Assume there are n courses offered by the university, where each course has one or no prerequisites. If course j is the prerequisite for course i, then we are only allowed to take course i after course j. And after taking each course i, we can get a reward r_i. Given the reward and prerequisite for each course, design an algorithm to find the maximum total reward we can get by taking m courses. The time complexity should be O(nm^2).

Assume there are n courses offered by the university, where each course has one or no prerequisites. If course j is the prerequisite for course i, then we are only allowed to take course i after course j. And after taking each course i, we can get a reward r_i. Given the reward and prerequisite for each course, design an algorithm to find the maximum total reward we can get by taking m courses. The time complexity should be O(nm^2).

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter14: Game Theory

Section14.3: Linear Programming And Zero-sum Games

Problem 9P

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Assume there are n courses offered by the university, where each course has one or no prerequisites.If course j is the prerequisite for course i, then we are only allowed to take course i after course j. And after taking each course i, we can get a reward r_i. Given the reward and prerequisite for each course, design analgorithm to find the maximum total reward we can get by taking m courses. The time complexity shouldbe O(nm^2). Please only answer if you have the algorithm
Consider the following version of Knapsack. Given are two weight limits Wi and W2, where WW2. Given are also n items (wi, ci), (W2, C2)..... (Wn, Cn), where w, is the weight and c the cost of the i-th item. We want to find a subset of these items of the highest cost, where the subset weights at least W₁ and at most W2. Give an O(nW₂) algorithm for this problem. (Recall that the cost (respectively weight) of a subset is the sum of the costs (respectively weights) of the items in the subset.)
Consider the following version of Knapsack. Given are two weight limits W1 and W2, whereW1 ≤ W2. Given are also n items (w1, c1),(w2, c2), . . . ,(wn, cn), where wiis the weight and cithe cost of the i-th item. We want to find a subset of these items of the highest cost, wherethe subset weights at least W1 and at most W2. Give an O(nW2) algorithm for this problem.(Recall that the cost (respectively weight) of a subset is the sum of the costs (respectivelyweights) of the items in the subset.)
a. Given n items, where each item has a weight and a value, and a knapsack that can carry at most W You are expected to fill in the knapsack with a subset of items in order to maximize the total value without exceeding the weight limit. For instance, if n = 6 and items = {(A, 10, 40), (B, 50, 30), (C, 40, 80), (D, 20, 60), (E, 40, 10), (F, 10, 60)} where each entry is represented as (itemIdi, weighti, valuei). Use greedy algorithm to solve the fractional knapsack problem. b. Given an array of n numbers, write a java or python program to find the k largest numbers using a comparison-based algorithm. We are not interested in the relative order of the k numbers and assuming that (i) k is a small constant (e.g., k = 5) independent of n, and (ii) k is a constant fraction of n (e.g., k = n/4). Provide the Big-Oh characterization of your algorithm.
In this problem we have n jobs j1, j2, ..., jn, each has an associated deadline d1, d2, ..., dn and profit p1, p2, ..., pn. Profit will only be awarded or earned if the job is completed before the deadline. We assume that each job takes 1 unit of time to complete. The objective is to earn maximum profit when only one job can be scheduled or processed at any given time.Provide the pseudocode of an algorithm to find the sequence of jobs to do with the maximum total profit. Also describe the main idea of your algorithm using plain language.[Hint: You can select the jobs in a greedy way. You can use the following example to help your analysis.] Job J1 J2 J3 J4 J5 Deadline 2 1 3 2 1 Profit 60 100 20 40 20 The best job sequence would be J2 →J1 →J3.
A school is creating class schedules for its students. The students submit their requested courses and then a program will be designed to find the optimal schedule for all students. The school has determined that finding the absolute best schedule cannot be solved in a reasonable time. Instead they have decided to use a simpler algorithm that produces a good but non-optimal schedule in a more reasonable amount of time. Which principle does this decision best demonstrate? A. Unreasonable algorithms may sometimes also be undecidable B. Heuristics can be used to solve some problems for which no reasonable algorithm exists C. Two algorithms that solve the same problem must also have the same efficiency D. Approximate solutions are often identical to optimal solutions
Consider the challenge of determining whether a witness questioned by a law enforcement agency is telling the truth. An innovative questioning system pegs two individuals against each other. A reliable witness can determine whether the other individual is telling the truth. However, an unreliable witness's testimony is questionable. Given all the possible outcomes from the given scenarios, we obtain the table below. This pairwise approach could then be applied to a larger pool of witnesses. Answer the following: 1) If at least half of the K witnesses are reliable, the number of pairwise tests needed is Θ(n). Show the recurrence relation that models the problem. Provide a solution using your favorite programming language, that solves the recurrence, using initial values entered by the user.
Write the formula for calculating the optimal cut of the text t of length n, where we have m cut points given in the field R.TIP: In the case of the first question, R = [3, 7, 17].C) Record the algorithm for calculating the optimal cut and estimate the time and complexity.
Not every greedy approach works. For activity selection problem, your book gives examples which shows that the following greedy strategies don't work: the earliest start time, the shortest-duration activity, and the activity with the fewest overlaps. Give concrete examples to show that the 0-1 knapsack problem doesn’t work for: (a) the heaviest item first, (b) the lightest item first, (c) the most expensive item first, and (d) the least expensive item first.
Calculate the optimal path from g to a using A* algorithm (as heuristic function, use the provided table)
Correct answer will be upvoted else downvoted. Let C={c1,c2,… ,cm} (c1<c2<… <cm) be the arrangement of individuals who hold cardboards of 'C'. Let P={p1,p2,… ,pk} (p1<p2<… <pk) be the arrangement of individuals who hold cardboards of 'P'. The photograph is acceptable if and provided that it fulfills the accompanying requirements: C∪P={1,2,… ,n} C∩P=∅. ci−ci−1≤ci+1−ci(1<i<m). pi−pi−1≥pi+1−pi(1<i<k). Given a cluster a1,… ,an, kindly track down the number of good photographs fulfilling the accompanying condition: ∑x∈Cax<∑y∈Pay. The appropriate response can be huge, so output it modulo 998244353. Two photographs are unique if and provided that there exists no less than one individual who holds a cardboard of 'C' in one photograph yet holds a cardboard of 'P' in the other. Input Each test contains numerous experiments. The main line contains the number of experiments t (1≤t≤200000). Depiction of the experiments follows. The…
Consider the following version of Knapsack. Given are two weight limits W 1 and W 2 , whereW 1 ≤ W 2 . Given are also n items (w 1 , c 1 ), (w 2 , c 2 ), . . . , (w n , c n ), where w i is the weight and c ithe cost of the i-th item. We want to find a subset of these items of the highest cost, wherethe subset weights at least W 1 and at most W 2 . Give an O(nW 2 ) algorithm for this problem.(Recall that the cost (respectively weight) of a subset is the sum of the costs (respectivelyweights) of the items in the subset.)