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zc3%BAJ+bPz&g;K3x-;e%`jCsx5et1vzzifmfkHrV{O|Du4PNgqaizp5ab=D2mzl23 zS6p}ZhynxteXz=^8NbC@U(443yVWuU+#$v>_N3EhYH}&6H zfbU@L%?8_AxR+94t)iIzHwxq|X0r%rj~?8j|NZ3xxK022U89#Q$^Q3vo~QW>q9lLB zcmY}TzXzRqgA%CU|2_-AKKbu=e%xP1|NBu9SpWb11FQId^@`NZ)x@9I5| 输入: +> words = ["oath","pea","eat","rain"] and board = +> [ +> ['o','a','a','n'], +> ['e','t','a','e'], +> ['i','h','k','r'], +> ['i','f','l','v'] +> ] +> +> 输出: ["eat","oath"] + +**说明:** + +你可以假设所有输入都由小写字母 `a-z` 组成。 + +**提示:** + +- 你需要优化回溯算法以通过更大数据量的测试。你能否早点停止回溯? +- 如果当前单词不存在于所有单词的前缀中,则可以立即停止回溯。什么样的数据结构可以有效地执行这样的操作?散列表是否可行?为什么? 前缀树如何?如果你想学习如何实现一个基本的前缀树,请先查看这个问题: [实现Trie(前缀树)](https://leetcode-cn.com/problems/implement-trie-prefix-tree/description/)。 + +#### 解题思路 + +假如在矩阵里遇到了一个字符 ”V”,而字典里根本就没有以 “V” 开头的字符串,则不需要将深度优先搜索进行下去,可以大大地提高搜索效率。 + +构建好了前缀树之后,每次从矩阵里的某个字符出发进行搜索的时候,同步地对前缀树进行对比,如果发现字符一直能被找到,就继续进行下去,一步一步地匹配,直到在前缀树里发现一个完整的字符串,把它输出即可。 + +#### 代码示例 + +```java +class Solution { + public List findWords(char[][] board, String[] words) { + //构建字典树 + Trie myTrie = new Trie(); + TrieNode root = myTrie.root; + //插入数据 + for (String word : words){ + myTrie.insert(word); + } + + //构建结果集容器 + List result = new LinkedList<>(); + //矩阵行数 + int m = board.length; + //矩阵列数 + int n = board[0].length; + //存储该节点是否访问 + boolean[][] visited = new boolean[n][m]; + //遍历整个二维数组 + for (int i = 0; i < m; i++) { + for (int j = 0; j < n; j++) { + find(board, visited, i, j, m, n, result, root); + } + } + return result; + } + + private void find(char[][] board, boolean[][] visited, int i, int j, int m, int n, List result, TrieNode cur) { + //边界判断以及是否已经访问判断 + if (i < 0 || i >= m || j < 0 || j >= n || visited[j][i]) + return; + //获取子节点状态,判断其是否有子节点 + cur = cur.child[board[i][j] - 'a']; + if (cur == null) { + return; + } + //修改节点状态,防止重复访问 + visited[j][i] = true; + //找到单词加入 + if (cur.isEnd) { + result.add(cur.val); + //找到单词后,修改字典树内叶子节点状态为false,防止出现重复单词 + cur.isEnd = false; + } + find(board, visited, i + 1, j, m, n, result, cur); + find(board, visited, i - 1, j, m, n, result, cur); + find(board, visited, i, j + 1, m, n, result, cur); + find(board, visited, i, j - 1, m, n, result, cur); + //最后修改节点状态为未访问状态 + visited[j][i] = false; + } + + + /** + * 字典树 + */ + class Trie { + //创建根节点 + TrieNode root = new TrieNode(); + + void insert(String s) { + TrieNode cur = root; + for (char c : s.toCharArray()) { + //判断是否存在该字符的节点,不存在则创建 + if (cur.child[c - 'a'] == null) { + cur.child[c - 'a'] = new TrieNode(); + cur = cur.child[c - 'a']; + } else + cur = cur.child[c - 'a']; + } + //遍历结束后,修改叶子节点的状态,并存储字符串 + cur.isEnd = true; + cur.val = s; + } + } + + /** + * 字典树节点 + */ + class TrieNode { + /** + * 存储最后节点的字符串 + */ + String val; + /** + * 根据字符排序,[a,b,c,……,z] + */ + TrieNode[] child = new TrieNode[26]; + /** + * 是否是最后叶子节点 + */ + boolean isEnd = false; + } + + public static void main(String[] args) { + Solution solution = new Solution(); + char[][] chars = {{'o','a','a','n'}, {'e','t','a','e'}, {'i','h','k','r'}, {'i','f','l','v'}}; + + String[] words = {"oath","pea","eat","rain"}; + List words1 = solution.findWords(chars, words); + + System.out.println(words1); + } +} +``` + + + +### [820. 