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 5.3, Problem 5E
Program Plan Intro
To prove that the probability that all elements are unique is at least
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Let T be a sorted array of n elements. An element x is said to be a majority element in T if the number of elements i, with T[i] = x, is greater than n/2.
Give an algorithm (code or pseudo-code) that can decide whether T includes a majority element (it cannot have more than one), and if so, find it. Your algorithm must run
in linear time.
Algorithm FHL Version of Schreier-Sims is O ( I f) 16), provided that the size of the initial generating set is O ( I ~ 14).
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.
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- For any input array of n 3-digit numbers, prove that Radix Sort is guaranteed to correctly sort these n numbers, with the algorithm running in O(n) time.arrow_forwardWhat is the obvious limit-to the problem of finding the median of a given array of n-elements? Is this obvious lower bound a TIGHT lower bound, why?arrow_forwardGiven an array of numbers X₁ = {x₁, x2, ..., n } an exchanged pair in X is a pair xi, xj such that i x¡ . Note that an element x; can be part of up to n - 1 exchanged pairs, and that the maximal possible number of exchanged pairs in X is n(n − 1)/2, which is achieved if the array is sorted in descending order. Give a divide-and-conquer algorithm that counts the number of exchanged pairs in X in O(nlogn) time.arrow_forward
- Let 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_forwardLet B = {a + bi|a, b ∈ N}. Prove that B is countable.arrow_forwardAlgorithm A excecutes an O(logn) time compuation for each entry of an n-element array. What is the worst case running time of Algorithm A?arrow_forward
- make algorithm for MatrixMultiplication (A1, A2, ... , An) pre-cond: An instance is a sequence of n matrices. post-cond: optSol is a bracketing that requires the fewest multiplications, and optCost is the resulting number of multiplications.arrow_forwardQuestion: use magic-pivot to obtain a deterministic quick-sort algorithim with the worst-case running time of O(n log n).arrow_forwardUse the Transform-and-Conquer algorithm design technique with Instance Simplification variant to design an O(nlogn) algorithm for the problem below. Show the pseudocode. Given a set S of n integers and another integer x, determine whether or not there exist two elements in S whose sum is exactly x.arrow_forward
- Suppose arrays A and B are both sorted in increasing order and both contain n elements. What is the time complexity to find the median of A U B, i.e., the set of elements in A or B?arrow_forwardConsider the problem of sorting an array A[1, ..., n] of integers. We presented an O(n log n)- time algorithm in class and, also, proved a lower bound of (nlogn) for any comparison-based algorithm.arrow_forwardWe apply the binary search on a 17-element ordered array. Assume that a givenkey appears in the array, and it is between the 5th element and the 13th element(inclusive). How many comparisons do we need for the average-case efficiency?arrow_forward
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