂度为 $O(1)$ 。 +哈希表通过建立「键 Key」和「值 Value」之间的映射,实现高效的元素查找。具体地,输入一个 Key ,在哈希表中查询并获取 Value ,时间复杂度为 $O(1)$ 。 -(图) +例如,给定一个包含 $n$ 个学生的数据库,每个学生有 "姓名 `name` ” 和 “学号 `id` ” 两项数据,希望实现一个查询功能:**输入一个学号,返回对应的姓名**,则可以使用哈希表实现。 + +![hash_map](hash_map.assets/hash_map.png) + +

Fig. 哈希表抽象表示

+ +## 哈希表优势 + +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: + +- **无序数组:** 每个元素为 `[学号, 姓名]` ; +- **有序数组:** 将 `1.` 中的数组按照学号从小到大排序; +- **链表:** 每个结点的值为 `[学号, 姓名]` ; +- **二叉搜索树:** 每个结点的值为 `[学号, 姓名]` ,根据学号大小来构建树; + +使用上述方法,各项操作的时间复杂度如下表所示(在此不做赘述,详解可见 [二叉搜索树章节](https://www.hello-algo.com/chapter_tree/binary_search_tree/#_6))。无论是查找元素、还是增删元素,哈希表的时间复杂度都是 $O(1)$ ,全面胜出! + +
+ +| | 无序数组 | 有序数组 | 链表 | 二叉搜索树 | 哈希表 | +| -------- | -------- | ----------- | ------ | ----------- | ------ | +| 查找元素 | $O(n)$ | $O(\log n)$ | $O(n)$ | $O(\log n)$ | $O(1)$ | +| 插入元素 | $O(1)$ | $O(n)$ | $O(1)$ | $O(\log n)$ | $O(1)$ | +| 删除元素 | $O(n)$ | $O(n)$ | $O(n)$ | $O(\log n)$ | $O(1)$ | + +
## 哈希表常用操作 @@ -18,19 +43,19 @@ Map map = new HashMap<>(); /* 添加操作 */ // 在哈希表中添加键值对 (key, value) -map.put(10001, "小哈"); -map.put(10002, "小啰"); -map.put(10003, "小算"); -map.put(10004, "小法"); -map.put(10005, "小哇"); +map.put(12836, "小哈"); +map.put(15937, "小啰"); +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(10002); +String name = map.get(15937); /* 删除操作 */ // 在哈希表中删除键值对 (key, value) -map.remove(10005); +map.remove(10583); ``` 遍历哈希表有三种方式,即 **遍历键值对、遍历键、遍历值**。 @@ -51,28 +76,6 @@ for (String val: map.values()) { } ``` -## 哈希表优势 - -给定一个包含 $n$ 个学生的数据库,每个学生有 "姓名 `name` ” 和 “学号 `id` ” 两项数据,希望实现一个查询功能,即 **输入一个学号,返回对应的姓名**,那么可以使用哪些数据结构来存储呢? - -- **无序数组:** 每个元素为 `[学号, 姓名]` ; -- **有序数组:** 将 `1.` 中的数组按照学号从小到大排序; -- **链表:** 每个结点的值为 `[学号, 姓名]` ; -- **二叉搜索树:** 每个结点的值为 `[学号, 姓名]` ,根据学号大小来构建树; -- **哈希表:** 以学号为 Key 、姓名为 Value 。 - -使用上述方法,各项操作的时间复杂度如下表所示(在此不做赘述,详解可见 [二叉搜索树章节](https://www.hello-algo.com/chapter_tree/binary_search_tree/#_6)),**哈希表全面胜出!** - -
- -| | 无序数组 | 有序数组 | 链表 | 二叉搜索树 | 哈希表 | -| ------------ | -------- | ----------- | ------ | ----------- | ------ | -| 查找指定元素 | $O(n)$ | $O(\log n)$ | $O(n)$ | $O(\log n)$ | $O(1)$ | -| 插入元素 | $O(1)$ | $O(n)$ | $O(1)$ | $O(\log n)$ | $O(1)$ | -| 删除元素 | $O(n)$ | $O(n)$ | $O(n)$ | $O(\log n)$ | $O(1)$ | - -
- ## 哈希函数 哈希表中存储元素的数据结构被称为「桶 Bucket」,底层实现可能是数组、链表、二叉树(红黑树),或是它们的组合。 @@ -87,10 +90,12 @@ for (String val: map.values()) { 以上述学生数据 `Key 学号 -> Value 姓名` 为例,我们可以将「哈希函数」设计为 $$ -f(x) = x \% 10000 +f(x) = x \% 100 $$ -(图) +![hash_function](hash_map.assets/hash_function.png) + +

Fig. 哈希函数

```java title="array_hash_map.java" /* 键值对 int->String */ @@ -107,16 +112,16 @@ class Entry { class ArrayHashMap { private List bucket; public ArrayHashMap() { - // 初始化一个长度为 10 的桶(数组) + // 初始化一个长度为 100 的桶(数组) bucket = new ArrayList<>(); - for (int i = 0; i < 10; i++) { + for (int i = 0; i < 100; i++) { bucket.add(null); } } /* 哈希函数 */ private int hashFunc(int key) { - int index = key % 10000; + int index = key % 100; return index; } @@ -146,8 +151,18 @@ class ArrayHashMap { ## 哈希冲突 -细心的同学可能会发现,哈希函数 $f(x) = x \% 10000$ 会在某些情况下失效。例如,当输入的 Key 为 10001, 20001, 30001, ... 时,哈希函数的计算结果都是 1 ,指向同一个 Value ,表明不同学号指向了同一个人,这明显是不对的。 +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 +$$ +f(12836) = f(20336) = 36 +$$ +导致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,我们将如何避免哈希冲突的问题留在下章讨论。 + +![hash_collision](hash_map.assets/hash_collision.png) + +

Fig. 哈希冲突

-上述现象被称为「哈希冲突 Hash Collision」,其会严重影响查询的正确性,我们将如何避免哈希冲突的问题留在下章讨论。 +综上所述,一个优秀的「哈希函数」应该具备以下特性: -(图) +- 尽量少地发生哈希冲突; +- 时间复杂度 $O(1)$ ,计算尽可能高效; +- 空间使用率高,即 “键值对占用空间 / 哈希表总占用空间” 尽可能大;