动态规划基础之LCS问题

LCS，即最长公共子序列。子序列的元素在原序列中的元素可以非连续，而子串的元素在原序列中的元素一定连续。LCS问题就是给定两个序列X和Y，找到它们的一个最长公共子序列。

例题如下

Common Subsequence

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 66085    Accepted Submission(s): 30723

Problem Description

A subsequence of a given sequence is the given sequence with some elements (possible none) left out. Given a sequence X = <x1, x2, ..., xm> another sequence Z = <z1, z2, ..., zk> is a subsequence of X if there exists a strictly increasing sequence <i1, i2, ..., ik> of indices of X such that for all j = 1,2,...,k, xij = zj. For example, Z = <a, b, f, c> is a subsequence of X = <a, b, c, f, b, c> with index sequence <1, 2, 4, 6>. Given two sequences X and Y the problem is to find the length of the maximum-length common subsequence of X and Y.
The program input is from a text file. Each data set in the file contains two strings representing the given sequences. The sequences are separated by any number of white spaces. The input data are correct. For each set of data the program prints on the standard output the length of the maximum-length common subsequence from the beginning of a separate line.

Sample Input

abcfbc abfcab programming contest abcd mnp

Sample Output

4 2 0

 

如果用暴力的思想，求出X的所有子序列，再验证是否为Y的子序列复杂度高达O(n2^m)，无法接受。而运用动态规划求LCS，复杂度为O(nm)。

我们设dp[i][j]为第一个数组到i，第二个数组到j的LCS的长度。往下考虑，如果s1[i]==s2[j]，则dp[i][j]=dp[i-1][j-1]+1，否则dp[i][j]=max(dp[i][j-1],dp[i-1][j])。状态转移方程列出，发现第i行只由第i-1行以及第i行决定。于是我们只要定义一个dp[2][m]的数组即可。

代码如下

#include<bits/stdc++.h>
using namespace std;
int dp[2][1005];
int main(){
    string s,t;
    while(cin>>s>>t){
        memset(dp,0,sizeof(dp));
        int n=s.size(),m=t.size();
        int last=0,next=1;
        for(int i=1;i<=n;i++){
            for(int j=1;j<=m;j++){
                if(s[i-1]==t[j-1])dp[next][j]=dp[last][j-1]+1;
                else dp[next][j]=max(dp[last][j],dp[next][j-1]);
                }
            swap(last,next);
        }
        printf("%d\n",dp[last][m]);
    }
    return 0;
}