什么是最小生成树?
给定一个连通图和无向图，该图的生成树是一个子图，该子图是一棵将所有顶点连接在一起的树。单个图可以具有许多不同的生成树。加权，连接，无向图的最小生成树(MST)或最小权重生成树是权重小于或等于其他所有生成树的权重的生成树。生成树的权重是赋予生成树的每个边缘的权重之和。
最小生成树有几个边?
最小生成树具有(V – 1)个边，其中V是给定图中顶点的数量。
最小生成树的应用是什么?
有关MST的应用，请参见此内容。
以下是使用Kruskal算法查找MST的步骤
1. Sort all the edges in non-decreasing order of their weight.
2. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If cycle is not formed, include this edge. Else, discard it.
3. Repeat step#2 until there are (V-1) edges in the spanning tree.
步骤2使用联合查找算法来检测周期。因此，我们建议您阅读以下文章作为先决条件。
联合查找算法|集合1(图形中的检测周期)
联合查找算法|第2组(按等级和路径压缩合并)
该算法是贪婪算法。贪婪的选择是选择重量最小的边沿，该边沿不会在到目前为止构造的MST中引起循环。让我们以一个示例来理解它：考虑下面的输入图。
该图包含9个顶点和14个边。因此，形成的最小生成树将具有(9 – 1)= 8个边。
After sorting:
Weight Src Dest
1 7 6
2 8 2
2 6 5
4 0 1
4 2 5
6 8 6
7 2 3
7 7 8
8 0 7
8 1 2
9 3 4
10 5 4
11 1 7
14 3 5现在从边缘排序列表中一一挑选所有边缘
1.拾取边缘7-6：不形成循环，将其包括在内。
2.拾取边缘8-2：不形成循环，将其包括在内。
3.拾取边缘6-5：不形成循环，将其包括在内。
4.拾取边缘0-1：不形成循环，将其包括在内。
5.拾取边缘2-5：不形成循环，将其包括在内。
6.拾取边8-6：由于包括该边会导致循环，因此请将其丢弃。
7.拾取边缘2-3：不形成循环，将其包括在内。
8.拾取边7-8：由于包括该边会导致循环，因此请将其丢弃。
9.拾取边0-7：不形成循环，将其包括在内。
10.选择边1-2：由于包含该边会导致循环，因此请将其丢弃。
11.抓边3-4：不形成任何循环，将其包括在内。
由于包含的边数等于(V – 1)，因此算法在此处停止。
下面是上述想法的实现：
C
// C program for Kruskal's algorithm to find Minimum
// Spanning Tree of a given connected, undirected and
// weighted graph
#include
#include
#include
// a structure to represent a weighted edge in graph
struct Edge {
int src, dest, weight;
};
// a structure to represent a connected, undirected
// and weighted graph
struct Graph {
// V-> Number of vertices, E-> Number of edges
int V, E;
// graph is represented as an array of edges.
// Since the graph is undirected, the edge
// from src to dest is also edge from dest
// to src. Both are counted as 1 edge here.
struct Edge* edge;
};
// Creates a graph with V vertices and E edges
struct Graph* createGraph(int V, int E)
{
struct Graph* graph = (struct Graph*)(malloc(sizeof(struct Graph)));
graph->V = V;
graph->E = E;
graph->edge = (struct Edge*)malloc(sizeof( struct Edge));
return graph;
}
// A structure to represent a subset for union-find
struct subset {
int parent;
int rank;
};
// A utility function to find set of an element i
// (uses path compression technique)
int find(struct 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(struct 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++;
}
}
// Compare two edges according to their weights.
