Problem 5. Let S be a perhaps multi-set of n integers. Give an algorithm to determine whether 5 has k identical integers. Your algorithm should terminate in O(n) expected time, regardless of :. For example, suppose that S = {75,123, 65, 75, 9, 9, 32,9, 93}. Then the answer is put no if k > 4. yes if k < 3,
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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).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.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 0
- There are n people who want to carpool during m days. On day i, some subset ???? of people want to carpool, and the driver di must be selected from si . Each person j has a limited number of days fj they are willing to drive. Give an algorithm to find a driver assignment 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 network flow. For example, for the following input with n = 3 and m = 3, the algorithm could assign Tom to Day 1 and Day 2, and Mark to Day 3. Person Day 1 Day 2 Day 3 Driving Limit 1 (Tom) x x x 2 2 (Mark) x x 1 3 (Fred) x x 0Today 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…Let G: (0,1}" {0,1)"+l be defined as follows: G(x....x) = (X1, X Ox2, X O x2 O X3, ..., X1 O X2 O ... O x X). Prove or G is a PRG. disprove
- Consider array A of n numbers. We want to design a dynamic programming algorithm to find the maximum sum of consecutive numbers with at least one number. Clearly, if all numbers are positive, the maximum will be the sum of all the numbers. On the other hand, if all of them are negative, the maximum will be the largest negative number. The complexity of your dynamic programming algorithm must be O(n2). However, the running time of the most efficient algorithm is O(n). Design the most efficient algorithm you can and answer the following questions based on it. To get the full points you should design the O(n) algorithm. However, if you cannot do that, still answer the following questions based on your algorithm and you will get partial credit. Write the recursion that computes the optimal solution recursively based on the solu- (a) tion(s) to subproblem(s). Briefly explain how it computes the solution. Do not forget the base case(s) of your recursion.5. You are given a set of n positive numbers A = {a₁,..., an} and a positive integer t. Design a dynamic programming algorithm running in O(nt) time that decides whether there exists a subset A' CA such that Σ x = t. Note that each element of A can be xЄA' used at most once. Is the run-time of your algorithm polynomial with respect to the size of the input?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…
- Given A = {1,2,3} and B={u,v}, determine. a. A X B b. B X BSuppose S = {1, 2,...,n} and f : S ! S. If R ⇢ S, define f(R) = {f(x) | x 2 R}.Device an O(n) algorithm for determining the largest R ⇢ S, such that f(R) = R.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.