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Chapter 5: Problem 41

Modify the Backtracking algorithm for the \(0-1\) Knapsack problem (Algorithm 5.7) to produce a solution in a variable-length list.

Short Answer

Expert verified
The Backtracking algorithm for the \(0-1\) Knapsack problem can be modified to output a variable-length list by starting with an empty list and appending included items as you iterate through available items. The list expands and shrinks dynamically as per the backtracking path in the algorithm.

Step by step solution

01

Understand the Existing Backtracking Algorithm for the Knapsack Problem

Initially, have a clear understanding of the existing algorithm. The algorithm works by creating a tree of solutions, starting with a root node, and attempting to add each item to the knapsack. If adding the item doesn't exceed the weight limit, a child node is created. This continues until all possible combinations of items have been tried.
02

Modification to Variable-Length List

Currently, the algorithm produces a solution in an array of fixed size. But, here it needs to output the solution in a variable-length list. So, instead of filling up a fixed array of size \(n\), you'll be appending the included weights and values into a list. Starting with an empty list, append the included items as you iterate through available items.
03

Implementing the Changes

Translate this logical modification into a programming language like Python or Java. Ensure to initialize the solution as an empty list at start and then append each chosen item during backtracking. The stopping condition remains the same i.e., when the total weight exceeds the capacity or when there are no more items to consider, the algorithm backtracks and tries a different combination.

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Most popular questions from this chapter

List three more applications of backtracking.

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 \(n=5 ?\) Can we 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.

Use the Backtracking algorithm for the \(0-1\) Knapsack problem (Algorithm 5.7) to maximize the profit for the following problem instance. Show the actions step by step. $$\begin{array}{ccccc} i & p_{i} & w_{i} & \frac{p_{i}}{w_{i}} & \\ 1 & \$ 20 & 2 & 10 & \\ 2 & \$ 30 & 5 & 6 & \\ 3 & \$ 35 & 7 & 5 & W=19 \\ 4 & \$ 12 & 3 & 4 & \\ 5 & \$ 3 & 1 & 3 & \end{array}$$

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.

Write a backtracking algorithm for the \(n\) -Queens problem that uses a version of procedure expand instead ofa version of procedure checknode.
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