[LeetCode] Encode String with Shortest Length 最短长度编码字符串

Given a non-empty string, encode the string such that its encoded length is the shortest.

The encoding rule is: k[encoded_string], where the encoded_string inside the square brackets is being repeated exactly k times.

Note:

1. k will be a positive integer and encoded string will not be empty or have extra space.
2. You may assume that the input string contains only lowercase English letters. The string's length is at most 160.
3. If an encoding process does not make the string shorter, then do not
   encode it. If there are several solutions, return any of them is fine.

Example 1:

Input: "aaa"
Output: "aaa"
Explanation: There is no way to encode it such that it is shorter than the input string, so we do not encode it.

Example 2:

Input: "aaaaa"
Output: "5[a]"
Explanation: "5[a]" is shorter than "aaaaa" by 1 character.

Example 3:

Input: "aaaaaaaaaa"
Output: "10[a]"
Explanation: "a9[a]" or "9[a]a" are also valid solutions, both of them have the same length = 5, which is the same as "10[a]".

Example 4:

Input: "aabcaabcd"
Output: "2[aabc]d"
Explanation: "aabc" occurs twice, so one answer can be "2[aabc]d".

Example 5:

Input: "abbbabbbcabbbabbbc"
Output: "2[2[abbb]c]"
Explanation: "abbbabbbc" occurs twice, but "abbbabbbc" can also be encoded to "2[abbb]c", so one answer can be "2[2[abbb]c]".

 

这道题让我们压缩字符串，把相同的字符串用中括号括起来，然后在前面加上出现的次数，感觉还是一道相当有难度的题呢。参考了网上大神的帖子才弄懂该怎么做，这道题还是应该用DP来做。我们建立一个二维的DP数组，其中dp[i][j]表示s在[i, j]范围内的字符串的缩写形式(如果缩写形式长度大于子字符串，那么还是保留子字符串)，那么如果s字符串的长度是n，最终我们需要的结果就保存在dp[0][n-1]中，然后我们需要遍历s的所有子字符串，对于任意一段子字符串[i, j]，我们\\我们以中间任意位置k来拆分成两段，比较dp[i][k]加上dp[k+1][j]的总长度和dp[i][j]的长度，将长度较小的字符串赋给dp[i][j]，然后我们要做的就是在s中取出[i, j]范围内的子字符串t进行合并。合并的方法是我们在取出的字符串t后面再加上一个t，然后在这里面寻找子字符串t的第二个起始位置，如果第二个起始位置小于t的长度的话，说明t包含重复字符串，举个例子吧，比如 t = "abab", 那么t+t = "abababab"，我们在里面找第二个t出现的位置为2，小于t的长度4，说明t中有重复出现，重复的个数为t.size()/pos = 2个，那么我们就要把重复的地方放入中括号中，注意中括号里不能直接放这个子字符串，而是应该从dp中取出对应位置的字符串，因为重复的部分有可能已经写成缩写形式了，比如题目中的例子5。再看一个例子，如果t = "abc"，那么t+t = "abcabc"，我们在里面找第二个t出现的位置为3，等于t的长度3，说明t中没有重复出现，那么replace就还是t。然后我们比较我们得到的replace和dp[i][j]中的字符串长度，把长度较小的赋给dp[i][j]即可，时间复杂度为O(n³)，空间复杂度为O(n²)，参见代码如下：

 

解法一：

class Solution {
public:
    string encode(string s) {
        int n = s.size();
        vector<vector<string>> dp(n, vector<string>(n, ""));
        for (int step = ; step <= n; ++step) {
            for (int i = ; i + step -  < n; ++i) {
                int j = i + step - ;
                dp[i][j] = s.substr(i, step);
                for (int k = i; k < j; ++k) {
                    string left = dp[i][k], right = dp[k + ][j];
                    if (left.size() + right.size() < dp[i][j].size()) {
                        dp[i][j] = left + right;
                    }
                }
                string t = s.substr(i, j - i + ), replace = "";
                auto pos = (t + t).find(t, );
                if (pos >= t.size()) replace = t;
                else replace = to_string(t.size() / pos) + '[' + dp[i][i + pos - ] + ']';
                if (replace.size() < dp[i][j].size()) dp[i][j] = replace;
            }
        }
        return dp[][n - ];
    }
};

根据热心网友iffalse的留言，我们可以优化上面的方法。如果t是重复的，是不是就不需要再看left.size() + right.size() < dp[i][j].size()了。例如t是abcabcabcabcabc, 最终肯定是5[abc]，不需要再看3[abc]+abcabc或者abcabc+3[abc]。对于一个本身就重复的字符串，最小的长度肯定是n[REPEATED]，不会是某个left+right。所以应该把k的那个循环放在t和replace那部分代码的后面。这样的确提高了一些运算效率的，参见代码如下：

解法二：

class Solution {
public:
    string encode(string s) {
        int n = s.size();
        vector<vector<string>> dp(n, vector<string>(n, ""));
        for (int step = ; step <= n; ++step) {
            for (int i = ; i + step -  < n; ++i) {
                int j = i + step - ;
                dp[i][j] = s.substr(i, step);
                string t = s.substr(i, j - i + ), replace = "";
                auto pos = (t + t).find(t, );
                if (pos < t.size()) {
                    replace = to_string(t.size() / pos) + "[" + dp[i][i + pos - ] + "]";
                    if (replace.size() < dp[i][j].size()) dp[i][j] = replace;
                    continue;
                }
                for (int k = i; k < j; ++k) {
                    string left = dp[i][k], right = dp[k + ][j];
                    if (left.size() + right.size() < dp[i][j].size()) {
                        dp[i][j] = left + right;
                    }
                }
            }
        }
        return dp[][n - ];
    }
};

类似题目：

Decode String

Number of Atoms

参考资料：

https://leetcode.com/problems/encode-string-with-shortest-length/

https://leetcode.com/problems/encode-string-with-shortest-length/discuss/95599/Accepted-Solution-in-Java

https://leetcode.com/problems/encode-string-with-shortest-length/discuss/95605/Easy-to-understand-C%2B%2B-O(n3)-solution

https://leetcode.com/problems/encode-string-with-shortest-length/discuss/95619/C%2B%2B-O(N3)-time-O(N2)-space-solution-using-memorized-dynamic-programming-with-detail-explanations

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