单词的压缩编码](https://leetcode-cn.com/problems/short-encoding-of-words/) + +#### 题目描述 + +给定一个单词列表,我们将这个列表编码成一个索引字符串 `S` 与一个索引列表 `A`。 + +例如,如果这个列表是 `["time", "me", "bell"]`,我们就可以将其表示为 `S = "time#bell#"` 和 `indexes = [0, 2, 5]`。 + +对于每一个索引,我们可以通过从字符串 `S` 中索引的位置开始读取字符串,直到 "`#`" 结束,来恢复我们之前的单词列表。 + +那么成功对给定单词列表进行编码的最小字符串长度是多少呢? + +**示例:** + +``` +输入: words = ["time", "me", "bell"] +输出: 10 +说明: S = "time#bell#" , indexes = [0, 2, 5] 。Copy to clipboardErrorCopied +``` + +**提示:** + +1. `1 <= words.length <= 2000` +2. `1 <= words[i].length <= 7` +3. 每个单词都是小写字母 。 + +#### 解题思路 + +因为我们需要知道单词列表里,哪些单词是其它某个单词的后缀。既然要求的是后缀,我们只要把单词的倒序插入字典树,再用字典树判断某个单词的逆序是否出现在字典树里就可以了。 + +比如示例中的 `["time", "me", "bell"]`的逆序就是 `["emit", "em", "lleb"]` 。我们可以发现 `em` 是 `emit` 的前缀。所以 `em` 就可以忽略了。我们必须要先插入单词长的数组,否则会有问题。比如如果我先插入了 `em` ,再插入 `emit` ,会发现两个都可以插入进去,很显然是不对的,所以在插入之前需要先根据单词的长度由长到短排序。 + +#### 代码示例 + +```java +class Solution { + public int minimumLengthEncoding(String[] words) { + int len = 0; + Trie trie = new Trie(); + // 先对单词列表根据单词长度由长到短排序 + Arrays.sort(words, (s1, s2) -> s2.length() - s1.length()); + // 单词插入trie,返回该单词增加的编码长度 + for (String word: words) { + len += trie.insert(word); + } + return len; + } +} + +// 定义trie +class Trie { + TrieNode root = new TrieNode(); + + public int insert(String word) { + TrieNode cur = root; + boolean isNew = false; + // 倒着插入单词 + for (int i = word.length() - 1; i >= 0; i--) { + int c = word.charAt(i) - 'a'; + if (cur.children[c] == null) { + isNew = true; // 是新单词 + cur.children[c] = new TrieNode(); + } + cur = cur.children[c]; + } + // 如果是新单词的话编码长度增加新单词的长度+1,否则不变。 + return isNew? word.length() + 1: 0; + } +} + +class TrieNode { + char val; + + TrieNode[] children = new TrieNode[26]; +} +``` \ No newline at end of file From 3659183299a83fe7385293a71bb75a56585f6950 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=E9=AD=8F=E6=99=AF=E4=BC=A6?= 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zc3%BAJ+bPz&g;K3x-;e%`jCsx5et1vzzifmfkHrV{O|Du4PNgqaizp5ab=D2mzl23 zS6p}ZhynxteXz=^8NbC@U(443yVWuU+#$v>_N3EhYH}&6H zfbU@L%?8_AxR+94t)iIzHwxq|X0r%rj~?8j|NZ3xxK022U89#Q$^Q3vo~QW>q9lLB zcmY}TzXzRqgA%CU|2_-AKKbu=e%xP1|NBu9SpWb11FQId^@`NZ)x@9I5| a abc bac bbc ca -2. 前缀树的根节点是空的 + - 所谓空,即只利用到这个节点的 children 属性,即只关心在这个字典里,有哪些打头的字符。 +### **性质** -3. 除了根节点,其他所有节点都是有可能是单词的结尾,叶子节点一定都是单词的结尾。 +1. 根节点不包含字符,除根节点外的每一个子节点都包含一个字符; +2. 从根节点到某一节点。路径上经过的字符连接起来,就是该节点对应的字符串; +3. 每个节点的所有子节点包含的字符都不相同。 -#### **实现** + -前缀树最基本的操作就是两个:创建和搜索。 +### **实现** -**1. 创建** - -- 遍历一遍输入的字符串,对每个字符串进行遍历 -- 从前缀树的根节点开始,将每个字符串加入到节点的 children 字符集当中。 -- 如果字符集已经包含了这个字符,则跳过。 -- 如果当前字符是字符串的最后一个,则把当前节点的 isEnd 标记为真。 +字典树最基本的操作就是两个:**创建和搜索**。 -由上,创建的方法很直观。 +**1. 创建** -前缀树真正强大的地方在于,每个节点还能用来保存额外的信息,比如可以用来记录拥有相同前缀的所有字符串。因此,当前用户输入某个前缀时,就能在 $O(1)$ 的时间内给出对应的推荐字符串。 +假如这个字典树中所存储的字符串都是小写字母,那么每个节点中最多可以包含的字符数量为:26**(如果出现大小写,或者数字等其他字符时,可以根据具体需求,修改 `children` 数组长度)**。那么我们可以直接使用一个长度为 26 的「子节点数组」来存储该节点下子节点所表示的字符。(可以根据具体需求,增加字段值。比如:该「字符」在该字典树中出现过的次数,该「字符串」是否携带一些附加信息,等) -trie的节点的定义就很好给出来了(trie的根节点定义成一个空节点): +**代码示例** ```java class TrieNode { - /** + /** * 存储最后节点的字符串 */ String val; /** * 根据字符排序,[a,b,c,……,z] */ - TrieNode[] child = new TrieNode[26]; + TrieNode[] children; /** * 是否是最后叶子节点 */ - boolean isEnd = false; + boolean isEnd; + + TrieNode() { + children = new TrieNode[26]; + isEnd = false; + } } ``` -插入单词的操作如下: + + + +那么插入操作思路如下: + +1. 遍历一遍输入的字符串,对每个字符串进行遍历; +2. 从前缀树的根节点开始,将每个字符串加入到节点的 children 字符集当中; + - 如果字符集已经包含了这个字符,则跳过; + - 如果当前字符是字符串的最后一个,则把当前节点的 isEnd 标记为真。 + +由上,创建的方法很直观。 + +**代码示例** ```java public void insert(String s) { TrieNode cur = root; for (char c : s.toCharArray()) { - //判断是否存在该字符的节点,不存在则创建 - if (cur.child[c - 'a'] == null) { - cur.child[c - 'a'] = new TrieNode(); - cur = cur.child[c - 'a']; - } else - cur = cur.child[c - 'a']; + // 判断是否存在该字符的子节点,不存在则创建 + if (cur.children[c - 'a'] == null) { + cur.children[c - 'a'] = new TrieNode(); + } + cur = cur.children[c - 'a']; } //遍历结束后,修改叶子节点的状态,并存储字符串 cur.isEnd = true; @@ -73,9 +85,30 @@ public void insert(String s) { } ``` -**2. 搜索** + + +**2.搜索** + +与创建方法类似,从字典树的根节点出发,逐个匹配输入的「前缀字符」,如果该字符存在就继续往下一层搜索,如果没遇到,就立即返回。 + +**代码示例** + +```java +public boolean search(String word) { + TrieNode cur = root; + for (char c : word.toCharArray()) { + // 查看当前节点中 children 数组内是否存在该「字符」 + cur = cur.children[c - 'a']; + if (cur == null) { + return false; + } + } + // 判断最终循环结束后当前节点是否为一个单词的结尾 + return cur.isEnd; +} +``` + -与创建方法类似,从前缀树的根节点出发,逐个匹配输入的前缀字符,如果遇到了就继续往下一层搜索,如果没遇到,就立即返回。 ## 例题分析 @@ -107,27 +140,24 @@ trie.search("app"); // 返回 true ```java class Trie { + TrieNode root; + class TrieNode { - TrieNode[] children; String val; + + TrieNode[] children; + + boolean isEnd = false; TrieNode() { children = new TrieNode[26]; } } - TrieNode root; - - /** - * Initialize your data structure here. - */ public Trie() { root = new TrieNode(); } - /** - * Inserts a word into the trie. - */ public void insert(String word) { TrieNode cur = root; for (char c : word.toCharArray()) { @@ -136,29 +166,21 @@ class Trie { } cur = cur.children[c - 'a']; } + cur.isEnd = true; cur.val = word; } - /** - * Returns if the word is in the trie. - */ public boolean search(String word) { TrieNode cur = root; for (char c : word.toCharArray()) { - if (cur.children[c - 'a'] == null) { - break; - } cur = cur.children[c - 'a']; + if (cur == null) { + return false; + } } - if (cur.val != null && cur.val.equals(word)) { - return true; - } - return false; + return cur.isEnd; } - /** - * Returns if there is any word in the trie that starts with the given prefix. - */ public boolean startsWith(String prefix) { TrieNode cur = root; for (char c : prefix.toCharArray()) { @@ -174,6 +196,8 @@ class Trie { ## 练习题 +字典树的应用比较明显,一般都是围绕子串、前缀、后缀,匹配或者计数,列举部分例题。 + ### [212. 单词搜索 II](https://leetcode-cn.com/problems/word-search-ii/) #### 题目描述 @@ -210,60 +234,63 @@ class Trie { 假如在矩阵里遇到了一个字符 ”V”,而字典里根本就没有以 “V” 开头的字符串,则不需要将深度优先搜索进行下去,可以大大地提高搜索效率。 -构建好了前缀树之后,每次从矩阵里的某个字符出发进行搜索的时候,同步地对前缀树进行对比,如果发现字符一直能被找到,就继续进行下去,一步一步地匹配,直到在前缀树里发现一个完整的字符串,把它输出即可。 +构建好了字典树之后,每次从矩阵里的某个字符出发进行搜索的时候,同步地对字典树进行对比,如果发现字符一直能被找到,就继续进行下去,一步一步地匹配,直到在前缀树里发现一个完整的字符串,把它输出即可。 #### 代码示例 ```java class Solution { public List findWords(char[][] board, String[] words) { - //构建字典树 - Trie myTrie = new Trie(); - TrieNode root = myTrie.root; - //插入数据 - for (String word : words){ - myTrie.insert(word); + // 构建字典树 + Trie trie = new Trie(); + // 根节点 + TrieNode root = trie.root; + // 插入数据 + for (String word : words) { + trie.insert(word); } - //构建结果集容器 + // 构建结果集容器 List result = new LinkedList<>(); - //矩阵行数 - int m = board.length; - //矩阵列数 - int n = board[0].length; - //存储该节点是否访问 - boolean[][] visited = new boolean[n][m]; - //遍历整个二维数组 - for (int i = 0; i < m; i++) { - for (int j = 0; j < n; j++) { - find(board, visited, i, j, m, n, result, root); + // 矩阵行数 + int row = board.length; + // 矩阵列数 + int col = board[0].length; + // 存储该节点是否访问 + boolean[][] visited = new boolean[col][row]; + // 遍历整个二维数组 + for (int i = 0; i < row; i++) { + for (int j = 0; j < col; j++) { + dfs(board, visited, i, j, row, col, result, root); } } return result; } - private void find(char[][] board, boolean[][] visited, int i, int j, int m, int n, List result, TrieNode cur) { - //边界判断以及是否已经访问判断 - if (i < 0 || i >= m || j < 0 || j >= n || visited[j][i]) + private void dfs(char[][] board, boolean[][] visited, int i, int j, int row, int col, List result, TrieNode cur) { + // 边界判断以及是否已经访问判断 + if (i < 0 || i >= row || j < 0 || j >= col || visited[j][i]) { return; - //获取子节点状态,判断其是否有子节点 + } + // 获取子节点状态,判断其是否有子节点 cur = cur.child[board[i][j] - 'a']; if (cur == null) { return; } - //修改节点状态,防止重复访问 + // 修改节点状态,防止重复访问 visited[j][i] = true; - //找到单词加入 + // 找到单词加入 if (cur.isEnd) { result.add(cur.val); - //找到单词后,修改字典树内叶子节点状态为false,防止出现重复单词 + // 找到单词后,修改字典树内叶子节点状态为false,防止出现重复单词 cur.isEnd = false; } - find(board, visited, i + 1, j, m, n, result, cur); - find(board, visited, i - 1, j, m, n, result, cur); - find(board, visited, i, j + 1, m, n, result, cur); - find(board, visited, i, j - 1, m, n, result, cur); - //最后修改节点状态为未访问状态 + // 会出现一个子串是另一个子串的前缀,例如:apple 和 apples + dfs(board, visited, i + 1, j, row, col, result, cur); + dfs(board, visited, i - 1, j, row, col, result, cur); + dfs(board, visited, i, j + 1, row, col, result, cur); + dfs(board, visited, i, j - 1, row, col, result, cur); + // 最后修改节点状态为未访问状态 visited[j][i] = false; } @@ -272,8 +299,11 @@ class Solution { * 字典树 */ class Trie { - //创建根节点 - TrieNode root = new TrieNode(); + TrieNode root; + + public Trie() { + this.root = new TrieNode(); + } void insert(String s) { TrieNode cur = root; @@ -281,9 +311,8 @@ class Solution { //判断是否存在该字符的节点,不存在则创建 if (cur.child[c - 'a'] == null) { cur.child[c - 'a'] = new TrieNode(); - cur = cur.child[c - 'a']; - } else - cur = cur.child[c - 'a']; + } + cur = cur.child[c - 'a']; } //遍历结束后,修改叶子节点的状态,并存储字符串 cur.isEnd = true; @@ -308,94 +337,16 @@ class Solution { */ boolean isEnd = false; } - - public static void main(String[] args) { - Solution solution = new Solution(); - char[][] chars = {{'o','a','a','n'}, {'e','t','a','e'}, {'i','h','k','r'}, {'i','f','l','v'}}; - - String[] words = {"oath","pea","eat","rain"}; - List words1 = solution.findWords(chars, words); - - System.out.println(words1); - } } ``` + +### [692. 前K个高频单词](https://leetcode-cn.com/problems/top-k-frequent-words/) -### [820. 单词的压缩编码](https://leetcode-cn.com/problems/short-encoding-of-words/) - -#### 题目描述 - -给定一个单词列表,我们将这个列表编码成一个索引字符串 `S` 与一个索引列表 `A`。 - -例如,如果这个列表是 `["time", "me", "bell"]`,我们就可以将其表示为 `S = "time#bell#"` 和 `indexes = [0, 2, 5]`。 - -对于每一个索引,我们可以通过从字符串 `S` 中索引的位置开始读取字符串,直到 "`#`" 结束,来恢复我们之前的单词列表。 - -那么成功对给定单词列表进行编码的最小字符串长度是多少呢? - -**示例:** - -``` -输入: words = ["time", "me", "bell"] -输出: 10 -说明: S = "time#bell#" , indexes = [0, 2, 5] 。Copy to clipboardErrorCopied -``` - -**提示:** - -1. `1 <= words.length <= 2000` -2. `1 <= words[i].length <= 7` -3. 每个单词都是小写字母 。 - -#### 解题思路 - -因为我们需要知道单词列表里,哪些单词是其它某个单词的后缀。既然要求的是后缀,我们只要把单词的倒序插入字典树,再用字典树判断某个单词的逆序是否出现在字典树里就可以了。 - -比如示例中的 `["time", "me", "bell"]`的逆序就是 `["emit", "em", "lleb"]` 。我们可以发现 `em` 是 `emit` 的前缀。所以 `em` 就可以忽略了。我们必须要先插入单词长的数组,否则会有问题。比如如果我先插入了 `em` ,再插入 `emit` ,会发现两个都可以插入进去,很显然是不对的,所以在插入之前需要先根据单词的长度由长到短排序。 +### [676. 实现一个魔法字典](https://leetcode-cn.com/problems/implement-magic-dictionary/) -#### 代码示例 - -```java -class Solution { - public int minimumLengthEncoding(String[] words) { - int len = 0; - Trie trie = new Trie(); - // 先对单词列表根据单词长度由长到短排序 - Arrays.sort(words, (s1, s2) -> s2.length() - s1.length()); - // 单词插入trie,返回该单词增加的编码长度 - for (String word: words) { - len += trie.insert(word); - } - return len; - } -} +### [面试题 17.17. 多次搜索](https://leetcode-cn.com/problems/multi-search-lcci/) -// 定义trie -class Trie { - TrieNode root = new TrieNode(); - public int insert(String word) { - TrieNode cur = root; - boolean isNew = false; - // 倒着插入单词 - for (int i = word.length() - 1; i >= 0; i--) { - int c = word.charAt(i) - 'a'; - if (cur.children[c] == null) { - isNew = true; // 是新单词 - cur.children[c] = new TrieNode(); - } - cur = cur.children[c]; - } - // 如果是新单词的话编码长度增加新单词的长度+1,否则不变。 - return isNew? word.length() + 1: 0; - } -} -class TrieNode { - char val; - - TrieNode[] children = new TrieNode[26]; -} -``` \ No newline at end of file