最小生成树（Minimum Spanning Tree）的原理及实现（Java）

最小生成树（Minimum Spanning Tree）：给定无向图中，边权重最小的生成树。
满足条件
生成树——包含图中所有顶点的树；
最小——边权重之和最小的生成树。
它是图中全局的概念，而SPT（Shortest Path Tree，最短路径树）是相对图中某一个初始节点而言的。

算法原理

Cut和Crossing edge

Cut——将图中所有顶点分入（这个好像是随便来的？）两个非空集合中；
Crossing Edge——连接的顶点分别位于不同的集合中。
在一次cut中，所有crossing edge权值最小的edge一定是MST中的一个边（证明过程很有趣）

Prims算法

1.随机选定graph中的一个顶点；
2.将它和graph中其他顶点分割为两个集合A和B；
3.在Crossing edge中找到权重最小（如果有多个，随便选一个）的edge连接的顶点；
4.重新分割，将3中发现的属于集合B的顶点归入A中；
重复3和4步骤，直至找到MST。步骤三的时间复杂度比较高，可以使用PQ进行优化，不过好难看懂，可以参考这个video。

Kruskals算法

1.将所有edge按照权重在PQ中按从小到大的顺序跟踪；
2.取出PQ跟踪的第一个edge，将其放入MST中，检查是否会形成环路，如果是，则丢弃该edge；
3.重复步骤2直至找到MST（V-1条边）。

java代码
使用如下所示的无向、有权图作为示例

MSTFind.java

import java.util.*;public class MSTFind {private Graph graph;private WQU t;MSTFind(Graph graph) {this.graph = graph;}public List<Graph.Edge> Prims() {/* 1.随机选定graph中的一个顶点；2.将它和graph中其他顶点分割为两个集合A和B；3.在Crossing edge中找到权重最小（如果有多个，随便选一个）的edge连接的顶点，将该edge归入MST中；4.重新分割，将3中发现的属于集合B的顶点归入A中；重复3和4步骤，直至找到MST（V-1条边）。*/Set<Integer> A = new HashSet<>();Set<Integer> B = new HashSet<>();List<Graph.Edge> MST = new ArrayList<>();A.add(0);for (int i = 0; i < graph.V(); i++) {if (!A.contains(i)) {B.add(i);}}while (MST.size() != graph.V() - 1) {Graph.Edge minEdge = findMinCrossEdge(A, B);MST.add(minEdge);cutSets(A, B, minEdge);}return MST;}private Graph.Edge findMinCrossEdge(Set<Integer> a, Set<Integer> b) {/* 找到权重最小的crossing edge*/Graph.Edge res = new Graph.Edge(0, new Graph.Pair(0, Integer.MAX_VALUE));for (int v : a) {for (Graph.Pair p : graph.adj(v)) {if (!a.contains(p.getAnotherV())) {if (p.getWeight() < res.getWeight()) {res = new Graph.Edge(v, p);}}}}return res;}private void cutSets(Set<Integer> a, Set<Integer> b, Graph.Edge e) {for (int v : e.getVs()) {if (!a.contains(v)) {a.add(v);b.remove(v);}}}public List<Graph.Edge> Kruskals() {/* 1.将所有edge按照权重在PQ中按从小到大的顺序跟踪；2.取出PQ跟踪的第一个edge，将其放入MST中，检查是否会形成环路，如果是，则丢弃该edge；3.重复步骤2直至找到MST（V-1条边）。*/PriorityQueue<Graph.Edge> PQ = new PriorityQueue<>(new selfComparator());PQ.addAll(graph.getEdges());t = new WQU(graph.V());List<Graph.Edge> MST = new ArrayList<>();while (MST.size() != graph.V() - 1) {Graph.Edge curEdge = PQ.remove();MST.add(curEdge);if (checkCircle(curEdge)) {MST.remove(MST.size() - 1);}}return MST;}private boolean checkCircle(Graph.Edge e) {/* 使用WQU检测是否存在环路*/int v = e.getV();int w = e.getT().getAnotherV();if (t.isConnected(v, w)) {return true;} else {t.connect(v, w);return false;}}public class selfComparator implements Comparator<Graph.Edge> {@Overridepublic int compare(Graph.Edge o1, Graph.Edge o2) {if (o1.getWeight() < o2.getWeight()) {return -1;} else if (o1.getWeight() > o2.getWeight()) {return 1;}return 0;}}public static void main(String[] args) {Graph test = new Graph(7);   // 按照实例初始化graphtest.addEdge(0, 1, 2);test.addEdge(0, 2, 1);test.addEdge(1, 2, 5);test.addEdge(1, 3, 11);test.addEdge(1, 4, 3);test.addEdge(2, 4, 1);test.addEdge(2, 5, 15);test.addEdge(3, 4, 2);test.addEdge(4, 5, 4);test.addEdge(3, 6, 1);test.addEdge(4, 6, 5);test.addEdge(5, 6, 1);MSTFind t = new MSTFind(test);System.out.println("Prims算法的MST：");for (Graph.Edge e : t.Prims()) {System.out.print(e.getVs() + " ");}System.out.println("\nKruskals算法的MST：");for (Graph.Edge e : t.Kruskals()) {System.out.print(e.getVs() + " ");}}
}