// Used in qsort() for sorting an array of edges
int myComp(const void* a, const void* b)
{
struct Edge* a1 = (struct Edge*)a;
struct Edge* b1 = (struct Edge*)b;
return a1->weight > b1->weight;
}
// The main function to construct MST using Kruskal's
// algorithm
void KruskalMST(struct Graph* graph)
{
int V = graph->V;
struct Edge
result[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
// 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
qsort(graph->edge, graph->E, sizeof(graph->edge[0]),
myComp);
// Allocate memory for creating V ssubsets
struct subset* subsets
= (struct subset*)malloc(V * sizeof(struct subset));
// Create V subsets with single elements
for (int v = 0; v < V; ++v) {
subsets[v].parent = v;
subsets[v].rank = 0;
}
// Number of edges to be taken is equal to V-1
while (e < V - 1 && i < graph->E) {
// Step 2: Pick the smallest edge. And increment
// the index for next iteration
struct Edge next_edge = graph->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
printf(
"Following are the edges in the constructed MST\n");
int minimumCost = 0;
for (i = 0; i < e; ++i)
{
printf("%d -- %d == %d\n", result[i].src,
result[i].dest, result[i].weight);
minimumCost += result[i].weight;
}
printf("Minimum Cost Spanning tree : %d",minimumCost);
return;
}
// Driver program to test above functions
int main()
{
/* Let us create following weighted graph
10
0--------1
| \ |
6| 5\ |15
| \ |
2--------3
4 */
int V = 4; // Number of vertices in graph
int E = 5; // Number of edges in graph
struct Graph* graph = createGraph(V, E);
// add edge 0-1
graph->edge[0].src = 0;
graph->edge[0].dest = 1;
graph->edge[0].weight = 10;
// add edge 0-2
graph->edge[1].src = 0;
graph->edge[1].dest = 2;
graph->edge[1].weight = 6;
// add edge 0-3
graph->edge[2].src = 0;
graph->edge[2].dest = 3;
graph->edge[2].weight = 5;
// add edge 1-3
graph->edge[3].src = 1;
graph->edge[3].dest = 3;
graph->edge[3].weight = 15;
// add edge 2-3
graph->edge[4].src = 2;
graph->edge[4].dest = 3;
graph->edge[4].weight = 4;
KruskalMST(graph);
return 0;
} Java
// 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
{
int src, dest, weight;
// Comparator function used for
// sorting edgesbased 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()
{
// Tnis will store the resultant MST
Edge result[] = new Edge[V];
// An index variable, used for result[]
int e = 0;
// An index variable, used for sorted edges
int i = 0;
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 = 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");
int minimumCost = 0;
for (i = 0; i < e; ++i)
{
System.out.println(result[i].src + " -- "
+ result[i].dest
+ " == " + result[i].weight);
minimumCost += result[i].weight;
}
System.out.println("Minimum Cost Spanning Tree "
+ minimumCost);
}
// Driver Code
public static void main(String[] args)
{
/* Let us create following weighted graph
10
0--------1
| \ |
6| 5\ |15
| \ |
2--------3
4 */
int V = 4; // Number of vertices in graph
int E = 5; // Number of edges in graph
Graph graph = new Graph(V, E);
// add edge 0-1
graph.edge[0].src = 0;
graph.edge[0].dest = 1;
graph.edge[0].weight = 10;
// add edge 0-2
graph.edge[1].src = 0;
graph.edge[1].dest = 2;
graph.edge[1].weight = 6;
// add edge 0-3
graph.edge[2].src = 0;
graph.edge[2].dest = 3;
graph.edge[2].weight = 5;
// add edge 1-3
graph.edge[3].src = 1;
graph.edge[3].dest = 3;
graph.edge[3].weight = 15;
// add edge 2-3
graph.edge[4].src = 2;
graph.edge[4].dest = 3;
graph.edge[4].weight = 4;
// Function call
graph.KruskalMST();
}
}
// This code is contributed by Aakash Hasija Python
# Python program for Kruskal's algorithm to find
# Minimum Spanning Tree of a given connected,
# undirected and weighted graph
from collections import defaultdict
# Class to represent a graph
class Graph:
def __init__(self, vertices):
self.V = vertices # No. of vertices
self.graph = [] # default dictionary
# to store graph
# function to add an edge to graph
def addEdge(self, u, v, w):
self.graph.append([u, v, w])
