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Boruvka’s algorithm | Greedy Algo-9

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  • Difficulty Level : Hard
  • Last Updated : 27 Jun, 2022
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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. 
 

Boruvka's algorithm Image 1

Initially MST is empty. Every vertex is singe component as highlighted in blue color in below diagram. 
 

Boruvka's algorithm Image 2

  
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}. 
 

Boruvka's algorithm Image 3

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

Boruvka's algorithm Image 4

  
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} 
 

Boruvka's algorithm Image 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. 

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
        # components 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 with weight 5 included in MST
Edge 0-1 with weight 10 included in MST
Edge 2-3 with weight 4 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.


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