Floyd算法（java）

一、Floyd算法简介

Floyd算法是求解每对顶点之间的最短路径，时间复杂度为O(∣V∣3)O(|V|^3)O(∣V∣3)。

二、Floyd算法思想

Floyd算法的基本思想是:

递推产生一个n阶方阵序列A−1,A0,...,Ak,...,An−1A^{-1},A^{0},...,A^{k},...,A^{n-1}A−1,A0,...,Ak,...,An−1，其中Ak[i][j]A^{k}[i][j]Ak[i][j]表示从顶点viv_{i}vi到顶点vjv_{j}vj的路径长度，k表示绕行第k个顶点的运算步骤。
初始时，对于任意两个顶点viv_{i}vi和vjv_{j}vj，若它们之间存在边，则以此边上的权值作为它们之间的最短路径长度；若它们之间不存在有向边，则以∞∞∞作为它们之间的最短路径长度。
以后逐步尝试在原路径中加入顶点k(k=0,1.…,n-1)作为中间顶点。若增加中间顶点后，得到的路径比原来的路径长度减少了，则以此新路径代替原路径。

算法描述如下:
定义一个n阶方阵序列A−1,A0,...,Ak,...,An−1A^{-1},A^{0},...,A^{k},...,A^{n-1}A−1,A0,...,Ak,...,An−1，其中，
A−1[i][j]=arcs[i][j]A^{-1}[i][j]=arcs[i][j]A−1[i][j]=arcs[i][j]
Ak[i][j]=Min{Ak−1[i][j],Ak−1[i][k]+Ak−1[k][j],k=0,1,...,n−1}A^{k}[i][j]=Min\begin{Bmatrix}{A^{k-1}[i][j],A^{k-1}[i][k]+A^{k-1}[k][j]},k=0,1,...,n-1\end{Bmatrix}Ak[i][j]=Min{Ak−1[i][j],Ak−1[i][k]+Ak−1[k][j],k=0,1,...,n−1}
式中，A0[i][j]A^{0}[i][j]A0[i][j]是从顶点viv_{i}vi到顶点vjv_{j}vj、中间顶点是v0v_{0}v0的最短路径的长度，Ak[i][j]A^{k}[i][j]Ak[i][j]是从顶点viv_{i}vi到顶点vjv_{j}vj、中间顶点的序号不大于k的最短路径的长度。Floyd算法是一个迭代的过程，每迭代一次，在从viv_{i}vi到顶点vjv_{j}vj的最短路径上就多考虑了一个顶点；经过n次迭代后，所得到的An−1[i][j]A^{n-1}[i][j]An−1[i][j]就是viv_{i}vi到vjv_{j}vj的最短路径长度，即方阵An−1A^{n-1}An−1中就保存了任意一对顶点之间的最短路径长度。

三、Floyd算法代码实现

package com.haiyang.algorithm.floyd;import com.sun.corba.se.impl.orbutil.graph.Graph;import java.util.Arrays;/*** @author haiYang* @create 2022-02-09 10:44*/
public class FloydAlgorithm {public static void main(String[] args) {// 测试看看图是否创建成功char[] vertex = {'A', 'B', 'C', 'D', 'E', 'F', 'G'};//创建邻接矩阵int[][] matrix = new int[vertex.length][vertex.length];final int N = 65535;matrix[0] = new int[]{0, 5, 7, N, N, N, 2};matrix[1] = new int[]{5, 0, N, 9, N, N, 3};matrix[2] = new int[]{7, N, 0, N, 8, N, N};matrix[3] = new int[]{N, 9, N, 0, N, 4, N};matrix[4] = new int[]{N, N, 8, N, 0, 5, 4};matrix[5] = new int[]{N, N, N, 4, 5, 0, 6};matrix[6] = new int[]{2, 3, N, N, 4, 6, 0};//创建 Graph 对象FGraph graph = new FGraph(vertex.length, matrix, vertex);//调用弗洛伊德算法graph.floyd();graph.show();graph.showAToB('C', 'D', vertex);}
}class FGraph {//顶点集合private char[] vertex;//邻接矩阵private int[][] arcs;//前驱矩阵private int[][] path;public FGraph(int vertexNum, int[][] arcs, char[] vertex) {this.vertex = vertex;this.arcs = arcs;this.path = new int[vertexNum][vertexNum];for (int i = 0; i < vertexNum; i++) {Arrays.fill(path[i], i);}}public void show() {System.out.println("==========邻接矩阵==========");for (int i = 0; i < arcs.length; i++) {for (int j = 0; j < arcs.length; j++) {System.out.print(arcs[i][j] + " ");}System.out.println();}System.out.println("==========前驱矩阵==========");for (int i = 0; i < path.length; i++) {for (int j = 0; j < path.length; j++) {System.out.print(vertex[path[i][j]] + " ");}System.out.println();}}public void floyd() {//保存距离int len = 0;//k为中间顶点for (int k = 0; k < arcs.length; k++) {//i为开始顶点for (int i = 0; i < arcs.length; i++) {//j为结束顶点for (int j = 0; j < arcs.length; j++) {len = arcs[i][k] + arcs[k][j];if (len < arcs[i][j]) {arcs[i][j] = len;//更新距离path[i][j] = path[k][j];//更新前驱矩阵}}}}}public void showAToB(char a, char b, char[] vertex) {int beginIndex = 0;int endIndex = 0;for (int i = 0; i < vertex.length; i++) {if (a == vertex[i]) {beginIndex = i;}if (b == vertex[i]) {endIndex = i;}}System.out.println(a + "到" + b + "的距离为：" + arcs[beginIndex][endIndex]);System.out.print("路径为：");showAToBPath(beginIndex, endIndex, vertex);System.out.print(b);}public void showAToBPath(int beginIndex, int endIndex, char[] vertex) {if (path[beginIndex][endIndex] == endIndex && path[beginIndex][endIndex] == beginIndex) {return;}showAToBPath(beginIndex, path[beginIndex][endIndex], vertex);System.out.print(vertex[path[beginIndex][endIndex]] + " ");}}