Kruskal 算法

Kruskal 算法是一种最小生成树算法，它以图为输入，并找到该图的边的子集，该子集

形成一棵树，该树包含每个顶点
在所有可以从图中形成的树中具有最小权重总和

Kruskal 算法的工作原理

它属于一类称为贪心算法的算法，这些算法通过找到局部最优来寻求全局最优。

我们从权重最低的边开始，然后不断添加边，直到达到目标。

实现 Kruskal 算法的步骤如下：

将所有边从低权重到高权重排序
选取权重最低的边并将其添加到生成树中。如果添加边产生了环，则拒绝此边。
继续添加边，直到我们覆盖所有顶点。

Kruskal 算法示例

Choose the edge with the least weight, if there are more than 1, choose anyone — 选择权重最小的边，如果有多个，则选择任意一个

Choose the next shortest edge and add it — 选择下一条最短的边并添加它

Choose the next shortest edge that doesn't create a cycle and add it — 选择下一条不会产生环的最短边并添加它

Repeat until you have a spanning tree — 重复直到您得到一棵生成树

Kruskal 算法伪代码

任何最小生成树算法都围绕着检查添加边是否会产生环。

找出这一点最常用的方法是称为并查集的算法。并查集算法将顶点划分为簇，并允许我们检查两个顶点是否属于同一簇，从而决定添加边是否会产生环。

KRUSKAL(G):
A = ∅
For each vertex v ∈ G.V:
    MAKE-SET(v)
For each edge (u, v) ∈ G.E ordered by increasing order by weight(u, v):
    if FIND-SET(u) ≠ FIND-SET(v):       
    A = A ∪ {(u, v)}
    UNION(u, v)
return A

Python、Java 和 C/C++ 示例

Python

Java

C++

# Kruskal's algorithm in Python


class Graph:
    def __init__(self, vertices):
        self.V = vertices
        self.graph = []

    def add_edge(self, u, v, w):
        self.graph.append([u, v, w])

    # Search function

    def find(self, parent, i):
        if parent[i] == i:
            return i
        return self.find(parent, parent[i])

    def apply_union(self, parent, rank, x, y):
        xroot = self.find(parent, x)
        yroot = self.find(parent, y)
        if rank[xroot] < rank[yroot]:
            parent[xroot] = yroot
        elif rank[xroot] > rank[yroot]:
            parent[yroot] = xroot
        else:
            parent[yroot] = xroot
            rank[xroot] += 1

    #  Applying Kruskal algorithm
    def kruskal_algo(self):
        result = []
        i, e = 0, 0
        self.graph = sorted(self.graph, key=lambda item: item[2])
        parent = []
        rank = []
        for node in range(self.V):
            parent.append(node)
            rank.append(0)
        while e < self.V - 1:
            u, v, w = self.graph[i]
            i = i + 1
            x = self.find(parent, u)
            y = self.find(parent, v)
            if x != y:
                e = e + 1
                result.append([u, v, w])
                self.apply_union(parent, rank, x, y)
        for u, v, weight in result:
            print("%d - %d: %d" % (u, v, weight))


g = Graph(6)
g.add_edge(0, 1, 4)
g.add_edge(0, 2, 4)
g.add_edge(1, 2, 2)
g.add_edge(1, 0, 4)
g.add_edge(2, 0, 4)
g.add_edge(2, 1, 2)
g.add_edge(2, 3, 3)
g.add_edge(2, 5, 2)
g.add_edge(2, 4, 4)
g.add_edge(3, 2, 3)
g.add_edge(3, 4, 3)
g.add_edge(4, 2, 4)
g.add_edge(4, 3, 3)
g.add_edge(5, 2, 2)
g.add_edge(5, 4, 3)
g.kruskal_algo()

// Kruskal's algorithm in Java

import java.util.*;

class Graph {
  class Edge implements Comparable<Edge> {
    int src, dest, weight;

    public int compareTo(Edge compareEdge) {
      return this.weight - compareEdge.weight;
    }
  };

