¿Qué es un árbol de expansión?
Un árbol de expansión es un subconjunto de un gráfico conectado G, donde todas las aristas están conectadas, es decir, podemos atravesar cualquier arista desde una arista particular con o sin intermediarios. Además, un árbol de expansión no debe tener ningún ciclo. Así podemos decir que si hay n vértices en un grafo conexo entonces el no. de aristas que puede tener un árbol de expansión es n-1.
¿Qué es el árbol de expansión mínimo?
Dado un grafo conexo y no dirigido, un árbol de expansión de ese grafo es un subgrafo que es un árbol y conecta todos los vértices entre sí. Un solo gráfico puede tener muchos árboles de expansión diferentes. Un árbol de expansión mínimo (MST) o un árbol de expansión de peso mínimo para un gráfico ponderado, conectado y no dirigido es un árbol de expansión con un peso menor o igual que el peso de cualquier otro árbol de expansión. El peso de un árbol de expansión es la suma de los pesos asignados a cada borde del árbol de expansión.
¿Cuántas aristas tiene un árbol de expansión mínimo?
Un árbol de expansión mínimo tiene (V – 1) aristas donde V es el número de vértices en el gráfico dado.
¿Cuáles son las aplicaciones del árbol de expansión mínimo?
Vea esto para aplicaciones de MST.
A continuación se muestran los pasos para encontrar MST usando el algoritmo de Kruskal
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<Edge> { 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() { // This 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 subsets 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 doesn'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
Python3
# 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 doesn'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 Yadav
C#
// C# Code for above approach using System; class Graph { // A class to represent a graph edge class Edge : IComparable<Edge> { 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 subsets 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 doesn'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++
#include <bits/stdc++.h> using namespace std; // DSU data structure // path compression + rank by union class DSU { int* parent; int* rank; public: DSU(int n) { parent = new int[n]; rank = new int[n]; for (int i = 0; i < n; i++) { parent[i] = -1; rank[i] = 1; } } // Find function int find(int i) { if (parent[i] == -1) return i; return parent[i] = find(parent[i]); } // union function void unite(int x, int y) { int s1 = find(x); int s2 = find(y); if (s1 != s2) { if (rank[s1] < rank[s2]) { parent[s1] = s2; rank[s2] += rank[s1]; } else { parent[s2] = s1; rank[s1] += rank[s2]; } } } }; class Graph { vector<vector<int> > edgelist; int V; public: Graph(int V) { this->V = V; } void addEdge(int x, int y, int w) { edgelist.push_back({ w, x, y }); } void kruskals_mst() { // 1. Sort all edges sort(edgelist.begin(), edgelist.end()); // Initialize the DSU DSU s(V); int ans = 0; cout << "Following are the edges in the " "constructed MST" << endl; for (auto edge : edgelist) { int w = edge[0]; int x = edge[1]; int y = edge[2]; // take that edge in MST if it does form a cycle if (s.find(x) != s.find(y)) { s.unite(x, y); ans += w; cout << x << " -- " << y << " == " << w << endl; } } cout << "Minimum Cost Spanning Tree: " << ans; } }; int main() { /* Let us create following weighted graph 10 0--------1 | \ | 6| 5\ |15 | \ | 2--------3 4 */ Graph g(4); g.addEdge(0, 1, 10); g.addEdge(1, 3, 15); g.addEdge(2, 3, 4); g.addEdge(2, 0, 6); g.addEdge(0, 3, 5); // int n, m; // cin >> n >> m; // Graph g(n); // for (int i = 0; i < m; i++) // { // int x, y, w; // cin >> x >> y >> w; // g.addEdge(x, y, w); // } g.kruskals_mst(); return 0; }
Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA