Problem 55 Although the Nearest Neighbor Me... [FREE SOLUTION]

Chapter 14: Problem 55

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.

Short Answer

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Kruskal's Algorithm ensures the smallest possible total weight for a spanning tree by consistently selecting the smallest edge that does not form a cycle with the already selected edges. This is in contrast to the Nearest Neighbor method that makes locally optimal decisions, which may not result in the smallest total weight for the Hamilton circuit.

Step by step solution

Understand the Problem Context

First, realize that we are dealing with a weighted graph. In such a graph, a spanning tree is a tree that connects all the vertices of the graph. The minimum spanning tree (MST) is the spanning tree with the smallest possible total weight. Kruskal's algorithm is a procedure to find this MST.

Describe Kruskal鈥檚 Algorithm

Kruskal's algorithm starts by sorting all the edges from the lowest weight to the highest. One by one, it adds the edge with the lowest weight that does not form a cycle with the already included edges. The algorithm keeps adding edges until we have a spanning tree, meaning all vertices are connected.

Describe why Kruskal's Algorithm Ensures Minimum Total Weight

The reason Kruskal's algorithm achieves a minimum spanning tree is due to its greedy nature. By always choosing the smallest possible edge, and only including larger edges when they don't create cycles, the algorithm ensures that the selected edges will have the lowest sum of edge weights for a spanning tree. This is guaranteed by cut property in graph theory, that states that given any cut, if the edge crossing the cut is safe edge, then it would be included in MST.

Compare with Nearest Neighbor Method

Unlike Kruskal's algorithm, the Nearest Neighbor method does not guarantee the smallest total weight for the Hamilton circuit. It works by starting at a random vertex, then at each step, it extends the current circuit by visiting the vertex closest to the current one. This could lead to locally optimal choices but globally suboptimal solutions, hence not always providing the minimum total weight.

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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.

Short Answer

Step by step solution

Understand the Problem Context

Describe Kruskal鈥檚 Algorithm

Describe why Kruskal's Algorithm Ensures Minimum Total Weight

Compare with Nearest Neighbor Method

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