1. Let \( S = \{1, 2, \ldots, n\} \) be a set of activities, where activity \( i \in S \) has a start time \( s_i \), a finish time \( f_i \), and a profit \( p_i \) associated with it. Design a dynamic programming algorithm for computing a set of compatible activities of maximum profit. Argue that the time complexity of your algorithm is \( O(n \log(n)) \). (You must start with the correct definitions, and then write a recurrence relation.)

Answered: Let S = {1,2...n} be a set activities,…

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

Problem 1PE

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1. Let \( S = \{1, 2, \ldots, n\} \) be a set of activities, where activity \( i \in S \) has a start time \( s_i \), a finish time \( f_i \), and a profit \( p_i \) associated with it. Design a dynamic programming algorithm for computing a set of compatible activities of maximum profit. Argue that the time complexity of your algorithm is \( O(n \log(n)) \). (You must start with the correct definitions, and then write a recurrence relation.)

Expert Solution

Step 1

Let p(j) be the maximum profit and i < j such that interval i does not overlap with j.

Let DPA be the dynamic programming algorithm solution.

Let n be the last interval.

The algorithm is as follows:

Sort the jobs by non-decreasing finish times.
For every i from 1 to n, determine the maximum profit of the activity from the set of activities[0..i]. Compare the inclusion of activity[i] for scheduling the exclusion of activity[i] to the schedule, and then take the maximum profit.

Use binary search to find the latest activity that does not conflict with activity[i].

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