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Chapter 14: Problem 56

Why is the Brute Force Method impractical for large numbers of vertices?

Short Answer

Expert verified
The Brute Force Method is impractical for large numbers of vertices due to its factorial time complexity. The processing power required to analyze every permutation of vertices quickly becomes excessive as the number of vertices increases.

Step by step solution

01

Understand Brute Force Method

The Brute Force Method in graph theory requires examining all possible routes or sequencings of vertices to find the shortest path or optimal solution. Essentially, it is a trial-and-error process.
02

Identify the Complexity of Brute Force Method

For a graph with 'n' vertices, the brute force method would require analyzing each possible permutation of vertices. Given that there are n! (n factorial) permutations of n vertices, the time complexity of this algorithm is O(n!) which is, growing much more rapidly even than an exponential function.
03

Explain Computational Limits for Large Vertices

The factorial growth of computational steps makes this approach impractical for large number of vertices. Even with modern computational power, processing time becomes excessive. For instance, a graph with 20 vertices would require 20! (approximately 2.43 x 10^18) operations which are unbearable with current computational capabilities.

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Most popular questions from this chapter

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 nine vertices and six edges.

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

How can you look at a graph and determine if it has a Hamilton circuit?

What is meant by the degree of a graph's vertex and how is it determined?

In Exercises 5-6, draw two equivalent graphs for each description. The vertices are \(A, B, C\), and \(D\). The edges are \(A D, B C, D C\), \(B B\), and \(D B\).
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