Floyd算法
Floyd算法
Dijkstra算法是用于解决单源最短路径问题的，Floyd算法则是解决点对之间最短路径问题的。Floyd算法的设计策略是动态规划，而Dijkstra採取的是贪心策略。当然，贪心算法就是动态规划的特例。
算法思想
点对之间的最短路径仅仅会有两种情况：
- 两点之间有边相连。weight(Vi,Vj)即是最小的。
- 通过还有一点：中介点，两点相连，使weight(Vi,Vv)+weight(Vv,Vj)最小。
故当Vv取全然部顶点后，Distance(Vi,Vj)就可以达到最小。Floyd算法的起点就是图的邻接矩阵。
思想极难得到，而有了思想，稍加经验就可以写出代码。向思想的开创者致敬。
代码
#include<iostream>
#include<iomanip>
#include<stack>
using namespace std;
#define MAXWEIGHT 100
#undef INFINITY
#define INFINITY 1000
class Graph
{
private://顶点数 int numV;//边数 int numE;//邻接矩阵 int **matrix;
public:Graph(int numV);//建图 void createGraph(int numE);//析构方法 ~Graph();//Floyd算法void Floyd();//打印邻接矩阵 void printAdjacentMatrix();//检查输入 bool check(int, int, int);
};//构造函数，指定顶点数目
Graph::Graph(int numV)
{//对输入的顶点数进行检測while (numV <= 0){cout << "顶点数有误！又一次输入 "; cin >> numV; } this->numV = numV; //构建邻接矩阵。并初始化 matrix = new int*[numV]; int i, j; for (i = 0; i < numV; i++) matrix[i] = new int[numV]; for (i = 0; i < numV; i++) for (j = 0; j < numV; j++) { if (i == j) matrix[i][i] = 0; else matrix[i][j] = INFINITY; } } void Graph::createGraph(int numE) { /* 对输入的边数做检測 一个numV个顶点的有向图，最多有numV*(numV - 1)条边 */ while (numE < 0 || numE > numV*(numV - 1)) { cout << "边数有问题！又一次输入 "; cin >> numE; } this->numE = numE; int tail, head, weight, i; i = 0; cout << "输入每条边的起点(弧尾)、终点(弧头)和权值" << endl; while (i < numE) { cin >> tail >> head >> weight; while (!check(tail, head, weight)) { cout << "输入的边不对！请又一次输入 " << endl; cin >> tail >> head >> weight; } matrix[tail][head] = weight; i++; } } Graph::~Graph() { int i; for (i = 0; i < numV; i++) delete[] matrix[i]; delete[]matrix; } /* 弗洛伊德算法 求各顶点对之间的最短距离 及其路径 */ void Graph::Floyd() { //为了不改动邻接矩阵，多用一个二维数组 int **Distance = new int*[numV]; int i, j; for (i = 0; i < numV; i++) Distance[i] = new int[numV]; //初始化 for (i = 0; i < numV; i++) for (j = 0; j < numV; j++) Distance[i][j] = matrix[i][j]; //prev数组 int **prev = new int*[numV]; for (i = 0; i < numV; i++) prev[i] = new int[numV]; //初始化prev for (i = 0; i < numV; i++) for (j = 0; j < numV; j++) { if (matrix[i][j] == INFINITY) prev[i][j] = -1; else prev[i][j] = i; } int d, v; for (v = 0; v < numV; v++) for (i = 0; i < numV; i++) for (j = 0; j < numV; j++) { d = Distance[i][v] + Distance[v][j]; if (d < Distance[i][j]) { Distance[i][j] = d; prev[i][j] = v; } } //打印Distance和prev数组 cout << "Distance..." << endl; for (i = 0; i < numV; i++) { for (j = 0; j < numV; j++) cout << setw(3) << Distance[i][j]; cout << endl; } cout << endl << "prev..." << endl; for (i = 0; i < numV; i++) { for (j = 0; j < numV; j++) cout << setw(3) << prev[i][j]; cout << endl; } cout << endl; //打印顶点对最短路径 stack<int> s; for (i = 0; i < numV; i++) { for (j = 0; j < numV; j++) { if (Distance[i][j] == 0); else if (Distance[i][j] == INFINITY) cout << "顶点 " << i << " 到顶点 " << j << " 无路径！
" << endl; else { s.push(j); v = j; do{ v = prev[i][v]; s.push(v); } while (v != i); //打印路径 cout << "顶点 " << i << " 到顶点 " << j << " 的最短路径长度是 " << Distance[i][j] << " ，其路径序列是..."; while (!s.empty()) { cout << setw(3) << s.top(); s.pop(); } cout << endl; } } cout << endl; } //释放空间 for (i = 0; i < numV; i++) { delete[] Distance[i]; delete[] prev[i]; } delete[]Distance; delete[]prev; } //打印邻接矩阵 void Graph::printAdjacentMatrix() { int i, j; cout.setf(ios::left); cout << setw(7) << " "; for (i = 0; i < numV; i++) cout << setw(7) << i; cout << endl; for (i = 0; i < numV; i++) { cout << setw(7) << i; for (j = 0; j < numV; j++) cout << setw(7) << matrix[i][j]; cout << endl; } } bool Graph::check(int tail, int head, int weight) { if (tail < 0 || tail >= numV || head < 0 || head >= numV || weight <= 0 || weight >= MAXWEIGHT) return false; return true; }
主函数int main()
{cout << "******Floyd***by David***" << endl;int numV, numE;cout << "建图..." << endl;cout << "输入顶点数 ";cin >> numV;Graph graph(numV);cout << "输入边数 ";cin >> numE;graph.createGraph(numE);cout << endl << "Floyd..." << endl;graph.Floyd();system("pause");return 0;
}
小结
若有所帮助，顶一个哦。
专栏文件夹：
- 数据结构与算法文件夹
- c指针
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