Floyd(動態規划)求解任意兩點間的最短路徑(圖解)

Floyd算法的精髓在於動態規划的思想,即每次找最優解時都建立在上一次最優解的基礎上,當算法執行完畢時一定是最優解

對於鄰接矩陣w,w保存最初始情況下任意兩點間的直接最短距離,但沒有加入中繼點進行考慮
如w[1][2]=20,即表示點1與點2的當前最短距離(直接距離)為20

對於路徑矩陣path,保存了點i到點j的最短路徑中下一個點的位置,
如path[1][2]=0,表示1->2的路徑中的下一個點為結點0

Floyd算法對所有中繼點在任意兩點中進行循環遍歷.即k從0-n時考慮(i->k,k->j)的路徑是否小於(i->j)的路,如果小於即更新鄰接矩陣w的值與path矩陣中的值,使其始終保持最短
圖解如下:

代碼用例:

代碼如下

點擊查看代碼

#include<iostream>
#include<fstream>
#include<vector>
using namespace std;
const int MAX = 999;
class Solution {
public:
	void GetPath(vector<vector<int>>vec,int n) {
		
		vector<vector<int>>path(n);

		//初始化
		for (int i = 0; i != n; i++)
			path[i].resize(n);

		for(int i=0;i!=n;i++)
			for (int j = 0; j != n; j++) {
				if (vec[i][j] != MAX)path[i][j] = j;
				else path[i][j] = -1;
			}

		for (int i = 0; i < n; i++)path[i][i] = -1;
		for(int k=0;k!=n;k++)
			for(int i=0;i!=n;i++)
				for (int j = 0; j != n; j++) {
					if (vec[i][k] + vec[k][j] < vec[i][j]) {
						vec[i][j] = vec[i][k] + vec[k][j];
						path[i][j] = path[i][k];
					}
				}
				
		for (int i = 0; i != n; i++)
		{
			cout << "\nStating from vertex: " << i << endl;
			bool flag = 0;
			for (int j = 0; j != n; j++)
				if (j != i && vec[i][j] < MAX) {
					flag = 1;
					cout << i << "->" << j << ":distance=" << vec[i][j] << ": " << i;
					int k = path[i][j];
					while (k != j) {
						cout << "->" << k;
						k = path[k][j];
					}
					cout << "->" << j << endl;
				}

			if (!flag)cout << "there's no path while starting from "<<i<<endl;
		}
	}
};
int main() {
	ifstream putIn("D:\\Input.txt", ios::in);
	int num;
	int finalCount = 0;
	putIn >> num;
	const int x = num;
	vector<vector<int>>myVector(num);
	//myVector:帶權鄰接矩陣
	for (int i = 0; i < num; i++)
		myVector[i].resize(num);
	for (int i = 0; i < num; i++)
		for (int j = 0; j < num; j++) {
			int temp;
			putIn >> temp;
			myVector[i][j] = temp;
		}
	Solution solution;
	cout << "Input文件中的鄰接矩陣為\n";
	for (auto x : myVector) {
		for (auto y : x)cout << y << '\t';
		cout << endl;
	}
	solution.GetPath(myVector,num);
	return 0;
}