We have discussed following topics on Minimum Spanning Tree.
Applications of Minimum Spanning Tree Problem
Kruskal’s Minimum Spanning Tree Algorithm
Prim’s Minimum Spanning Tree Algorithm
In this post, Boruvka’s algorithm is discussed. Like Prim’s and Kruskal’s, Boruvka’s algorithm is also a Greedy algorithm. Below is complete algorithm.
1) Input is a connected, weighted and un-directed graph.
2) Initialize all vertices as individual components (or sets).
3) Initialize MST as empty.
4) While there are more than one components, do following
for each component.
a) Find the closest weight edge that connects this
component to any other component.
b) Add this closest edge to MST if not already added.
5) Return MST.
Below is the idea behind above algorithm (The idea is same as Prim’s MST algorithm).
A spanning tree means all vertices must be connected. So the two disjoint subsets (discussed above) of vertices must be connected to make a Spanning Tree. And they must be connected with the minimum weight edge to make it a Minimum Spanning Tree.
Let us understand the algorithm with below example.
Initially MST is empty. Every vertex is singe component as highlighted in blue color in below diagram.
For every component, find the cheapest edge that connects it to some other component.
Component Cheapest Edge that connects
it to some other component
{0} 0-1
{1} 0-1
{2} 2-8
{3} 2-3
{4} 3-4
{5} 5-6
{6} 6-7
{7} 6-7
{8} 2-8
The cheapest edges are highlighted with green color. Now MST becomes {0-1, 2-8, 2-3, 3-4, 5-6, 6-7}.
After above step, components are {{0,1}, {2,3,4,8}, {5,6,7}}. The components are encircled with blue color.
We again repeat the step, i.e., for every component, find the cheapest edge that connects it to some other component.
Component Cheapest Edge that connects
it to some other component
{0,1} 1-2 (or 0-7)
{2,3,4,8} 2-5
{5,6,7} 2-5
The cheapest edges are highlighted with green color. Now MST becomes {0-1, 2-8, 2-3, 3-4, 5-6, 6-7, 1-2, 2-5}
At this stage, there is only one component {0, 1, 2, 3, 4, 5, 6, 7, 8} which has all edges. Since there is only one component left, we stop and return MST.
Implementation:
Below is implementation of above algorithm. The input graph is represented as a collection of edges and union-find data structure is used to keep track of components.
C/C++
// Boruvka's algorithm to find Minimum Spanning // Tree of a given connected, undirected and // weighted graph #include <stdio.h> // 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 as a collection of edges. 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. Edge* edge; }; // A structure to represent a subset for union-find struct subset { int parent; int rank; }; // Function prototypes for union-find (These functions are defined // after boruvkaMST() ) int find(struct subset subsets[], int i); void Union(struct subset subsets[], int x, int y); // The main function for MST using Boruvka's algorithm void boruvkaMST(struct Graph* graph) { // Get data of given graph int V = graph->V, E = graph->E; Edge *edge = graph->edge; // Allocate memory for creating V subsets. struct subset *subsets = new subset[V]; // An array to store index of the cheapest edge of // subset. The stored index for indexing array 'edge[]' int *cheapest = new int[V]; // Create V subsets with single elements for (int v = 0; v < V; ++v) { subsets[v].parent = v; subsets[v].rank = 0; cheapest[v] = -1; } // Initially there are V different trees. // Finally there will be one tree that will be MST int numTrees = V; int MSTweight = 0; // Keep combining components (or sets) until all // compnentes are not combined into single MST. while (numTrees > 1) { // Everytime initialize cheapest array for (int v = 0; v < V; ++v) { cheapest[v] = -1; } // Traverse through all edges and update // cheapest of every component for (int i=0; i<E; i++) { // Find components (or sets) of two corners // of current edge int set1 = find(subsets, edge[i].src); int set2 = find(subsets, edge[i].dest); // If two corners of