  // Union
  class subset {
    int parent, rank;
  };

  int vertices, edges;
  Edge edge[];

  // Graph creation
  Graph(int v, int e) {
    vertices = v;
    edges = e;
    edge = new Edge[edges];
    for (int i = 0; i < e; ++i)
      edge[i] = new Edge();
  }

  int find(subset subsets[], int i) {
    if (subsets[i].parent != i)
      subsets[i].parent = find(subsets, subsets[i].parent);
    return subsets[i].parent;
  }

  void Union(subset subsets[], int x, int y) {
    int xroot = find(subsets, x);
    int yroot = find(subsets, y);

    if (subsets[xroot].rank < subsets[yroot].rank)
      subsets[xroot].parent = yroot;
    else if (subsets[xroot].rank > subsets[yroot].rank)
      subsets[yroot].parent = xroot;
    else {
      subsets[yroot].parent = xroot;
      subsets[xroot].rank++;
    }
  }

  // Applying Krushkal Algorithm
  void KruskalAlgo() {
    Edge result[] = new Edge[vertices];
    int e = 0;
    int i = 0;
    for (i = 0; i < vertices; ++i)
      result[i] = new Edge();

    // Sorting the edges
    Arrays.sort(edge);
    subset subsets[] = new subset[vertices];
    for (i = 0; i < vertices; ++i)
      subsets[i] = new subset();

    for (int v = 0; v < vertices; ++v) {
      subsets[v].parent = v;
      subsets[v].rank = 0;
    }
    i = 0;
    while (e < vertices - 1) {
      Edge next_edge = new Edge();
      next_edge = edge[i++];
      int x = find(subsets, next_edge.src);
      int y = find(subsets, next_edge.dest);
      if (x != y) {
        result[e++] = next_edge;
        Union(subsets, x, y);
      }
    }
    for (i = 0; i < e; ++i)
      System.out.println(result[i].src + " - " + result[i].dest + ": " + result[i].weight);
  }

  public static void main(String[] args) {
    int vertices = 6; // Number of vertices
    int edges = 8; // Number of edges
    Graph G = new Graph(vertices, edges);

    G.edge[0].src = 0;
    G.edge[0].dest = 1;
    G.edge[0].weight = 4;

    G.edge[1].src = 0;
    G.edge[1].dest = 2;
    G.edge[1].weight = 4;

    G.edge[2].src = 1;
    G.edge[2].dest = 2;
    G.edge[2].weight = 2;

    G.edge[3].src = 2;
    G.edge[3].dest = 3;
    G.edge[3].weight = 3;

    G.edge[4].src = 2;
    G.edge[4].dest = 5;
    G.edge[4].weight = 2;

    G.edge[5].src = 2;
    G.edge[5].dest = 4;
    G.edge[5].weight = 4;

    G.edge[6].src = 3;
    G.edge[6].dest = 4;
    G.edge[6].weight = 3;

    G.edge[7].src = 5;
    G.edge[7].dest = 4;
    G.edge[7].weight = 3;
    G.KruskalAlgo();
  }
}

// Kruskal's algorithm in C

#include <stdio.h>

#define MAX 30

typedef struct edge {
  int u, v, w;
} edge;

typedef struct edge_list {
  edge data[MAX];
  int n;
} edge_list;

edge_list elist;

int Graph[MAX][MAX], n;
edge_list spanlist;

void kruskalAlgo();
int find(int belongs[], int vertexno);
void applyUnion(int belongs[], int c1, int c2);
void sort();
void print();

