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 9.2, Problem 4E
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To describe a sequence of partitions that results in a worst-case performance of RANDOMIZED-SELECT for the array
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The worst-case scenario for the fast sort algorithm occurs when each partition consists of a single empty subarray, all partitions are of equal size, and the selection of the pivot element is indeterminate. Are the elements in the initial array completely randomized?
Let A be a random permutation of [a,b,c,d,e,f,g,h]. Determine the probability that exactly 12 comparisons are required by Merge Sort to sort the input array A. Clearly and carefully justify your answer.
Randomized quicksort compares individual pairs of elements but it does not necessarily compare every element to every other element. When the input is the array [2, 9, 5, 4, 6], what is the probability that randomized quicksort compares 2 and 4 directly to each other? Give an exact answer.
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- Let a be some constant, independent of the input array length n, strictly between 0 and 1/2. What is the probability that, with a randomly chosen pivot element, the Partition function produces a split in which the size of both the resulting subproblems is at least a • narrow_forwardLet A be a random permutation of [1, 2, 3, 4, 5, 6, 7, 8]. Determine the probability that exactly 12 comparisons are required by Merge Sort to sort the input array A. Clearly and carefully justify your answer.arrow_forwardWhen we apply the mergesort on a 12-element array, how many comparisons do we need for the worst case? Ans:arrow_forward
- Build a bottom-up mergesort that makes use of the array's order by carrying out the following steps each time it needs to locate two arrays to merge: Find the first element in an array that is smaller than its predecessor, then locate the next, and finally merge them to form a sorted subarray. Consider the array size and the number of maximal ascending sequences in the array while analysing the running time of this technique.arrow_forwardWrite a bottom-up mergesort that makes use of the array's order by carrying out the following steps each time it needs to locate two arrays to merge: locate the first entry in an array that is smaller than its predecessor, then locate the next, and finally merge them to form a sorted subarray. Consider the array size and the number of maximal ascending sequences in the array while analysing the running time of this algorithm.arrow_forwardWrite a bottom-up mergesort that makes use of the array's order by carrying out the following steps each time it needs to locate two arrays to merge: locate the first element in an array that is smaller than its predecessor, then locate the next, and finally merge them to form a sorted subarray. Consider the array size and the number of maximal ascending sequences in the array while analysing the running time of this method.arrow_forward
- Write a recurrence equation for the number of comparisons T(n) needed to process MergeSort on an input array of size n. Answer: Paragraph V B I U ►|| O + v .…..arrow_forwardGiven 2 sorted arrays (in increasing order), find a path through the intersection that produces the maximum sum and return the maximum sum. That is, we can switch from one array to another array only atcommon elements. If no intersection element is present, we need to take the sum of all elements from the array with greater sum. Sample Input:61 5 10 15 20 2552 4 5 9 15Sample Output :81arrow_forwardIn Quicksort, when PARTITION is called on an array with n elements, we require n − 1 comparisons, since we must compare the pivot element to each of the other n − 1 elements. If the input array is A = [1,2,3,4,5,6,7], show that Quicksort requires a total of 21 comparisons.arrow_forward
- Prove that when running quicksort on an array with N distinct items, the probability of comparing the i th and j th largest items is 2/(j i).arrow_forwardThe analysis of the expected running time of randomized Quicksort assumes that all element values are distinct. In this problem, we examine what happens when they are not, i.e., there exist same-valued elements in the array to be sorted. The Quicksort algorithm relies on the following partition algorithm, which finds a pivot randomly, and then put all the numbers less than or equal to the pivot in the left, and put all the numbers greater than the pivot in the right, and then return the pivot location, as well as the left and right sublists for recursive calls. In this algorithm, the list to be sorted is A, and we use p and r to denote the left-most and right-most indices of the currently processing subarray, respectively. For example, in the initial call of the Quicksort algorithm, we will let p = 0 and r = n−1, which correspond to the whole original array. The PARTITION(A, p, r) procedure returns an index q such that each element of A[p : q −1] is less than or equal to A[q] and each…arrow_forwardGiven an unsorted array A of size N that contains only positive integers, find a continuous sub-array that adds to a given number S and return the left and right index(1-based indexing) of that subarray. In case of multiple subarrays, return the subarray indexes which come first on moving from left to right. Note:- You have to return an ArrayList consisting of two elements left and right. In case no such subarray exists return an array consisting of element -1. Code please.arrow_forward
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