current edge belong to // same set, ignore current edge if (set1 == set2) continue; // Else check if current edge is closer to previous // cheapest edges of set1 and set2 else { if (cheapest[set1] == -1 || edge[cheapest[set1]].weight > edge[i].weight) cheapest[set1] = i; if (cheapest[set2] == -1 || edge[cheapest[set2]].weight > edge[i].weight) cheapest[set2] = i; } } // Consider the above picked cheapest edges and add them // to MST for (int i=0; i<V; i++) { // Check if cheapest for current set exists if (cheapest[i] != -1) { int set1 = find(subsets, edge[cheapest[i]].src); int set2 = find(subsets, edge[cheapest[i]].dest); if (set1 == set2) continue; MSTweight += edge[cheapest[i]].weight; printf("Edge %d-%d included in MST\n", edge[cheapest[i]].src, edge[cheapest[i]].dest); // Do a union of set1 and set2 and decrease number // of trees Union(subsets, set1, set2); numTrees--; } } } printf("Weight of MST is %d\n", MSTweight); return; } // Creates a graph with V vertices and E edges struct Graph* createGraph(int V, int E) { Graph* graph = new Graph; graph->V = V; graph->E = E; graph->edge = new Edge[E]; return graph; } // 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++; } } // 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; boruvkaMST(graph); return 0; } // Thanks to Anukul Chand for modifying above code. |
Python
# Boruvka'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 boruvkaMST(self): parent = []; rank = []; # An array to store index of the cheapest edge of # subset. It store [u,v,w] for each component cheapest =[] # Initially there are V different trees. # Finally there will be one tree that will be MST numTrees = self.V MSTweight = 0 # Create V subsets with single elements for node in range(self.V): parent.append(node) rank.append(0) cheapest =[-1] * self.V # Keep combining components (or sets) until all # compnentes are not combined into single MST while numTrees > 1: # Traverse through all edges and update # cheapest of every component for i in range(len(self.graph)): # Find components (or sets) of two corners # of current edge u,v,w = self.graph[i] set1 = self.find(parent, u) set2 = self.find(parent ,v) # If two corners of current edge belong to # same set, ignore current edge. Else check if # current edge is closer to previous # cheapest edges of set1 and set2 if set1 != set2: if cheapest[set1] == -1 or cheapest[set1][2] > w : cheapest[set1] = [u,v,w] if cheapest[set2] == -1 or cheapest[set2][2] > w : cheapest[set2] = [u,v,w] # Consider the above picked cheapest edges and add them # to MST for node in range(self.V): #Check if cheapest for current set exists if cheapest[node] != -1: u,v,w = cheapest[node] set1 = self.find(parent, u) set2 = self.find(parent ,v) if set1 != set2 : MSTweight += w self.union(parent, rank, set1, set2) print ("Edge %d-%d with weight %d included in MST" % (u,v,w)) numTrees = numTrees - 1 #reset cheapest array cheapest =[-1] * self.V print ("Weight of MST is %d" % MSTweight) 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) g.boruvkaMST() #This code is contributed by Neelam Yadav |
Output:
Edge 0-3 included in MST Edge 0-1 included in MST Edge 2-3 included in MST Weight of MST is 19
Interesting Facts about Boruvka’s algorithm:
1) Time Complexity of Boruvka’s algorithm is O(E log V) which is same as Kruskal’s and Prim’s algorithms.
2) Boruvka’s algorithm is used as a step in a faster randomized algorithm that works in linear time O(E).
3) Boruvka’s algorithm is the oldest minimum spanning tree algorithm was discovered by Boruuvka in 1926, long before computers even existed. The algorithm was published as a method of constructing an efficient electricity network.
Exercise:
The above code assumes that input graph is connected and it fails if a disconnected graph is given. Extend the above algorithm so that it works for a disconnected graph also and produces a forest.
References:
http://en.wikipedia.org/wiki/Bor%C5%AFvka%27s_algorithm
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
