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Prim算法最小生成树c++实现
标签(空格分隔): Prim 算法 c++
最近离散课上讲到了Prim算法实现最小生成树,现在用c++来实现。本文代码来源于乔帮主伪代码。
简述
对于给定的带权图G,为了使得生成树T的权w(T)最小,我们可以使用贪心算法。
求最小生成树的贪心法:
1.选择一个初始点
2.为了使得生成树的权极小化,选择
3.选择{
4.重复以上步骤,直到所以点都连在一起,得到生成树
代码实现
#include <iostream>
#include <vector>
#include <limits>
#include <map>
using namespace std;
typedef float Weight;
typedef int Vertex;
typedef pair<Vertex, Vertex> Edge;
typedef vector<Edge> Edges;
typedef map<Vertex, bool> Vertices;
typedef vector<vector<Weight> > GraphMatrix;
typedef vector<pair<Vertex, Weight> > Labels;
float inf = numeric_limits<float>::max();
//求V的最小权邻点
Vertex nearest_neighbour(Vertices& vertices, Labels& label) {
Vertex u;
int n = vertices.size();
Weight w = inf;
for (int i = 0; i < n; i++) {
if (!vertices[i] && label[i].second < w) {
w = label[i].second;
u = i;
}
}
return u;
}
pair<Edges, Weight> prim(GraphMatrix& g, Vertex s) {
int n = g.size();
Vertices V;
for (int i = 0; i < n; i++)
V[i] = false;
V[s] = true;
Edges E;
Weight w = 0; // weight of MST
Labels L(n);
for (int i = 0; i < n; i++)
L[i] = make_pair(s, g[s][i]);
for (int i = 0; i < n - 1; i++) {
int u = nearest_neighbour(V, L); // 找到最小权的连接点
V[u] = true;
E.push_back(make_pair(u, L[u].first));
w += g[u][L[u].first];
for (int j = 0; j < n; j++) {
if (!V[j] && (g[u][j] < inf) && (g[u][j] < L[j].second))
L[j] = make_pair(u, g[u][j]);
}
}
return make_pair(E, w);
}
int main() {
int numberOfVertices;
cin >> numberOfVertices;
GraphMatrix graph(numberOfVertices);
for (int i = 0; i < numberOfVertices; i++) {
for (int j = 0; j < numberOfVertices; j++) {
Weight w;
cin >> w; // 如果两点没有连通,则输入0
if (w == 0) w = inf;
graph[i].push_back(w);
}
}
pair<Edges, Weight> MST = prim(graph, 0);
cout << "The weight of MST is: " << MST.second << endl;
int n = MST.first.size();
cout << "The edges of MST are: " << endl;
for (int i = 0; i < n; i++)
cout << "(" << MST.first[i].first << ", " << MST.first[i].second << ")" << endl;
return 0;
}
输入案例
Input
( 其中0代表两点不连通 )
8
0 3 2 0 0 0 0 0
3 0 3 6 0 0 0 0
2 3 0 0 5 5 0 0
0 6 0 0 2 0 0 2
0 0 5 2 0 4 3 0
0 0 5 0 4 0 4 0
0 0 0 0 3 4 0 6
0 0 0 2 0 0 6 0
Output
The weight of MST is: 21
The edges of MST are:
(2, 0)
(1, 0)
(4, 2)
(3, 4)
(7, 3)
(6, 4)
(5, 4)