Minimum Spanning Tree

找出一个边的集合可以用最小的代价连接所有的点。

Kruskal

可视化

Sort E edges by increasing weight
T = {}
for (i = 0; i < edgeList.length; i++)
  if adding e = edgelist[i] does not form a cycle
    add e to T
  else ignore e
MST = T

如何来判断新加的边不会形成一个cycle，最简单的方法就是用Union-Find这个数据结构了。

java 实现

// Java program for Kruskal's algorithm to find Minimum Spanning Tree
// of a given connected, undirected and weighted graph
import java.util.*;
import java.lang.*;
import java.io.*;

class Graph {
    // A class to represent a graph edge
    class Edge implements Comparable<Edge> {
        int src, dest, weight;

        // Comparator function used for sorting edges based on
        // their weight
        public int compareTo(Edge compareEdge) {
            return this.weight-compareEdge.weight;
        }
    }

    // A class to represent a subset for union-find
    class subset {
        int parent, rank;
    };

    int V, E;    // V-> no. of vertices & E->no.of edges
    Edge edge[]; // collection of all edges

    // Creates a graph with V vertices and E edges
    Graph(int v, int e) {
        V = v;
        E = e;
        edge = new Edge[E];
        for (int i=0; i<e; ++i)
            edge[i] = new Edge();
    }

    // A utility function to find set of an element i
    // (uses path compression technique)
    int find(subset subsets[], int i) {
        // find root and make root as parent of i (path compression)
        if (subsets[i].parent != i)
            subsets[i].parent = find(subsets, subsets[i].parent);

        return subsets[i].parent;
    }

    // A function that does union of two sets of x and y
    // (uses union by rank)
    void Union(subset subsets[], int x, int y)
    {
        int xroot = find(subsets, x);
        int yroot = find(subsets, y);

        // Attach smaller rank tree under root of high rank tree
        // (Union by Rank)
        if (subsets[xroot].rank < subsets[yroot].rank)
            subsets[xroot].parent = yroot;
        else if (subsets[xroot].rank > subsets[yroot].rank)
            subsets[yroot].parent = xroot;

        // If ranks are same, then make one as root and increment
        // its rank by one
        else {
            subsets[yroot].parent = xroot;
            subsets[xroot].rank++;
        }
    }

    // The main function to construct MST using Kruskal's algorithm
    void KruskalMST()
    {
        Edge result[] = new Edge[V];  // Tnis will store the resultant MST
        int e = 0;  // An index variable, used for result[]
        int i = 0;  // An index variable, used for sorted edges
        for (i=0; i<V; ++i)
            result[i] = new Edge();

        // Step 1:  Sort all the edges in non-decreasing order of their
        // weight.  If we are not allowed to change the given graph, we
        // can create a copy of array of edges
        Arrays.sort(edge);

        // Allocate memory for creating V ssubsets
        subset subsets[] = new subset[V];
        for(i=0; i<V; ++i)
            subsets[i]=new subset();

        // Create V subsets with single elements
        for (int v = 0; v < V; ++v)
        {
            subsets[v].parent = v;
            subsets[v].rank = 0;
        }

        i = 0;  // Index used to pick next edge

        // Number of edges to be taken is equal to V-1
        while (e < V - 1)
        {
            // Step 2: Pick the smallest edge. And increment the index
            // for next iteration
            Edge next_edge = new Edge();
            next_edge = edge[i++];

            int x = find(subsets, next_edge.src);
            int y = find(subsets, next_edge.dest);

            // If including this edge does't cause cycle, include it
            // in result and increment the index of result for next edge
            if (x != y)
            {
                result[e++] = next_edge;
                Union(subsets, x, y);
            }
            // Else discard the next_edge
        }

        // print the contents of result[] to display the built MST
        System.out.println("Following are the edges in the constructed MST");
        for (i = 0; i < e; ++i)
            System.out.println(result[i].src+" -- "+result[i].dest+" == "+
                               result[i].weight);
    }
}