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Chapter 5: Problem 22

List some of the practical applications that are representable in terms of the \(m-\) Coloring Problem.

Short Answer

Expert verified
Some practical applications of the \(m-\) Coloring Problem include Map Coloring, Scheduling Problems (for example, class scheduling in schools and universities), and Frequency Assignment in telecommunication industry to avoid interference.

Step by step solution

01

Understanding the \(m-\) Coloring Problem

In graph theory, the \(m-\) coloring problem is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color. There are only \(m\) colors available for this task.
02

Application 1: Map Coloring

The concept of the \(m-\) Coloring Problem is used in Map Coloring. For example, you want to color a map in such a way that no two adjacent regions have the same color. This problem can be posed as a \(m-\) Coloring Problem where each region represents a vertex of the graph. The rule that no two adjacent vertices can share the same color ensures that no two neighboring regions will have the same color.
03

Application 2: Scheduling Problems

When it comes to scheduling tasks, \(m-\) Coloring Problem finds its application very well. For example, in time-tabling for schools or universities, we need to ensure that no two classes, which include common students or teachers, are scheduled at the same time. This requirement can be modeled as the \(m-\) Coloring Problem, where vertices represent classes and an edge connects two vertices if they have any common students or teachers. The task then is to color these vertices with minimum colors so that no two adjacent vertices (or coinciding classes) are assigned the same color (or scheduled at the same time).
04

Application 3: Frequency Assignment

The \(m-\) Coloring Problem finds its application in the telecommunications industry in frequency assignment problems. Each tower needs to be assigned a frequency such that no two towers within some geographic proximity are assigned the same frequency to avoid interference. This ends up becoming an \(m-\) Coloring Problem where each tower represents a vertex of the graph and an edge connects two vertices if they are within a certain proximity threshold. The aim is to assign frequencies represented by colors to these vertices such that no two adjacent vertices have the same color, ensuring no interference.

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Most popular questions from this chapter

Use the Backtracking Algorithm for the Sum-of-Subsets Problem (Algorithm 5.4) to find all combinations of the following numbers that sum to \(W=52\) \(w_{1}=2 \quad w_{2}=10 \quad w_{3}=13 \quad w_{4}=17 \quad w_{3}=22 \quad w_{6}=42\) Show the actions step by step.

Write an algorithm for the 2 -coloring problem whose time complexity is not worst-case exponential in \(n\)

Suppose that to color a graph properly we choose a starting vertex and a color to color as many vertices as possible. Then we select a new color and a new uncolored vertex to color as many more vertices as possible. We repeat this process until all the vertices of the graph are colored or all the colors are exhausted. Write an algorithm for this greedy approach to color a graph of \(n\) vertices. Analyze this algorithm and show the results using order notation.

Given an \(n \times n \times n\) cube containing \(n^{3}\) cells, we are to place \(n\) queens in the cube so that no two queens challenge each other (so that no two queens are in the same row, column, or diagonal). Can the \(n\) -Queens Algorithm (Algorithm 5.1 ) be extended to solve this problem? If so, write the algorithm and implement it on your system to solve problem instances in which \(n=4\) and \(n=8\)

Suppose we have a solution to the \(n\) -Queens Problem instance in which \(n=4,\) Can we extend this solution to find a solution to the problem instance in which \(n=5,\) then use the solutions for \(n=4\) and \(n=5\) to construct a solution to the instance in which \(n=6,\) and continue this dynamic programming approach to find a solution to any instance in which \(n>4 ?\) Justify your answer.
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