P 算法与 K 算法作者：Grey
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博客园：P 算法与 K 算法
CSDN：P 算法与 K 算法
说明P 算法和 K 算法主要用来解决最小生成树问题，即：不破坏连通性删掉某些边 ， 使得整体的权重最小 。
测评链接：牛客-最小生成树
K 算法【P 算法与 K 算法】K 算法使用的核心数据结构是并查集，然后将边权值排序 。
1）总是从权值最小的边开始考虑，依次考察权值依次变大的边
2）当前的边要么进入最小生成树的集合，要么丢弃
3）如果当前的边进入最小生成树的集合中不会形成环，就要当前边
4）如果当前的边进入最小生成树的集合中会形成环，就不要当前边
5）考察完所有边之后，最小生成树的集合也得到了
边存在小根堆里面，保证每次弹出的都是权重最小的值
点存在并查集中，每次加入一个边 ， 就把两个边的点 union
完整代码如下
import java.util.Arrays;import java.util.Comparator;import java.util.Scanner;public class Main {public static void main(String[] args) {Scanner in = new Scanner(System.in);int n = in.nextInt();int m = in.nextInt();int[][] graph = new int[m][3];for (int i = 0; i < m; i++) {// fromgraph[i][0] = in.nextInt();// tograph[i][1] = in.nextInt();// weightgraph[i][2] = in.nextInt();}System.out.println(k(graph, n));in.close();}// k算法生成最小生成树public static int k(int[][] graph, int n) {UnionFind uf = new UnionFind(n);Arrays.sort(graph, Comparator.comparingInt(o -> o[2]));int ans = 0;for (int[] edge : graph) {if (!uf.same(edge[0], edge[1])) {uf.union(edge[0], edge[1]);ans += edge[2];}}return ans;}public static class UnionFind {private final int[] parent;private final int[] size;private final int[] help;public UnionFind(int n) {parent = new int[n + 1];size = new int[n + 1];help = new int[n + 1];for (int i = 1; i < n; i++) {parent[i] = i;size[i] = 1;}}public boolean same(int a, int b) {return find(a) == find(b);}private int find(int a) {int index = 0;while (a != parent[a]) {help[index++] = a;a = parent[a];}index--;while (index > 0) {parent[help[index--]] = a;}return a;}public void union(int a, int b) {int f1 = find(a);int f2 = find(b);if (f1 != f2) {int size1 = size[f1];int size2 = size[f2];if (size1 > size2) {parent[f2] = f1;size[f2] = 0;size[f1] = size1 + size2;} else {parent[f1] = f2;size[f1] = 0;size[f2] = size1 + size2;}}}}}P 算法1）可以从任意节点出发来寻找最小生成树
2）某个点加入到被选取的点中后，解锁这个点出发的所有新的边
3）在所有解锁的边中选最小的边，然后看看这个边会不会形成环
4）如果会，不要当前边，继续考察剩下解锁的边中最小的边，重复3）
5）如果不会 ， 要当前边，将该边的指向点加入到被选取的点中 ， 重复2）
6）当所有点都被选取，最小生成树就得到了
完整代码如下
import java.util.*;public class Main {public static Set<Edge> P(Graph graph) {// 解锁的边进入小根堆PriorityQueue<Edge> priorityQueue = new PriorityQueue<>(Comparator.comparingInt(o -> o.weight));// 哪些点被解锁出来了HashSet<Node> nodeSet = new HashSet<>();Set<Edge> result = new HashSet<>(); // 依次挑选的的边在result里for (Node node : graph.nodes.values()) { // 随便挑了一个点// node 是开始点if (!nodeSet.contains(node)) {nodeSet.add(node);for (Edge edge : node.edges) { // 由一个点，解锁所有相连的边priorityQueue.add(edge);}while (!priorityQueue.isEmpty()) {Edge edge = priorityQueue.poll(); // 弹出解锁的边中，最小的边Node toNode = edge.to; // 可能的一个新的点if (!nodeSet.contains(toNode)) { // 不含有的时候，就是新的点nodeSet.add(toNode);result.add(edge);for (Edge nextEdge : toNode.edges) {priorityQueue.add(nextEdge);}}}}// 如果有森林 ， 就不能break，如果没有森林，就可以break//break;}return result;}public static class Graph {public HashMap<Integer, Node> nodes;public HashSet<Edge> edges;public Graph(int n) {nodes = new HashMap<>();edges = new HashSet<>(n);}}public static class Node {public int value;public int in;public int out;public ArrayList<Node> nexts;public ArrayList<Edge> edges;public Node(int value) {this.value = https://www.huyubaike.com/biancheng/value;in = 0;out = 0;nexts = new ArrayList<>();edges = new ArrayList<>();}}public static class Edge {public int weight;public Node from;public Node to;public Edge(int weight, Node from, Node to) {this.weight = weight;this.from = from;this.to = to;}}public static void main(String[] args) {Scanner in = new Scanner(System.in);int n = in.nextInt();int m = in.nextInt();Graph graph = new Graph(n);for (int i = 0; i < m; i++) {int from = in.nextInt();int to = in.nextInt();int weight = in.nextInt();if (!graph.nodes.containsKey(from)) {graph.nodes.put(from, new Node(from));}if (!graph.nodes.containsKey(to)) {graph.nodes.put(to, new Node(to));}Node fromNode = graph.nodes.get(from);Node toNode = graph.nodes.get(to);Edge fromToEdge = new Edge(weight, fromNode, toNode);Edge toFromEdge = new Edge(weight, toNode, fromNode);fromNode.nexts.add(toNode);fromNode.out++;fromNode.in++;toNode.out++;toNode.in++;fromNode.edges.add(fromToEdge);toNode.edges.add(toFromEdge);graph.edges.add(fromToEdge);graph.edges.add(toFromEdge);}Set