Greedy Algorithms | Set 9 (Boruvka’s algorithm)

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 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.
4

 
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}.
1
After above step, components are {{0,1}, {2,3,4,8}, {5,6,7}}. The components are encircled with blue color.
13

 
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}
14

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)
    {
        // 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[set1] == -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,
                       edge[cheapest[i]].weight);

                // 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;
}

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

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