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Consider a variant of the activity-selection problem which asks for maximizing the product of the length of selected activities instead of the number of selected activities. For example, for the activities in the example below, an optimal solution selects activities as and a with value 3 x 2 = 6. a1 9₂ = 2 a) Devise a DP program to solve this variant of the activity selection problem. It suffices to complete the first two steps of the DP algorithm. That is, define subproblems, describe the optimal value for a larger subproblem as a function of the optimal value for smaller subproblems, and write down a recursive formula for the optimal value of a subproblem (remember to include the base case). As before, assume activities are sorted by their finish time. You may use pred(i) to denote the index of the last interval that finishes before a, starts (e.g., pred(as) = as in the above example). Assume indices start at 1, and let pred(i) = 0 if no activity ends before ai. b) Emma suggests the following greedy algorithm for the problem: repeatedly select the longest interval that does not overlap previously-selected intervals. Prove or disprove whether Emma's algorithm is optimal. You must either prove that the problem admits greedy-choice property or present a counter-example in which Emma's algorithm is not optimal.

Consider a variant of the activity-selection problem which asks for maximizing the product of the length of selected activities instead of the number of selected activities. For example, for the activities in the example below, an optimal solution selects activities as and a with value 3 x 2 = 6. a1 9₂ = 2 a) Devise a DP program to solve this variant of the activity selection problem. It suffices to complete the first two steps of the DP algorithm. That is, define subproblems, describe the optimal value for a larger subproblem as a function of the optimal value for smaller subproblems, and write down a recursive formula for the optimal value of a subproblem (remember to include the base case). As before, assume activities are sorted by their finish time. You may use pred(i) to denote the index of the last interval that finishes before a, starts (e.g., pred(as) = as in the above example). Assume indices start at 1, and let pred(i) = 0 if no activity ends before ai. b) Emma suggests the following greedy algorithm for the problem: repeatedly select the longest interval that does not overlap previously-selected intervals. Prove or disprove whether Emma's algorithm is optimal. You must either prove that the problem admits greedy-choice property or present a counter-example in which Emma's algorithm is not optimal.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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Consider a variant of the activity-selection problem which asks for maximizing the product of the length of selected activities instead of the number of selected activities. For example, for the activities in the example below, an optimal solution selects activities az and as with value 3 x 2 = 6. a1 1 4₂ = 2 43 = 3 a4 = 1 as = 1 46 = 2 a) Devise a DP program to solve this variant of the activity selection problem. It suffices to complete the first two steps of the DP algorithm. That is, define subproblems, describe the optimal value for a larger subproblem as a function of the optimal value for smaller subproblems, and write down a recursive formula for the optimal value of a subproblem (remember to include the base case). As before, assume activities are sorted by their finish time. You may use pred(i) to denote the index of the last interval that finishes before a; starts (e.g., pred(as) = as in the above example). Assume indices start at 1, and let pred(i) = 0 if no activity ends before ai. b) Emma suggests the following greedy algorithm for the problem: repeatedly select the longest interval that does not overlap previously-selected intervals. Prove or disprove whether Emma's algorithm is optimal. You must either prove that the problem admits greedy-choice property or present a counter-example in which Emma's algorithm is not optimal.
Transcribed Image Text:Consider a variant of the activity-selection problem which asks for maximizing the product of the length of selected activities instead of the number of selected activities. For example, for the activities in the example below, an optimal solution selects activities az and as with value 3 x 2 = 6. a1 1 4₂ = 2 43 = 3 a4 = 1 as = 1 46 = 2 a) Devise a DP program to solve this variant of the activity selection problem. It suffices to complete the first two steps of the DP algorithm. That is, define subproblems, describe the optimal value for a larger subproblem as a function of the optimal value for smaller subproblems, and write down a recursive formula for the optimal value of a subproblem (remember to include the base case). As before, assume activities are sorted by their finish time. You may use pred(i) to denote the index of the last interval that finishes before a; starts (e.g., pred(as) = as in the above example). Assume indices start at 1, and let pred(i) = 0 if no activity ends before ai. b) Emma suggests the following greedy algorithm for the problem: repeatedly select the longest interval that does not overlap previously-selected intervals. Prove or disprove whether Emma's algorithm is optimal. You must either prove that the problem admits greedy-choice property or present a counter-example in which Emma's algorithm is not optimal.
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