Simulate the following scenario to find the maximum profit and the optimal set of items using 0/1 Knapsack algorithm. Knapsack Size Item 15 Profit Weight A 25 5 В 33 4 15 2 D 20 3 21
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- Consider 7 items along their respective weights and values I = I1, I2, I3, I4, I5, I6, I7W = 2,3,5,7,1,4,1V = 10,5,15,7,6,18,3 The capacity of knapsack W=15. Find the optimal solution of the fractional knapsackproblem. What is the complexity of finding the solution of fractional knapsack problem?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.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.)
- 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.)Assume you are given four objects, and you should fit as many of them into a knapsack of size 15as possible. The resulting value of the objects should be as large as possible.1. Please select your objects using the Greedy principle according to some greedy criterion youmay choose yourself (please make clear which criterion you’ve used). Is this solution optimal?2. Please explain and apply the principle of dynamic programming using the example above.OverlapGreedy(A) S <- A Best <- 0 while (S != 0) begin Select an activity a within S that overlaps with minimum number of other activities in S Best <- Best U {a} Delete all activities from S that overlap with a end Does OverlapGreedy always compute the optimum solution? If yes, provide a proof. If no, provide a counter example.
- 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 problem of providing change for an amount of n-bahts while using the smallest possible quantity of coins. You may assume that each coin’s value is integer. Assume that the coins available for use have values that can be expressed as powers of c. To clarify, the denominations of the coins can be written as c0,c1,..., ck , where c is an integer greater than 1, and k is a positive integer. Prove that the greedy algorithm always yields an optimal solution.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.
- Find an optimal solution to the fractional knapsack problem for an Instancewith number of items 5, Capacity of the sack W=10, profit associated with the items (p1,p2,p3,p4,p5)= (20,5,15,10,30) and weight associated with eachitem (w1,w2,w3,w4,w5)= (2,3,5,7,8).Apply the bottom-up dynamic programming algorithm to the followinginstance of the knapsack problem: Item Weight Value 1 3 $25 2 2 $30 3 1 $15 4 4 $40 5 5 $50 Capacity W = 8.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