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最新推荐文章于 2025-08-14 19:21:09 发布

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最新推荐文章于 2025-08-14 19:21:09 发布

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文章标签： java 数据结构与算法

原文链接：http://www.cnblogs.com/ACRykl/p/8298143.html

本文介绍了一个经典的计算机科学问题——最长公共子序列问题，并提供了一种基于动态规划的高效解决方案。通过递推公式，文章详细阐述了如何计算两个字符串的最长公共子序列长度。

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Common Subsequence

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 43355 Accepted Submission(s): 20013

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

分析：动态规划。c[i][j]记录"x₀,x₁, ...,x_n-2"与"y₀, y₁, ..., y_n-2"的最长公共子序列的长度，

递推公式如下：

#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
char a[2000],b[2000];
int dp[1000][1000];
int main()
{
    while(scanf("%s%s",a,b)!=EOF)
    {
        memset(dp,0,sizeof(dp));
        int N=strlen(a),M=strlen(b);
        for(int i=0;i<N;i++)
        {
            for(int j=0;j<M;j++)
            if(a[i]==b[j]) dp[i+1][j+1]=dp[i][j]+1;
            else dp[i+1][j+1]=max(dp[i][j+1],dp[i+1][j]);
        }
        printf("%d\n",dp[N][M]);
    }
    return 0;
}