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Chapter 5: Problem 34
List three more applications of backtracking.
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Write a backtracking algorithm for the \(n\) -Queens Problem that uses a version of procedure expand instead of a version of procedure checknode.
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.
Write an algorithm for the 2 -coloring problem whose time complexity is not worst-case exponential in \(n\)
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.
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 \(n=8\)
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