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Chapter 4: Problem 9

Assume that in a network of computers any two computers can be linked. Given a cost estimate for each possible link, should Algorithm 4.1 (Prim's algorithm) or Algorithm 4.2 (Kruskal's algorithm) be used? Justify your answer.

Short Answer

Expert verified
The choice between Prim's and Kruskal's algorithm depends on the nature of the network. Prim's algorithm is usually more efficient on dense graphs, while Kruskal's is better for sparse graphs. However, if the cost to evaluate or compare links is high, Kruskal's algorithm could be the better option since it only considers each link once.

Step by step solution

01

Understanding The Context

Firstly, we need to define what we're trying to achieve - to connect all computers in the network with the minimum cost. Any two computers can be linked and we are given a cost estimate for each possible link.
02

Prim's Algorithm Explanation

This algorithm works by 'growing' the spanning tree from a starting node by always adding the nearest node outside the tree. It deals with the graph as a whole and should be used when we have a dense graph where each node is connected to many others.
03

Kruskal's Algorithm Explanation

Kruskal's algorithm operates by sorting all the edges from the lowest cost to the highest, then adding them one at a time to the tree if they don't form a cycle with the edges already included. It treats each node as an isolated tree and should be used in the case of a sparse graph where each node isn't connected to many others.
04

Choosing the Appropriate Algorithm

The choice of algorithm actually depends on the nature of the network. Neither algorithm is superior in all scenarios, but generally, Prim's tends to perform better on dense graphs, while Kruskal's performs better on sparse graphs. However, if the cost of calculating or comparing the links is considerably high, Kruskal's might be better since it considers each link only once, unlike Prim's which re-assesses after adding each node.

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