Graph.java

import java.util.*;public class Graph {/* 无向、有权图*/private static List<Pair>[] ver;private static Set<Edge> edges;Graph(int v) {/* Create empty graph with v vertices*/ver = new List[v];edges = new HashSet<>();}public static class Pair {private int anotherV;private int weight;Pair(int v, int weight) {this.anotherV = v;this.weight = weight;}public int getAnotherV() {return anotherV;}public int getWeight() {return weight;}}public static class Edge {private int v;private Pair t;Edge(int v, Pair t) {this.v = v;this.t = t;}public int getV() {return v;}public int getWeight() {return t.getWeight();}public List<Integer> getVs() {List<Integer> res = new ArrayList<>();res.add(v);res.add(t.getAnotherV());return res;}}public void addEdge(int v, int w, int weight) {/* add an edge v-w with weight*/Pair pairV = new Pair(w, weight);Pair pairW = new Pair(v, weight);if (ver[v] == null) {ver[v] = new ArrayList<>();}if (ver[w] == null) {ver[w] = new ArrayList<>();}ver[v].add(pairV);ver[w].add(pairW);}Iterable<Pair> adj(int v) {/* vertices adjacent to v*/return ver[v];}public int V() {/* number of vertices*/return ver.length;}public int E() {/* number of edges*/int res = 0;for (int i = 0; i < ver.length; i++) {res += ver[i].size();}return res / 2;}public static void main(String[] args) {Graph test = new Graph(7);   // 按照实例初始化graphtest.addEdge(0, 1, 2);test.addEdge(0, 2, 1);test.addEdge(1, 2, 5);test.addEdge(1, 3, 11);test.addEdge(1, 4, 3);test.addEdge(2, 4, 1);test.addEdge(2, 5, 15);test.addEdge(3, 4, 2);test.addEdge(4, 5, 4);test.addEdge(3, 6, 1);test.addEdge(4, 6, 5);test.addEdge(5, 6, 1);for (int i = 0; i < test.V(); i++) {List<Integer> adj = new ArrayList<>();for (Pair p : test.adj(i)) {adj.add(p.getAnotherV());}System.out.println("和" + i + "邻接的顶点为：" + adj);}System.out.println("顶点的数量：" + test.V());   // 顶点的数量应该为5System.out.println("边的的数量：" + test.E());   // 边的数量应该为6}
}

Kruskals算法中检测环路的算法参考Graph中环路检测（DFS&WQU）及实现（Java）
WQU的实现参考加权快速联合（Weighted Quick Union）算法原理及实现（Java）

To be a sailor of the world bound for all ports.