LPS（最长回文子序列）

　（注意：我发现最长回文子序列(Longest Palindromic Subsequence)问题与最长回文子串(Longest Palindromic Substring)不一样，子序列不要求下标一定连续的，但子串下标一定是连续递增的，《算法导论》中要求的是最长回文子序列，所以与LeetCode-Longest Palindromic Substring不同，这里我做的是书中的问题）

　　这里对最优子结构和算法参考思路：

http://blog.****.net/u012243115/article/details/41010913

　　代码：

#include "stdafx.h"
#include <iostream>
#include <vector>
#include <string>
#include <minmax.h>

class Solution {
public:
	int LongestPalindromicSubsequence(std::string s)
	{
		if (s.empty())
			return 0;

		std::vector<std::vector<int> > dp(s.size(), std::vector<int>(s.size()));
		int LPSLength = 1;
		for (int i = s.size() - 1; i >= 0; i--)
		{
			dp[i][i] = 1;
			for (int j = i + 1; j < s.size(); j++)
			{
				if (s[i] == s[j])
					dp[i][j] = dp[i + 1][j - 1] + 2;
				else
					dp[i][j] = max(dp[i + 1][j], dp[i][j - 1]);
				LPSLength = max(LPSLength, dp[i][j]);
			}
		}
		return LPSLength;
	}
};

int main()
{
	std::vector<std::string> strs = { "character","inileveltnt","bilibili","abcef","rever" };

	for (auto s : strs)
	{
		std::cout << "字符串:" << s << std::endl;
		std::cout << "结果:" << Solution().LongestPalindromicSubsequence(s) << std::endl;
	}

	getchar();
	return 0;
}

　　LeetCode上的提交详情：

　　改成这样就提高到了51ms:

class Solution {
public:
    int longestPalindromeSubseq(string s) {
        if (s.empty())
			return 0;

		vector<vector<int> > dp(s.size(), vector<int>(s.size()));
		for (int i = s.size() - 1; i >= 0; i--)
		{
			dp[i][i] = 1;
			for (int j = i + 1; j < s.size(); j++)
			{
				if (s[i] == s[j])
					dp[i][j] = dp[i + 1][j - 1] + 2;
				else
					dp[i][j] = max(dp[i + 1][j], dp[i][j - 1]);
			}
		}
		return dp[0][s.size() - 1];
    }
};

　　提交详情：

　　还有更快的22ms的解决方案：

class Solution {
public:
    int longestPalindromeSubseq(string s) {
        if (s.size() == 0) return 0;
        vector<int> dp(s.size(), 0);
        for (int i = s.size() - 1; i >=0; i--) {
            int prev = 1;
            for (int j = i + 1; j < s.size(); j++) {
                int curr;
                if (s[i] == s[j]) curr = 2 + dp[j - 1];
                else curr = max(prev, dp[j]);
                dp[j - 1] = prev;
                prev = curr;
            }
            dp[s.size() - 1] = prev;
        }
        return dp[s.size() -1];
    }
};

　　也是dp思路，但空间复杂度优化为O(n)。