Scramble String

描述

Given a string s1, we may represent it as a binary tree by partitioning it to two non-empty substrings recursively.

Below is one possible representation of s1 = "great":

    great
   /    \
  gr    eat
 / \    /  \
g   r  e   at
           / \
          a   t

To scramble the string, we may choose any non-leaf node and swap its two children.

For example, if we choose the node "gr" and swap its two children, it produces a scrambled string "rgeat".

    rgeat
   /    \
  rg    eat
 / \    /  \
r   g  e   at
           / \
          a   t

We say that "rgeat" is a scrambled string of "great".

Similarly, if we continue to swap the children of nodes "eat" and "at", it produces a scrambled string "rgtae".

    rgtae
   /    \
  rg    tae
 / \    /  \
r   g  ta  e
       / \
      t   a

We say that "rgtae" is a scrambled string of "great".

Given two strings s1 and s2 of the same length, determine if s2 is a scrambled string of s1.

分析

首先想到的是递归（即深搜），对两个string进行分割，然后比较四对字符串。代码虽然简单，但是复杂度比较高。有两种加速策略，一种是剪枝，提前返回；一种是加缓存，缓存中间结果，即memorization（翻译为记忆化搜索）。

剪枝可以五花八门，要充分观察，充分利用信息，找到能让节点提前返回的条件。例如，判断两个字符串是否互为scamble，至少要求每个字符在两个字符串中出现的次数要相等，如果不相等则返回false。

加缓存，可以用数组或HashMap。本题维数较高，用HashMap，map和unordered_map均可。

既然可以用记忆化搜索，这题也一定可以用动规。设状态为f[n][i][j]，表示长度为n，起点为s1[i]和起点为s2[j]两个字符串是否互为scramble，则状态转移方程为

f[n][i][j]} =  (f[k][i][j] && f[n-k][i+k][j+k]) 
            || (f[k][i][j+n-k] && f[n-k][i+k][j])

递归

// Scramble String
// 递归，会超时，仅用来帮助理解
// 时间复杂度O(n^6)，空间复杂度O(1)
class Solution {
public:
    bool isScramble(const string& s1, const string& s2) {
        return isScramble(s1.begin(), s1.end(), s2.begin());
    }
private:
    typedef string::iterator Iterator;
    bool isScramble(Iterator first1, Iterator last1, Iterator first2) {
        auto length = distance(first1, last1);
        auto last2 = next(first2, length);

        if (length == 1) return *first1 == *first2;

        for (int i = 1; i < length; ++i)
            if ((isScramble(first1, first1 + i, first2)
                 && isScramble(first1 + i, last1, first2 + i))
                    || (isScramble(first1, first1 + i, last2 - i)
                            && isScramble(first1 + i, last1, first2)))
                return true;

        return false;
    }
};

动规

// Scramble String
// 动规，时间复杂度O(n^3)，空间复杂度O(n^3)
class Solution {
public:
    bool isScramble(const string& s1, const string& s2) {
        const int N = s1.size();
        if (N != s2.size()) return false;

        // f[n][i][j]，表示长度为n，起点为s1[i]和
        // 起点为s2[j]两个字符串是否互为scramble
        bool f[N + 1][N][N];
        fill_n(&f[0][0][0], (N + 1) * N * N, false);

        for (int i = 0; i < N; i++)
            for (int j = 0; j < N; j++)
                f[1][i][j] = s1[i] == s2[j];

        for (int n = 1; n 

递归+剪枝

// Scramble String
// 递归+剪枝
// 时间复杂度O(n^6)，空间复杂度O(1)
class Solution {
public:
    bool isScramble(const string& s1, const string& s2) {
        return isScramble(s1.begin(), s1.end(), s2.begin());
    }
private:
    typedef string::const_iterator Iterator;
    bool isScramble(Iterator first1, Iterator last1, Iterator first2) {
        auto length = distance(first1, last1);
        auto last2 = next(first2, length);
        if (length == 1) return *first1 == *first2;

        // 剪枝，提前返回
        int A[26]; // 每个字符的计数器
        fill(A, A + 26, 0);
        for(int i = 0; i < length; i++) A[*(first1+i)-'a']++;
        for(int i = 0; i < length; i++) A[*(first2+i)-'a']--;
        for(int i = 0; i < 26; i++) if (A[i] != 0) return false;

        for (int i = 1; i < length; ++i)
            if ((isScramble(first1, first1 + i, first2)
                 && isScramble(first1 + i, last1, first2 + i))
                    || (isScramble(first1, first1 + i, last2 - i)
                            && isScramble(first1 + i, last1, first2)))
                return true;

        return false;
    }
};

备忘录法

typedef string::const_iterator Iterator;
typedef tuple Key;
// 定制一个哈希函数
namespace std {
template<> struct hash {
    size_t operator()(const Key & x) const {
        Iterator first1, last1, first2;
        tie(first1, last1, first2) = x;

        int result = *first1;
        result = result * 31 + *last1;
        result = result * 31 + *first2;
        result = result * 31 + *(next(first2, distance(first1, last1)-1));
        return result;
    }
};
}

// Scramble String
// 递归+unordered_map做cache，比map快
// 时间复杂度O(n^3)，空间复杂度O(n^3)
class Solution {
public:
    unordered_map cache;

    bool isScramble(const string& s1, const string& s2) {
        cache.clear();
        return isScramble(s1.begin(), s1.end(), s2.begin());
    }

    bool isScramble(Iterator first1, Iterator last1, Iterator first2) {
        auto length = distance(first1, last1);
        auto last2 = next(first2, length);

        if (length == 1)
            return *first1 == *first2;

        for (int i = 1; i < length; ++i)
            if ((getOrUpdate(first1, first1 + i, first2)
                    && getOrUpdate(first1 + i, last1, first2 + i))
                    || (getOrUpdate(first1, first1 + i, last2 - i)
                            && getOrUpdate(first1 + i, last1, first2)))
                return true;

        return false;
    }

    bool getOrUpdate(Iterator first1, Iterator last1, Iterator first2) {
        auto key = make_tuple(first1, last1, first2);
        auto pos = cache.find(key);

        return (pos != cache.end()) ?
                pos->second : (cache[key] = isScramble(first1, last1, first2));
    }
};