Chapter 5: Problem 34
List three more applications of backtracking.
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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 \(n=8\)
Apply the Backtracking Algorithm for the \(n\) -Quecns Problem (Algorithm 5.1) to the problem instance in which \(n=8\), and show the actions step by step. Draw the pruned state space tree produced by this algorithm up to the point where the first solution is found.
List some of the practical applications that are representable in terms of the \(m-\) Coloring Problem.
Write a backtracking algorithm for the \(n\) -Queens Problem that uses a version of procedure expand instead of a version of procedure checknode.
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.
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