not handwritten Design a divide and conquer algorithm to solve the following problem in Θ(n log n) time: You are given a sequence A[1..n] of n numbers. Find i and j with i < j that maximizes A[j] − A[i].
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Design a divide and conquer
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- 7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).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?
- Let m be a randomly chosen non-negative integer having at most n decimal digits, i.e. an integer in the range 0 sms 10" - 1. Consider the following problem: determine m by asking only 5- way questions, i.e. questions with at most 5 possible responses. For instance, one could ask which of 5 specific sets m belongs to. Prove that any algorithm restricted to such questions, and which correctly solves this problem, runs in time Q(n).We have N jobs and N workers to do these jobs. It is known at what cost each worker will do each job (as a positive numerical value). We want to assign jobs to workers in such a way that the total cost of completion of all jobs is minimal among other possible alternative assignments. For this problem, write the algorithm as pseudocode, whose input is a matrix representing worker/job costs, and the output is a list of tuples showing which work will be done by which worker, and that tries to reach the solution with GREEDY technique. Explain in what sense your algorithm exhibits greedy behavior. What is the time complexity of your algorithm? Interpret if your algorithm always produces the best (optimum) result for each instance of the problem.1: Given a fixed integer B (B ≥ 2), we demonstrate that any integer N (N ≥ 0) can bewritten in a unique way in the form of the sum of p+1 terms as follows:N = a0 + a1×B + a2×B2 + … + ap×Bpwhere all ai, for 0 ≤ i ≤ p, are integer such that 0 ≤ ai ≤ B-1.The notation apap-1…a0 is called the representation of N in base B. Notice that a0 is theremainder of the Euclidean division of N by B. If Q is the quotient, a1 is the remainder of theEuclidean division of Q by B, etc.1. Write an algorithm that generates the representation of N in base B. 22. Compute the time complexity of your algorithm.
- 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…You are standing in front of an infinitely long straight fence; that is, the fence extends infinitely to your left and to your right. The fence has a single gate in it but you do not know where it is. Your goal is to minimize the distance you need to walk in order to find the gate. If n (which is unknown) is the distance to the gate in yards, design and an analyze an efficient algorithm for finding the gate in terms of n.Find the relation between the following functions: f(n) = log n and g(n) = Vn. (Square root for n)Hint: you may use L'Hopital's Theorem. For function f(n)=log n and time t=1 second, determine the largest size n of a problem that can be solved in time t, assume that the algorithm to solve the problem takes f(n) microseconds. Suppose you have algorithms with the two running times listed below. Suppose you have a computer that can perform 6 operations per second, and you need to compute a result in at most an hour of computation. For each of the algorithms, what is the largest input size n for which you would be able toget the result within an hour for:a) n^3b)10n^2
- 38. Suppose that, in a divide-and-conquer algorithm, we always divide an instance of size n of a problem into n subinstances of size n/3, and the dividing and combining steps take linear time. Write a recurrence equation for the running time T(n), and solve this recurrence equation for T(n). Show your solution in order notation.There are n people who want to carpool during m days. On day i, some subset si ofpeople want to carpool, and the driver di must be selected from si . Each person j hasa limited number of days fj they are willing to drive. Give an algorithm to find a driverassignment di ∈ si each day i such that no person j has to drive more than their limit fj. (The algorithm should output “no” if there is no such assignment.) Hint: Use networkflow.For example, for the following input with n = 3 and m = 3, the algorithm could assignTom to Day 1 and Day 2, and Mark to Day 3. Person Day 1 Day 2 Day 3 Limit 1 (Tom) x x x 2 2 (Mark) x x 1 3 (Fred) x x 0How many lines does his algorithm print? Write a recurrence and solve it. function printaton(n: an integer power of 2) { if n > 1 { printaton(n/2) printaton(n/2) printaton(n/2) for i = 1 to n ^ 4 do printline("are we done yet?") } } Use Master's theorem to obtain an asymptotic solution. Derive an exact solution by expanding the recurrence.