// Applying Krushkal Algo
void kruskalAlgo() {
  int belongs[MAX], i, j, cno1, cno2;
  elist.n = 0;

  for (i = 1; i < n; i++)
    for (j = 0; j < i; j++) {
      if (Graph[i][j] != 0) {
        elist.data[elist.n].u = i;
        elist.data[elist.n].v = j;
        elist.data[elist.n].w = Graph[i][j];
        elist.n++;
      }
    }

  sort();

  for (i = 0; i < n; i++)
    belongs[i] = i;

  spanlist.n = 0;

  for (i = 0; i < elist.n; i++) {
    cno1 = find(belongs, elist.data[i].u);
    cno2 = find(belongs, elist.data[i].v);

    if (cno1 != cno2) {
      spanlist.data[spanlist.n] = elist.data[i];
      spanlist.n = spanlist.n + 1;
      applyUnion(belongs, cno1, cno2);
    }
  }
}

int find(int belongs[], int vertexno) {
  return (belongs[vertexno]);
}

void applyUnion(int belongs[], int c1, int c2) {
  int i;

  for (i = 0; i < n; i++)
    if (belongs[i] == c2)
      belongs[i] = c1;
}

// Sorting algo
void sort() {
  int i, j;
  edge temp;

  for (i = 1; i < elist.n; i++)
    for (j = 0; j < elist.n - 1; j++)
      if (elist.data[j].w > elist.data[j + 1].w) {
        temp = elist.data[j];
        elist.data[j] = elist.data[j + 1];
        elist.data[j + 1] = temp;
      }
}

// Printing the result
void print() {
  int i, cost = 0;

  for (i = 0; i < spanlist.n; i++) {
    printf("\n%d - %d : %d", spanlist.data[i].u, spanlist.data[i].v, spanlist.data[i].w);
    cost = cost + spanlist.data[i].w;
  }

  printf("\nSpanning tree cost: %d", cost);
}

int main() {
  int i, j, total_cost;

  n = 6;

  Graph[0][0] = 0;
  Graph[0][1] = 4;
  Graph[0][2] = 4;
  Graph[0][3] = 0;
  Graph[0][4] = 0;
  Graph[0][5] = 0;
  Graph[0][6] = 0;

  Graph[1][0] = 4;
  Graph[1][1] = 0;
  Graph[1][2] = 2;
  Graph[1][3] = 0;
  Graph[1][4] = 0;
  Graph[1][5] = 0;
  Graph[1][6] = 0;

  Graph[2][0] = 4;
  Graph[2][1] = 2;
  Graph[2][2] = 0;
  Graph[2][3] = 3;
  Graph[2][4] = 4;
  Graph[2][5] = 0;
  Graph[2][6] = 0;

  Graph[3][0] = 0;
  Graph[3][1] = 0;
  Graph[3][2] = 3;
  Graph[3][3] = 0;
  Graph[3][4] = 3;
  Graph[3][5] = 0;
  Graph[3][6] = 0;

  Graph[4][0] = 0;
  Graph[4][1] = 0;
  Graph[4][2] = 4;
  Graph[4][3] = 3;
  Graph[4][4] = 0;
  Graph[4][5] = 0;
  Graph[4][6] = 0;

  Graph[5][0] = 0;
  Graph[5][1] = 0;
  Graph[5][2] = 2;
  Graph[5][3] = 0;
  Graph[5][4] = 3;
  Graph[5][5] = 0;
  Graph[5][6] = 0;

  kruskalAlgo();
  print();
}

// Kruskal's algorithm in C++

#include <algorithm>
#include <iostream>
#include <vector>
using namespace std;

#define edge pair<int, int>

class Graph {
   private:
  vector<pair<int, edge> > G;  // graph
  vector<pair<int, edge> > T;  // mst
  int *parent;
  int V;  // number of vertices/nodes in graph
   public:
  Graph(int V);
  void AddWeightedEdge(int u, int v, int w);
  int find_set(int i);
  void union_set(int u, int v);
  void kruskal();
  void print();
};
Graph::Graph(int V) {
  parent = new int[V];

  //i 0 1 2 3 4 5
  //parent[i] 0 1 2 3 4 5
  for (int i = 0; i < V; i++)
    parent[i] = i;

