Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 9.3, Problem 8E
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Suppose we are given two sorted arrays A and B which each contain n elements. Give an O(log n) time divide-and-conquer algorithm which finds the median of A ∪ B.
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Sample Input:61 5 10 15 20 2552 4 5 9 15Sample Output :81
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Design an algorithm which counts the total number of inversions between A and B that runs in O(n log n) time.
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Introduction to Algorithms
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- Given an integer array [A], efficiently find a descending sorted triplet such that: A[i]> A[i]> A[k] and 0 <=i<j <k <n, where n is the array size. For example, Input: A[] = {9, 1, 6, 7, 3, 4, 5) Output: Any one of the following triplets: (9, 7,5) (9, 7, 4) (9, 7, 3) (9, 6, 5) (9, 6, 4) (9, 6, 3) a) Design a brute-force algorithm to solve this problem complexity of your solution b) Design a more efficient algorithm to do the same task with less complexity and analyse the complexity of your solution. c) Develop a python code to implement your efficient algorithm. depend on the correctness of the code, indentation, comments, test-case] d) Prepare a brief report (250 words) comparing the two algorithmsarrow_forwardLet M(n) be the minimum number of comparisons needed to sort an array A with exactly n ele- ments. For example, M(1) = 0, M(2) = 1, and M(4) = 4. If n is an even number, clearly explain why M(n) = 2M(n/2) + n/2.arrow_forward1. Consider a problem of computing the prefix average of a sequence of numbers stored in an array P consisting of P integers. We want to compute an array P such that P is the average of P for P . Consider the two algorithms related to the above problem. Answer the questions given belowarrow_forward
- An array of n elements is almost sorted if and only if every element is at most k spots away from its actual location. Assuming that you can only perform pairwise comparisons, formally prove that any algorithm which can sort almost sorted arrays must have running time S2(n log k), You may assume that n is a multiple of k.arrow_forwardGiven two arrays A and B of equal size N, the task is to find if given arrays are equal or not. Two arrays are said to be equal if both of them contain same set of elements, arrangements (or permutation) of elements may be different though.Note : If there are repetitions, then counts of repeated elements must also be same for two array to be equal. Example 1: Input: N = 5 A[] = {1,2,5,4,0} B[] = {2,4,5,0,1} Output: 1.arrow_forwardYou are given an unsorted array A[1, ..., n].We know that the maximum (or minimum) element in the array can be found in (n-1) comparisons.Write an algorithm that finds both the maximum and the minimum element in the array in ~(3n/2) comparisons, i.e., (3n/2) + THETA(1) comparisons.arrow_forward
- Let B be an array that is of size n >= 6 containing integers from 1 to n-5, inclusive, with exactly five repeated. Describe an O(n) algorithm for finding the five integers in B that are repeated.arrow_forward(c) Consider an n x n array ARR stored in memory consisting of 0’s and 1’s such that, in a row of ARR, all 0’s comes before any of 1’s in the row. Write an algorithm having complexity O(n), if exists, that finds the row that contains the most 0’s. Step by step explain your algorithm with an illustrative example.arrow_forwardAlgorthim of how to find the largest element missing in an unsorted array of n positive integers and the running time of the algorthim is in O(n). Example of this is that the n=6 array C= [5,90,8,6,26,9] The largest element missing in the array is 7arrow_forward
- We have k sorted arrays, and we know in total there are n elements in these k arrays. Design an algorithm to sort all n items in one single array. What is the time complexity?arrow_forwardFind the complexity of the function used to find the kth smallest integer in an unordered array of integers: int selectkth(int a[], int k, int n) {int i, j, mini, tmp; for (i = 0; i < k; i++) { mini = i; for (j = i+1; j < n; j++) if (a[j]<a[mini]) mini = j; tmp = a[i]; a[i] = a[mini]; a[mini] = tmp; }return a[k-1];}arrow_forwardshows how to conduct a Fibonacci search. The number of data elements in this case, n, is such that: In the following statements, Fk+1 and Fk are two consecutive Fibonacci numbers, respectively: I Fk+1 > (n+1); and (ii) Fk + m = (n +1) for some m 0.arrow_forward
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