最小生成树的Kruskal算法

来源：互联网发布：游戏模型制作软件编辑：程序博客网时间：2024/06/05 14:55

Kruskal算法的主要思想是按照边的权重顺序（从小到大）处理他们，将边加入最小生成树中，加入的边不会和已经加入的边构成环，直到树中含有V-1条边为止。

Kruskal算法能够计算任意加权连通图的最小生成树。

数据结构：

用优先队列来对边按照权重排序

用并查集(之前博客里提到过的union-find)来识别会形成环的边

用链表来保存最小生成树的所有边

时间复杂度ElgE

-UF.h 并查集

#ifndef __UF_H__#define __UF_H__class UF {private:int*id;// 父节点数组int*sz;// 各个根节点对应的分量的大小intcount;// 连通分量的数量public:UF(int n);~UF() { delete[] id; delete[] sz; }int getCount()const { return count; }bool connected(int p, int q)const { return find(p) == find(q); }int find(int p)const;void myUnion(int p, int q);};UF::UF(int n) {id = new int[n];sz = new int[n];count = n;for (int i = 0; i < n; ++i) {id[i] = i;sz[i] = 1;}}int UF::find(int p)const {while (p != id[p]) {p = id[p];}return p;}void UF::myUnion(int p, int q) {int pID = find(p);int qID = find(q);if (pID == qID) return;if (sz[pID] < sz[qID]) {sz[qID] += sz[pID];id[pID] = qID;}else {sz[pID] += sz[qID];id[qID] = pID;}--count;}#endif

-KruskalMST.h

#ifndef __KRUSKAL_MST_H__#define __KRUSKAL_MST_H__#include "UF.h"#include "Edge.h"#include "EdgeWeightedGraph.h"#include <list>#include <vector>#include <functional>#include <queue>class KruskalMST {private:std::list<Edge> mst;double weight;public:KruskalMST(const EdgeWeightedGraph& G) {weight = 0.0;int vNum = G.getV();// 构造最小堆std::priority_queue<Edge, std::vector<Edge>, std::greater<Edge> > pq;for (Edge e : G.edges()) {pq.push(e);}UF uf(vNum);int v, w;Edge e;while(mst.size() != vNum - 1) {e = pq.top();pq.pop();v = e.either();w = e.other(v);if (uf.connected(v, w)) continue;uf.myUnion(v, w);// 添加进最小生成树mst.push_back(e);weight += e.getWeight();}}std::list<Edge> edges()const { return mst; }double getWeight()const { return weight; }};#endif

-main.cpp 测试用例

#include "KruskalMST.h"#include "EdgeWeightedGraph.h"#include <iostream>#include <functional>using namespace std;int main(){int vNum, eNum;cin >> vNum >> eNum;EdgeWeightedGraph G(vNum);int v, w;double weight;for (int i = 0; i < eNum; ++i) {cin >> v >> w >> weight;G.addEdge(Edge(v, w, weight));}KruskalMST kruskalMST(G);for (Edge e : kruskalMST.edges()) {cout << e.either() << "-"<< e.other(e.either()) << " "<< e.getWeight() << endl;}cout << "total weight: " << kruskalMST.getWeight();return 0;}

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