# A utility function to find set of an element i
# (uses path compression technique)
def find(self, parent, i):
if parent[i] == i:
return i
return self.find(parent, parent[i])
# A function that does union of two sets of x and y
# (uses union by rank)
def union(self, parent, rank, x, y):
xroot = self.find(parent, x)
yroot = self.find(parent, y)
# Attach smaller rank tree under root of
# high rank tree (Union by Rank)
if rank[xroot] < rank[yroot]:
parent[xroot] = yroot
elif rank[xroot] > rank[yroot]:
parent[yroot] = xroot
# If ranks are same, then make one as root
# and increment its rank by one
else:
parent[yroot] = xroot
rank[xroot] += 1
# The main function to construct MST using Kruskal's
# algorithm
def KruskalMST(self):
result = [] # This will store the resultant MST
# An index variable, used for sorted edges
i = 0
# An index variable, used for result[]
e = 0
# 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 graph
self.graph = sorted(self.graph,
key=lambda item: item[2])
parent = []
rank = []
# Create V subsets with single elements
for node in range(self.V):
parent.append(node)
rank.append(0)
# Number of edges to be taken is equal to V-1
while e < self.V - 1:
# Step 2: Pick the smallest edge and increment
# the index for next iteration
u, v, w = self.graph[i]
i = i + 1
x = self.find(parent, u)
y = self.find(parent, v)
# If including this edge does't
# cause cycle, include it in result
# and increment the indexof result
# for next edge
if x != y:
e = e + 1
result.append([u, v, w])
self.union(parent, rank, x, y)
# Else discard the edge
minimumCost = 0
print ("Edges in the constructed MST")
for u, v, weight in result:
minimumCost += weight
print("%d -- %d == %d" % (u, v, weight))
print("Minimum Spanning Tree" , minimumCost)
# Driver code
g = Graph(4)
g.addEdge(0, 1, 10)
g.addEdge(0, 2, 6)
g.addEdge(0, 3, 5)
g.addEdge(1, 3, 15)
g.addEdge(2, 3, 4)
# Function call
g.KruskalMST()
# This code is contributed by Neelam YadavC#
// C# Code for above approach
using System;
class Graph {
// A class to represent a graph edge
class Edge : IComparable {
public 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
public class subset
{
public 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()
{
// This will store the
// resultant MST
Edge[] result = new Edge[V];
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
Array.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
Console.WriteLine("Following are the edges in "
+ "the constructed MST");
int minimumCost = 0
for (i = 0; i < e; ++i)
{
Console.WriteLine(result[i].src + " -- "
+ result[i].dest
+ " == " + result[i].weight);
minimumCost += result[i].weight;
}
Console.WriteLine("Minimum Cost Spanning Tree"
+ minimumCost);
Console.ReadLine();
}
// Driver Code
public static void Main(String[] args)
{
/* Let us create following weighted graph
10
0--------1
| \ |
6| 5\ |15
| \ |
2--------3
4 */
int V = 4; // Number of vertices in graph
int E = 5; // Number of edges in graph
Graph graph = new Graph(V, E);
// add edge 0-1
graph.edge[0].src = 0;
graph.edge[0].dest = 1;
graph.edge[0].weight = 10;
// add edge 0-2
graph.edge[1].src = 0;
graph.edge[1].dest = 2;
graph.edge[1].weight = 6;
// add edge 0-3
graph.edge[2].src = 0;
graph.edge[2].dest = 3;
graph.edge[2].weight = 5;
// add edge 1-3
graph.edge[3].src = 1;
graph.edge[3].dest = 3;
graph.edge[3].weight = 15;
// add edge 2-3
graph.edge[4].src = 2;
graph.edge[4].dest = 3;
graph.edge[4].weight = 4;
// Function call
graph.KruskalMST();
}
}
// This code is contributed by Aakash Hasija C++
// C++ program for Kruskal's algorithm
// to find Minimum Spanning Tree of a
// given connected, undirected and weighted
// graph
#include
using namespace std;
// a structure to represent a
// weighted edge in graph
class Edge {
public:
int src, dest, weight;
};
// a structure to represent a connected,
// undirected and weighted graph
class Graph {
public:
// V-> Number of vertices, E-> Number of edges
int V, E;
// graph is represented as an array of edges.
// Since the graph is undirected, the edge
// from src to dest is also edge from dest
// to src. Both are counted as 1 edge here.
Edge* edge;
};
// Creates a graph with V vertices and E edges
Graph* createGraph(int V, int E)
{
Graph* graph = new Graph;
graph->V = V;
graph->E = E;
graph->edge = new Edge[E];
return graph;
}
// A structure to represent a subset for union-find
class subset {
public:
int parent;
int rank;
};
// 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++;
}
}
// Compare two edges according to their weights.
// Used in qsort() for sorting an array of edges
int myComp(const void* a, const void* b)
{
Edge* a1 = (Edge*)a;
Edge* b1 = (Edge*)b;
return a1->weight > b1->weight;
}
// The main function to construct MST using Kruskal's
// algorithm
void KruskalMST(Graph* graph)
{
int V = graph->V;
Edge result[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
// 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
qsort(graph->edge, graph->E, sizeof(graph->edge[0]),
myComp);
// Allocate memory for creating V ssubsets
subset* subsets = new subset[(V * sizeof(subset))];
// Create V subsets with single elements
for (int v = 0; v < V; ++v)
{
subsets[v].parent = v;
subsets[v].rank = 0;
}
// Number of edges to be taken is equal to V-1
while (e < V - 1 && i < graph->E)
{
// Step 2: Pick the smallest edge. And increment
// the index for next iteration
Edge next_edge = graph->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
cout << "Following are the edges in the constructed "
"MST\n";
int minimumCost = 0;
for (i = 0; i < e; ++i)
{
cout << result[i].src << " -- " << result[i].dest
<< " == " << result[i].weight << endl;
minimumCost = minimumCost + result[i].weight;
}
// return;
cout << "Minimum Cost Spanning Tree: " << minimumCost
<< endl;
}
// Driver code
int main()
{
/* Let us create following weighted graph
10
0--------1
| \ |
6| 5\ |15
| \ |
2--------3
4 */
int V = 4; // Number of vertices in graph
int E = 5; // Number of edges in graph
Graph* graph = createGraph(V, E);
// add edge 0-1
graph->edge[0].src = 0;
graph->edge[0].dest = 1;
graph->edge[0].weight = 10;
// add edge 0-2
graph->edge[1].src = 0;
graph->edge[1].dest = 2;
graph->edge[1].weight = 6;
// add edge 0-3
graph->edge[2].src = 0;
graph->edge[2].dest = 3;
graph->edge[2].weight = 5;
// add edge 1-3
graph->edge[3].src = 1;
graph->edge[3].dest = 3;
graph->edge[3].weight = 15;
// add edge 2-3
graph->edge[4].src = 2;
graph->edge[4].dest = 3;
graph->edge[4].weight = 4;
// Function call
KruskalMST(graph);
return 0;
}
// This code is contributed by rathbhupendra 输出
Following are the edges in the constructed MST
2 -- 3 == 4
0 -- 3 == 5
0 -- 1 == 10
Minimum Cost Spanning Tree: 19时间复杂度： O(ElogE)或O(ElogV)。边缘排序需要O(ELogE)时间。排序后，我们遍历所有边缘并应用find-union算法。查找和联合操作最多需要O(LogV)时间。因此，总体复杂度为O(ELogE + ELogV)时间。 E的值最多为O(V 2 )，因此O(LogV)与O(LogE)相同。因此，整体时间复杂度为O(ElogE)或O(ElogV)