【动态规划】最长公共子序列

一、引言

二、最长公共子序列

三、解题思路

四、代码

int LSCLen(char* arr, char* brr, int m, int n)
{
	//if (nullptr == arr || nullptr == brr) return 0;
	if (m == 0 || n == 0) return 0;
	else
	{
		if (arr[m] == brr[n])
		{
			return LSCLen(arr, brr, m - 1, n - 1) + 1;
		}
		else
		{
			int max1 = LSCLen(arr, brr, m - 1, n );
			int max2 = LSCLen(arr, brr, m, n - 1);
			return max1 > max2 ? max1 : max2;
		}
	}
}

int main()
{
	char x[] = "#ABCBDAB";
	char y[] = "#BDCABA";
	int m = strlen(x) - 1;
	int n = strlen(y) - 1;
	cout << LSCLen(x, y, m, n);
	return 0;
}

五、简单优化

#include<iostream>
#include<vector>
using namespace std;

int LSCLen(char* arr, char* brr, int m, int n, vector<vector<int>>& dp)
{
	//if (nullptr == arr || nullptr == brr) return 0;
	if (m == 0 || n == 0) return 0;
	else if(dp[m][n] != 0)
	{
		return dp[m][n];
	}
	else
	{
		if (arr[m] == brr[n])
		{
			dp[m][n] =  LSCLen(arr, brr, m - 1, n - 1, dp) + 1;
		}
		else
		{
			int max1 = LSCLen(arr, brr, m - 1, n, dp);
			int max2 = LSCLen(arr, brr, m, n - 1, dp);
			dp[m][n] = max1 > max2 ? max1 : max2;
		}
	}
	//存放就是m n规模下的最长公共子序列长度
	return dp[m][n];
}

void PrintVector(vector<vector<int>>& vc)
{
	for (const auto& x : vc)
	{
		for (const auto& y : x)
		{
			cout << y << " ";
		}
		cout << endl;
	}
}

int main()
{
	char x[] = "#ABCBDAB";
	//          01234567
	char y[] = "#BDCABA";
	//          0123456
	int m = strlen(x);
	int n = strlen(y);

	vector<vector<int>> dp;
	dp.resize(m);
	for (int i = 0; i < m; ++i)
	{
		dp[i].resize(n, 0);
	}
	cout << LSCLen(x, y, m - 1, n - 1, dp) << endl;
	
	//打印dp数组中的值
	PrintVector(dp);
	return 0;
}

六、最终输出

#include<iostream>
#include<vector>
using namespace std;

int LSCLen(char* arr, char* brr, int m, int n, vector<vector<int>>& dp, vector<vector<int>>& sp)
{
	//if (nullptr == arr || nullptr == brr) return 0;
	if (m == 0 || n == 0) return 0;
	else if(dp[m][n] != 0)
	{
		return dp[m][n];
	}
	else
	{
		if (arr[m] == brr[n])
		{
			dp[m][n] =  LSCLen(arr, brr, m - 1, n - 1, dp, sp) + 1;
			//斜线移动 1
			sp[m][n] = 1;
		}
		else
		{
			int max1 = LSCLen(arr, brr, m - 1, n, dp, sp);
			int max2 = LSCLen(arr, brr, m, n - 1, dp, sp);
			if (max1 > max2)
			{
				dp[m][n] = max1;
				//左移动
				sp[m][n] = 2;
			}
			else
			{
				dp[m][n] = max2;
				//上移动
				sp[m][n] = 3;
			}
		}
	}
	//存放就是m n规模下的最长公共子序列长度
	return dp[m][n];
}

void PrintVector(vector<vector<int>>& vc)
{
	for (const auto& x : vc)
	{
		for (const auto& y : x)
		{
			cout << y << " ";
		}
		cout << endl;
	}
	cout << endl;
}
void LCS(vector<vector<int>>& sp, char* arr, int i, int j)
{
	if (i == 0 || j == 0) return;
	//斜线移动
	if (sp[i][j] == 1)
	{
		LCS(sp, arr, i - 1, j - 1);
		cout << arr[i];
	}
	//向左移动
	else if (sp[i][j] == 2)
	{
		LCS(sp, arr, i - 1, j);
	}
	//向上移动
	else
	{
		LCS(sp, arr, i, j - 1);
	}
}


int main()
{
	char x[] = "#ABCBDAB";
	//          01234567
	char y[] = "#BDCABA";
	//          0123456
	int m = strlen(x);
	int n = strlen(y);

	vector<vector<int>> dp, sp;
	dp.resize(m);
	sp.resize(m);
	for (int i = 0; i < m; ++i)
	{
		dp[i].resize(n, 0);
		sp[i].resize(n, 0);
	}
	cout << LSCLen(x, y, m - 1, n - 1, dp, sp) << endl;
	
	//打印dp数组中的值
	PrintVector(dp);
	//打印sp方向数组
	PrintVector(sp);
	//根据sp打印最长公共子序列
	LCS(sp, x, m - 1, n - 1);
	return 0;
}