[LeetCode] 425. Word Squares 单词平方

Given a set of words (without duplicates), find all word squares you can build from them.

A sequence of words forms a valid word square if the kth row and column read the exact same string, where 0 ≤ k < max(numRows, numColumns).

For example, the word sequence ["ball","area","lead","lady"] forms a word square because each word reads the same both horizontally and vertically.

b a l l
a r e a
l e a d
l a d y

Note:

There are at least 1 and at most 1000 words.
All words will have the exact same length.
Word length is at least 1 and at most 5.
Each word contains only lowercase English alphabet a-z.

Example 1:

Input:
["area","lead","wall","lady","ball"]

Output:
[
  [ "wall",
    "area",
    "lead",
    "lady"
  ],
  [ "ball",
    "area",
    "lead",
    "lady"
  ]
]

Explanation:
The output consists of two word squares. The order of output does not matter (just the order of words in each word square matters).

Example 2:

Input:
["abat","baba","atan","atal"]

Output:
[
  [ "baba",
    "abat",
    "baba",
    "atan"
  ],
  [ "baba",
    "abat",
    "baba",
    "atal"
  ]
]

Explanation:
The output consists of two word squares. The order of output does not matter (just the order of words in each word square matters).

这道题是之前那道 Valid Word Square 的延伸，由于要求出所有满足要求的单词平方，所以难度大大的增加了，不要幻想着可以利用之前那题的解法来暴力破解，OJ 不会答应的。那么根据以往的经验，对于这种要打印出所有情况的题的解法大多都是用递归来解，那么这题的关键是根据前缀来找单词，如果能利用合适的数据结构来建立前缀跟单词之间的映射，使得我们能快速的通过前缀来判断某个单词是否存在，这是解题的关键。对于建立这种映射，这里主要有两种方法，一种是利用 HashMap 来建立前缀和所有包含此前缀单词的集合之前的映射，第二种方法是建立前缀树 Trie，顾名思义，前缀树专门就是为这种问题设计的。首先来看第一种方法，用 HashMap 来建立映射的方法，就是取出每个单词的所有前缀，然后将该单词加入该前缀对应的集合中去，然后建立一个空的 nxn 的 char 矩阵，其中n为单词的长度，目标就是来把这个矩阵填满，从0开始遍历，先取出长度为0的前缀，即空字符串，由于在建立映射的时候，空字符串也和每个单词的集合建立了映射，然后遍历这个集合，用遍历到的单词的i位置字符，填充矩阵 mat[i][i]，然后j从 i+1 出开始遍历，对应填充矩阵 mat[i][j] 和 mat[j][i]，然后根据第j行填充得到的前缀，到哈希表中查看有没单词，如果没有，就 break 掉，如果有，则继续填充下一个位置。最后如果 j==n 了，说明第0行和第0列都被填好了，再调用递归函数，开始填充第一行和第一列，依次类推，直至填充完成，参见代码如下：

解法一：

class Solution {
public:
    vector<vector<string>> wordSquares(vector<string>& words) {
        vector<vector<string>> res;
        unordered_map<string, set<string>> m;
        int n = words[0].size();
        for (string word : words) {
            for (int i = 0; i < n; ++i) {
                string key = word.substr(0, i);
                m[key].insert(word);
            }
        }
        vector<vector<char>> mat(n, vector<char>(n));
        helper(0, n, mat, m, res);
        return res;
    }
      void helper(int i, int n, vector<vector<char>>& mat, unordered_map<string, set<string>>& m, vector<vector<string>>& res) {
            if (i == n) {
                vector<string> out;
                for (int j = 0; j < n; ++j) out.push_back(string(mat[j].begin(), mat[j].end()));
                res.push_back(out);
                return;
            }
            string key = string(mat[i].begin(), mat[i].begin() + i);
        for (string str : m[key]) { 
            mat[i][i] = str[i];
            int j = i + 1;
            for (; j < n; ++j) {
                mat[i][j] = str[j];
                mat[j][i] = str[j];
                if (!m.count(string(mat[j].begin(), mat[j].begin() + i + 1))) break;
            }
            if (j == n) helper(i + 1, n, mat, m, res);
        }
    }
};

下面来看建立前缀树 Trie 的方法，这种方法的难点是看能不能熟练的写出 Trie 的定义，还有构建过程，以及后面在递归函数中，如果利用前缀树来快速查找单词的前缀，总之，这道题是前缀树的一种经典的应用，能白板写出来就说明基本上已经掌握了前缀树了，参见代码如下：

解法二：

class Solution {
public:
    struct TrieNode {
        vector<int> indexs;
        vector<TrieNode*> children;
        TrieNode(): children(26, nullptr) {}
    };
    TrieNode* buildTrie(vector<string>& words) {
        TrieNode *root = new TrieNode();
        for (int i = 0; i < words.size(); ++i) {
            TrieNode *t = root;
            for (int j = 0; j < words[i].size(); ++j) {
                if (!t->children[words[i][j] - 'a']) {
                    t->children[words[i][j] - 'a'] = new TrieNode();
                }
                t = t->children[words[i][j] - 'a'];
                t->indexs.push_back(i);
            }
        }
        return root;
    }
    vector<vector<string>> wordSquares(vector<string>& words) {
        TrieNode *root = buildTrie(words);
        vector<string> out(words[0].size());
        vector<vector<string>> res;
        for (string word : words) {
            out[0] = word;
            helper(words, 1, root, out, res);
        }
        return res;
    }
    void helper(vector<string>& words, int level, TrieNode* root, vector<string>& out, vector<vector<string>>& res) {
        if (level >= words[0].size()) {
            res.push_back(out);
            return;
        }
        string str = "";
        for (int i = 0; i < level; ++i) {
            str += out[i][level];
        }
        TrieNode *t = root;
        for (int i = 0; i < str.size(); ++i) {
            if (!t->children[str[i] - 'a']) return;
            t = t->children[str[i] - 'a'];
        }
        for (int idx : t->indexs) {
            out[level] = words[idx];
            helper(words, level + 1, root, out, res);
        }
    }
};