Problem 1. Assume we have an array A = {a₁,...,an) of n elements, where it is guaranteed a majority element exists. For example, (2, 1, 1) has a majority element 1. a. Describe a divide and conquer method to solve this problem. b. Write the run time as a recurrence sequence. c. Solve this recurrence sequence using any method introduced so far (expansion, generative function, master theorem).
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- Find a sorted matrix: The input is a real number x and a matrix A[1..n, 1..m] of nm real numbers, with each row A[i, 1..m] and column A[1..n, j] sorted. The objective is to locate the largest array entry A[i, j] that is less than or equal to x, or to report that all components of A are greater than x. Create and test an iterative method that analyses as few matrix elements as feasible. Be cautious if you assume a simple binary search would fix the problem. Find a sorted matrix: The input is a real number x and a matrix A[1..n, 1..m] of nm real numbers, with each row A[i, 1..m] and column A[1..n, j] sorted. The objective is to locate the largest array entry A[i, j] that is less than or equal to x, or to report that all components of A are greater than x. Create and test an iterative method that analyses as few matrix elements as feasible. Be cautious if you assume a simple binary search would fix the problem.For the problems given below, determine whether it is more efficient to use a divide and conquer strategy or a dynamic programming strategy, explain your reason. Give the recursion formula for each. (3*4=12) Find the average of a given set of unsorted numbers. Find whether there is a subset of integers in a given set that adds up to 8 Given a set of numbers, can they set be partitioned into two groups such that the sum of each group is equal; i.e 1,5,11,5 can be partitioned to 1,5,5 and 11Show how quicksort can be made to run in O(nlogn) time in the worst case. Assume the input array is A[0:n-1] and all elements in A are distinct. Write your answer as pseudo-code and use plain language to explain the idea of your algorithm. (Hint: you can use any algorithm helper functions in parts of the designed algorithm. If you use an algorithm, you can directly call the function name in the pseudo-code without expanding the details of the helper function, as long as you clearly explain the meaning of the helper function.)
- Give a clear description of an efficient algorithm for finding the k smallest elements of a very large n-element vector. Compare its running time with that of other plausible ways of achieving the same result, including that of applying k times your solution for part (a). [Note that in part (a) the result of the function consists of one element, whereas here it consists of k elements. As above, you may assume for simplicity that all the elements of the vector are different.]Problem 4: Let S = {s1, s2, . . . , sn} and T = {t1, t2, . . . , tm}, n ≤ m, be two sets of integers. (i) Describe a deterministic algorithm that checks whether S is a subset of T. What is the running time of your algorithm? (ii) Devise an algorithm that uses a hash table of length n to test whether S is a subset of T. What is the expected running time of your algorithm?Consider the problem of finding the smallest and largest elements in an array of n numbers. For solving this problem, design, respectively, a brute-force algorithm, a divide-and-conquer algorithm, and a presorting-based (transform-and-conquer) algorithm. determine and compare the efficiency in Big-O notation of the above three algorithms.
- Problem 2: Search an ArrayConsidering the search problem, we have a list of n integers A = ⟨v1, v2, · · · vn⟩. We wantto design an algorithm to check whether an item v exists or not such that it should returneither the index, i, if it was found or −1 otherwise, when not found.a) Implement the algorithm in Pythonb) Empirically, show the performance curve of the algorithm using time measurements.c) Using the basics of the theoretical analysis, write the complexity of its worst-casetime.d) Prove your algorithm is correct—it terminates with the valid output in every caseQuestion 2) 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.USING PYTHON A tridiagonal matrix is one where the only nonzero elements are the ones on the main diagonal (i.e., ai,j where j = i) and the ones immediately above and belowit(i.e.,ai,j wherej=i+1orj=i−1). Write a function that solves a linear system whose coefficient matrix is tridiag- onal. In this case, Gauss elimination can be made much more efficient because most elements are already zero and don’t need to be modified or added. Please show steps and explain.
- You are given an array of positive integers S[1, . . . , n]. You are initially positioned at the array’s first index 1, and each element S[i] represents your maximum length of jump at position i. For example, you can jump from position i to any position among i + 1, . . . , i + S[i]. Return true if you can reach the last index, or false otherwise. Precisely define the subproblem.Provide the recurrence equation.Describe the algorithm in pseudocode to compute the optimal value.Describe the algorithm in pseudocode to print out an optimal solution.Problem 2: Finding an element.(i) Let n be an even positive integer. Let A be an array containing n not necessarydifferent integers. Elements in A are not necessary sorted. Describe an algorithmwhich in time O(n) tests whether there exists an element x in A that appears in Aat least n/2 times.(ii) Let S be a set containing n integers. An element x ∈ S is an approximate median ofS, if|{y ∈ S | y < x}| ≥ n/4and |{y ∈ S | y > x}| ≥ n/4Given an array A containing n different integers, not necessary sorted, describe analgorithm which in time O(n) finds all approximate medians of the set {A[i] | i =1, 2, . . . , n}.Problem 4: Divide and conquer.(i) Let A be an array containing n different integers. Elements in A are sorted inincreasing order. Given an integer x describe an algorithm which in time O(log n)finds an index i of x in A, that is, A[i] = x, where the length n of A is unknown(you may assume that the implementation of an array is such that for A[i], wherei > n, a special character ∞ is returned).(ii) Let A be an array containing n different integers. Elements in A are not necessarysorted. Given an integer x describe an algorithm which in time O(n log n) testswhether there exist two elements u and v in A such that their sum is equal to x, thatis, u + v = x. The length n of A is known.