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Chapter 14: Problem 50

In your own words, briefly describe how to find the minimum spanning tree using Kruskal's Algorithm.

Short Answer

Expert verified
Kruskal's Algorithm finds the minimum spanning tree by initializing an empty set for the tree, arranging all edges in ascending order, then continuously adding the shortest edge that doesn't form a cycle until the set holds (V - 1) edges where V is the number of vertices.

Step by step solution

01

Arrange All the Edges in Ascending Order

Start by listing all the edges in the graph in an increasing order based on their weights. Create a set that includes all the vertices in the graph.
02

Initializing the Minimum Spanning Tree (MST)

Initialize an empty set 'MST' which will store the edges included in the MST.
03

Adding the Shortest Edge

Add the smallest edge in the graph to the 'MST' set. Check that it doesn't form a cycle with the tree obtained so far. If cycle is not formed, include this edge, else, discard it.
04

Repeat Previous Step

Repeat Step 3 until there are (V - 1) edges in the MST set, where V is the number of vertices in the graph. The reason we look for (V - 1) edges in the MST is that for a connected, undirected graph with V vertices, a tree (and therefore also a Spanning Tree and Minimum Spanning Tree) will have (V - 1) edges.
05

Completion of Kruskal Algorithm

Once there are (V - 1) edges in the MST set, the algorithm is complete. The MST set now contains the edges of the minimum spanning tree.

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Most popular questions from this chapter

In Exercises 15-18, determine the number of Hamilton circuits in a complete graph with the given number of vertices. 3

Although the Nearest Neighbor Method does not always give the Hamilton circuit for which the sum of the weights is a minimum, Kruskal's Algorithm always gives the spanning tree with the smallest possible total weight.

Describe a situation involving relationships that can be modeled with a graph.

Group members should determine a relationship that exists among some, but not all, members. Did some of you know one another before the course began? Do some of you have the same academic major? Be as creative as possible in determining this relationship. Then create a graph that serves as a model for describing this relationship.

In Exercises 11-16, a graph with no loops or more than one edge between any two vertices is described. Which one of the following applies to the description? i. The described graph is a tree. ii. The described graph is not a tree. iii. The described graph may or may not be a tree. The graph has four vertices and three edges.
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