Graph Algorithms

Question

What is the maximum and minimum number of edges to be considered in krushkal's algorithm with an example for both cases.

What I thought was since the Krushkal's algorithm is for finding minimum spanning tree the maximum number of edges is (V-1) where V is the number of vertices. Adding one more edge would result in a cycle in the graph. How can we obtain at a minimum value ?

Answer 1

A tree of N vertices always has N-1 edges. Consequently you have to consider at least N-1 edges during Kruskal's algorithm. An example may be a graph which is a tree.

Answer 2

Kruskal's algorithm stops when you've added V - 1 edges to your MST, so this is the minimum that have to be considered. This happens when the lowest value V - 1 edges of your graph do not form a cycle, and they will be added one after the other by the algorithm, after which it will stop.

For example, consider a complete graph with edges with cost 1 , which is minimum in the graph, between node 1 and every other node. Make all the other edges have cost 2 .

The worst case is when you must inspect every edge (of which there are O(V^2) ) until you finally select V - 1 . This means that you have to force a lot of cycles to be created before the last edge is added.

Consider a complete graph again. Have the V - 2 edges between node 1 and V - 2 nodes have cost 1, which is minimum in the graph. These will be selected first. Now let node k be the one that is not part of a selected edge, so that is left out of the graph. Have the edge between node k and node 1 have the largest cost. This will cause it to be inspected and added to the MST last, forcing the algorithm to inspect all O(V^2) edges before building the MST.

Remember the Kruskal's algorithm processes edges in increasing order of their cost, rejecting edges that would form a cycle if added to the MST we are building.