We are to complete a set of n jobs using m identical machines. Each job has a known duration. The goal is to schedule the jobs on m machines so that the time to finish the last job is as small as possible. Once a job is scheduled on a machine, it can’t be stopped and restarted. Any job can run on any machine. A greedy algorithm to solve this problem is as follows: sort jobs in decreasing order of duration. Schedule jobs one by one, choosing the machine where it can start the earliest. a. For m = 3 and n = 6 where the durations are <9, 12, 3, 8, 6, 5>, what is the finish time of the last job using greedy schedule? b. Exhibit an input for which the greedy schedule is NOT optimal

We are to complete a set of n jobs using m identical machines. Each job has a known duration. The goal is to schedule the jobs on m machines so that the time to finish the last job is as small as possible. Once a job is scheduled on a machine, it can’t be stopped and restarted. Any job can run on any machine. A greedy algorithm to solve this problem is as follows: sort jobs in decreasing order of duration. Schedule jobs one by one, choosing the machine where it can start the earliest. a. For m = 3 and n = 6 where the durations are <9, 12, 3, 8, 6, 5>, what is the finish time of the last job using greedy schedule? b. Exhibit an input for which the greedy schedule is NOT optimal

C++ Programming: From Problem Analysis to Program Design

8th Edition

ISBN:9781337102087

Author:D. S. Malik

Publisher:D. S. Malik

Chapter15: Recursion

Section: Chapter Questions

Problem 18SA

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You are given X number of projects with their start and end times. Devise a greedy algorithm that determines the max number of projects that can be performed by one person with the following assumptions: - A person can only work on a single project at a time. - Start and finish times are integers.
Consider a currency system that has the following coins and their values: dollar (100cents), quarter (25 cents), dime (10 cents), nickel (5 cents) and 1-cent coins. (A unit-valuecoin is always required). Suppose we want to give a change of value n cents in such a waythat the total number of coins n is minimized. Design a greedy algorithm to solve thisproblem.
we consider a problem where we are given a set of coins andour task is to form a sum of money n using the coins. The values of the coins arecoins = {c1, c2,..., ck}, and each coin can be used as many times we want. Whatis the minimum number of coins needed?For example, if the coins are the euro coins (in cents){1,2,5,10,20,50,100,200}and n = 520, we need at least four coins. The optimal solution is to select coins200+200+100+20 whose sum is 520.
A. One greedy strategy to solve the knapsack problem is to consider the items in order of decreasing values of where Vi and Si are the value and the size of item i. For each item, add it to the knapsack if it fits otherwise ignore it and move to the next item in the ordered list. Write a complete pseudocode for this strategy. The pseudocode should print the details of the solution (selected items, their total value and size). B. Show that this strategy does not necessarily yield optimal solutions. Take the example of exercise 1 and change the knapsack capacity to 15 (The value of the optimal solution is 54)..
There are n different sizes of boxes, from 1 to n. There is an unlimited supply of boxes of each size t, each with a value vt . A box of size t can hold several smaller boxes of sizes a1, a2, . . . , ak as long as the sum of sizes a1 + a2 + . . . + ak is strictly less than t. Each of these boxes may be filled with yet more boxes, and so on. Design an algorithm which runs in O(n2 ) time and finds the maximum value that can be attained by taking one box, potentially with smaller boxes nested inside it.
Find the minimum number of platforms in a train station to avoid delays in the arrival of a train. In other words, given a list of train arrival and departure times, determine in principle what would be the number of operating platforms to avoid delays in the arrival of a train. What would be the greedy strategy if we used a greedy algorithm to solve the problem? To exemplify, use the following information:Arrival = {2.00,2.10,3.00,3.20,3.50,5.00}Output = {2.30,3.40,3.20,4.30,4.00,5.20}
The classic example of following a greedy algorithm is making change. Let’s say you buy some items at the store and the change from your purchase is 63 cents. How does the clerk determine the change to give you? If the clerkfollows a greedy algorithm, he or she gives you two quarters, a dime, and three pennies. That is the smallest number of coins that will equal 63 cents (given that we don’t allow fifty-cent pieces). It has been proven that an optimal solution for coin changing can always be found using the current American denominations of coins. However, if we introduce some other denomination to the mix, the greedy algorithm doesn’t produce an optimal solution.Write a program that uses a greedy algorithm to make change (this code assumes change of less than one dollar).
Let's say there are n villages, {X1, . . . , Xn} on the country-road and we aim to build K < n restaurants to cover them. Each restaurant has to be built in a village, and we hope to minimize the average distance from each village to the closest restaurant. Please give an algorithm to compute the optimal way to place these K restaurants. The algorithm should run in O(k * n^2) time. Solutions with slightly higher time complexity also accepted.
Consider the following problem. The input consists of n skiers with heights p1, ··· , pn , and n skies with heights s1, ··· , sn. The problem is to assign each skier a ski to to minimize the average difference between the height of a skier and his/her assigned ski. That is, if the ith skier is given the Alpha(i)th ski, then you want to minimize: 1/ffln ∑i=1 to n |pi- s↵(i)| (a) Consider the following greedy algorithm. Find the skier and ski whose height di↵erence is minimized. Assign this skier this ski. Repeat the process until every skier has a ski. Prove of disprove that this algorithm is correct. (b) Consider the following greedy algorithm. Give the shortest skier the shortest ski, give the second shortest skier the second shortest ski, give the third shortest skier the third shortest ski, etc.Prove of disprove that this algorithm is correct. HINT: One of the above greedy algorithms is correct and one is incorrect for the other. The proof of correctness must be done…
Professor JAson uses the following algorithm for merging k sorted lists, each having n/k elements. He takes the first list and merges it with the second list using a linear-time algorithm for merging two sorted lists, such as the merging algorithm used in mergesort. Then, he merges the resulting list of 2n/k elements with the third list, merges the list of 3n/k elements that results with the fourth list, and so forth, until he ends up with a single sorted list of all n elements. Analyze the worst-case running time of the professor’s algorithm in terms of n and k.
The classic example of following a greedy algorithm is making change. Let’ssay you buy some items at the store and the change from your purchase is63 cents. How does the clerk determine the change to give you? If the clerkfollows a greedy algorithm, he or she gives you two quarters, a dime, andthree pennies. That is the smallest number of coins that will equal 63 cents(given that we don’t allow fifty-cent pieces).It has been proven that an optimal solution for coin changing can alwaysbe found using the current American denominations of coins. However, if weintroduce some other denomination to the mix, the greedy algorithm doesn’tproduce an optimal solution.Write a program that uses a greedy algorithm to make change (this codeassumes change of less than one dollar):
The Counting Coins problem is to count to a desired value by choosing the least possible coins and the greedy approach will force the algorithm to pick the largest possible coin. If we are provided coins of GH₵ 1, 2, 3 and 7 outline the procedure for counting the following coins based on the Greedy Algorithm.i. GH₵ 40ii. GH₵ 27