Assuming you possess a total of 'm' dollars, and are accompanied by a group of 'n' friends. For every friend i, where i ranges from 1 to n, the price P[i] of the candy that would bring contentment to the respective friend is known. The objective is to devise a method for allocating a sum of m dollars in a manner that maximizes the number of contented friends. Propose an O(n log n) time greedy algorithm for determining the monetary allocation to be assigned to each friend.
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Assuming you possess a total of 'm' dollars, and are accompanied by a group of 'n' friends. For every friend i, where i ranges from 1 to n, the price P[i] of the candy that would bring contentment to the respective friend is known. The objective is to devise a method for allocating a sum of m dollars in a manner that maximizes the number of contented friends. Propose an O(n log n) time greedy
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- You have m dollars and a group of n friends. For each friend 1 ≤ i ≤n, you know the price P[i] of the piece of candy that would make your friend happy. You want to find a way to distribute the m dollars such that as many of your friends as possible are happy. Design an O(n log n) time greedy algorithm to find how much money you will allocate each friend.group of n people are lying on the beach. The beach is represented by the real line R and the location of the i-th person is some integer x; e Z. Your task is to prevent people from getting sunburned by covering them with umbrellas. Each umbrella corresponds to a closed interval I = [a, a+ L] of length Le N, and the i-th person is covered by that umbrella if r; e I. Design a greedy algorithm for covering all people with the minimum number of umbrellas. The input consists of the integers x1,..., Xn, and L. The output of your algorithm should be the positions of umbrellas. A For example, if the input is x1 = 1, x2 = 3, x3 = 5, and L = 2, then an optimum solution is the set of two umbrellas placed at positions 2 and 5, covering intervals [1,3] and [4, 6]. %3D %3! %3D 3 4 5 6 1 2 The running time of your algorithm should be polynomial in n.(a) A student has been asked to put some parcels on a shelf. The parcels all weigh different amounts, and the shelf has a maximum safe loading weight capacity of 100 Kg. The weight of parcels are as follows (in kg): parcel 1 2 3 4 5 6 7 weight (kg) 8 50 2 15 4 5 20 The student has been asked to load the maximum weight possible parcels on the shelf subject to the maximum safe loading weight. State two possible approaches for a greedy algorithm solution to solve this problem. In each case, state clearly the result you would get from applying that approach to this problem, stating whether the solution is optimal or not. If 4 your answer does not produce an optimal solution, what algorithm could be emploved to find one?
- An electrician has wired n lights, all initially on, so that: 1) light 1 can always be turned on/off, and 2) for k > 1, light k cannot be turned either on or off unless light k – 1 is on and all preceding lights are off for k > 1. The question we want to explore is the following: how many moves are required to turn all n lights off? For n = 5, a solution sequence has been worked out below. Fill in the missing entries. The lights are counted from left to right, so the first bit is the first light, and so on. 11111 01111 11011 10011 00010 10010 11010We 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 given three ropes with lengths n₁, n2, and n3. Our goal is to find the smallest value k such that we can fully cover the three ropes with smaller ropes of lengths 1,2,3,...,k (one rope from each length). For example, as the figure below shows, when n₁ = 5, n₂ 7, and n3 = 9, it is possible to cover all three ropes with smaller ropes of lengths 1, 2, 3, 4, 5, 6, that is, the output should be k = 6. = Devise a dynamic-programming solution that receives the three values of n₁, n2, and n3 and outputs k. It suffices to show Steps 1 and 2 in the DP paradigm in your solution. In Step 1, you must specify the subproblems, and how the value of the optimal solutions for smaller subproblems can be used to describe those of large subproblems. In Step 2, you must write down a recursive formula for the minimum number of operations to reconfigure. Hint: You may assume the value of k is guessed as kg, and solve the decision problem that asks whether ropes of lengths n₁, n2, n3 can be covered by…
- A student has been asked to put some parcels on a shelf. The parcels all weigh different amounts, and the shelf has a maximum safe loading weight capacity of 150 Kg. The weight of parcels are as follows (in Kg): The student has been asked to load the maximum weight possible parcels on the shelf subject to the maximum safe loading weight. State two possible approaches for a greedy algorithm solution to solve this problem. In each case, state clearly the result you would get from applying that approach to this problem, stating whether the solution is optimal or not. If your answer does not produce an optimal solution, what algorithm could be employed to find one?Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you ll in the values is the correct one.Today is Max's birthday. He has ordered a rectangular fruit cake which is divided into N x M pieces. Each piece of the cake contains a different fruit numbered from 1 to N*M. He has invited K friends, each of whom have brought a list of their favorite fruit choices. A friend goes home happy if the piece he receives is of his favorite fruit. Note that each friend can receive only one piece of cake. Design a way for Max to find the maximum number of friends he can make happy. Input The first line of the input consists of an integer - numOfFriends, representing the number of friends(k). The next Klines consist of X+1 space-separated integers, where the first integer represents the count of choices of the th friend followed by X space-separated integers representing the fruits he likes. The next line of the input consists of an integer - numN, representing the number of rows. The next line of the input consists of an integer - numM, representing the number of columns. Output Print an…
- There are n students who studied at a late-night study for final exam. The time has come to order pizzas. Each student has his own list of required toppings (e.g. mushroom, pepperoni, onions, garlic, sausage, etc). Everyone wants to eat at least half a pizza, and the topping of that pizza must be in his reqired list. A pizza may have only one topping. How to compute the minimum number of pizzas to order to make everyone happy?Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Prove that the coin changing problem exhibits optimal substructure. Design a recursive backtracking (brute-force) algorithm that returns the minimum number of coins needed to make change for n cents for any set of k different coin denominations. Write down the pseudocode and prove that your algorithm is correct.Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…