  G.clear();
  T.clear();
}
void Graph::AddWeightedEdge(int u, int v, int w) {
  G.push_back(make_pair(w, edge(u, v)));
}
int Graph::find_set(int i) {
  // If i is the parent of itself
  if (i == parent[i])
    return i;
  else
    // Else if i is not the parent of itself
    // Then i is not the representative of his set,
    // so we recursively call Find on its parent
    return find_set(parent[i]);
}

void Graph::union_set(int u, int v) {
  parent[u] = parent[v];
}
void Graph::kruskal() {
  int i, uRep, vRep;
  sort(G.begin(), G.end());  // increasing weight
  for (i = 0; i < G.size(); i++) {
    uRep = find_set(G[i].second.first);
    vRep = find_set(G[i].second.second);
    if (uRep != vRep) {
      T.push_back(G[i]);  // add to tree
      union_set(uRep, vRep);
    }
  }
}
void Graph::print() {
  cout << "Edge :"
     << " Weight" << endl;
  for (int i = 0; i < T.size(); i++) {
    cout << T[i].second.first << " - " << T[i].second.second << " : "
       << T[i].first;
    cout << endl;
  }
}
int main() {
  Graph g(6);
  g.AddWeightedEdge(0, 1, 4);
  g.AddWeightedEdge(0, 2, 4);
  g.AddWeightedEdge(1, 2, 2);
  g.AddWeightedEdge(1, 0, 4);
  g.AddWeightedEdge(2, 0, 4);
  g.AddWeightedEdge(2, 1, 2);
  g.AddWeightedEdge(2, 3, 3);
  g.AddWeightedEdge(2, 5, 2);
  g.AddWeightedEdge(2, 4, 4);
  g.AddWeightedEdge(3, 2, 3);
  g.AddWeightedEdge(3, 4, 3);
  g.AddWeightedEdge(4, 2, 4);
  g.AddWeightedEdge(4, 3, 3);
  g.AddWeightedEdge(5, 2, 2);
  g.AddWeightedEdge(5, 4, 3);
  g.kruskal();
  g.print();
  return 0;
}

Kruskal 算法 vs Prim 算法

Prim 算法是另一种流行的最小生成树算法，它使用不同的逻辑来查找图的 MST。Prim 算法不从边开始，而是从一个顶点开始，并不断添加不在树中的权重最低的边，直到覆盖所有顶点。

Kruskal 算法的复杂度

Kruskal 算法的时间复杂度为：O(E log E)。

Kruskal 算法的应用

用于布置电线
在计算机网络中（局域网连接）

热门教程

热门实例

参考资料

认证课程

成为一名认证的 Python
程序员。

热门教程

参考资料

热门实例

DSA 简介

数据结构 (I)

数据结构 (II)

基于树的 DSA (I)

基于树的 DSA (II)

基于图的 DSA

排序和搜索算法

贪婪算法

动态规划

其他算法

Kruskal 算法

Kruskal 算法的工作原理

Kruskal 算法示例

Kruskal 算法伪代码

Python、Java 和 C/C++ 示例

Kruskal 算法 vs Prim 算法

Kruskal 算法的复杂度

Kruskal 算法的应用

目录

热门教程

热门实例

参考资料

认证课程

成为一名认证的 Python程序员。

热门教程

参考资料

热门实例

DSA 简介

数据结构 (I)

数据结构 (II)

基于树的 DSA (I)

基于树的 DSA (II)

基于图的 DSA

排序和搜索算法

贪婪算法

动态规划

其他算法

DSA 教程

Kruskal 算法

Kruskal 算法的工作原理

Kruskal 算法示例

Kruskal 算法伪代码

Python、Java 和 C/C++ 示例

Kruskal 算法 vs Prim 算法

Kruskal 算法的复杂度

Kruskal 算法的应用

目录

相关教程

成为一名认证的 Python
程序员。