[solution] longest common subsequence

subject

Title Description

Find two strings that contain only lowercase letters $X,Y$ The longest common subsequence length of .

Input format

The first line is two positive integers $m,n（m<1000,n<1000）$, There are two $X,Y$ The length of two strings .

The second line is a length of $m$ A string containing only lowercase letters .

The third line is a $n$ A string containing only lowercase letters .

Output format

Output a positive integer according to the requirements of the topic .

Answer key

#include<iostream>
#include<cmath>
#include<algorithm>
using namespace std;
const int N = 1010;
int f[N][N];
char X[N],Y[N];
int main(){
    
    int m,n;
    cin >> m >> n;
    cin >> X + 1 >> Y + 1;
    for(int i = 1;i <= m;i++)
        for(int j = 1;j <= n;j++)
            if(X[i] == Y[j]) f[i][j] = f[i - 1][j - 1] + 1;
            else f[i][j] = max(f[i][j - 1],f[i - 1][j]);
    cout << f[m][n] << endl;
    return 0;
}

This problem is an introduction to dynamic programming .

Please make sure you understand what the sequence is before you understand this problem .

for instance ：
For strings abcdefgh,abc、bcd It's his subsequence ,acfg、dgh It is also its subsequence .
Elements in a subsequence can be discontinuous in the original sequence .

Be similar to Fibonacci sequence This question , This problem also uses an array $f$ Store something we need state .

$f[i][j]$ Express $X$ Before $i$ Characters and $Y$ Before $j$ The longest common subsequence of characters .

—— Don't ask why you can say so , That's right .
Let's assume that it can be expressed in this way , that $f[m][n]$ This is the value we want to output .

So let's see $f[i][j]$ How exactly can it be derived .

Similar to Fibonacci $f[i]=f[i-1]+f[i-2]$, our $f[i][j]$ Can we also draw from the smaller data scale ？

The answer is yes .

$$f[i][j]=\{\begin{array}{c}f[i-1][j-1]+1,X[i]=Y[j]\\ max(f[i-1][j],f[i][j-1]),X[i]\ne Y[i]\end{array}$$

To be continued ...

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Originality is not easy. , Thank you for your support ！