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 2.2, Problem 2E
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To write the pseudo code of selection sort and also describe the loop invariants and the running time complexity in best and worst case.
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Consider sorting n numbers stored in array A by first finding the smallest elementof A and exchanging it with the element in A[1]. Then find the second smallestelement of A, and exchange it with A[2]. Continue in this manner for the first n-1elements of A. Write pseudocode for this algorithm, which is known as selectionsort. What loop invariant does this algorithm maintain? Why does it need to run foronly the first n - 1 elements, rather than for all n elements? Give the best-case andworst-case running times of selection sort in Θ -notation.
Write an algorithm that sorts a list of n items by dividing it into three sublists of about n/3 items, sorting each sublist recursively and merging the three sorted sublists. Analyze your algorithm, and give the results under order notation.
Consider sorting n numbers stored in array A[1:n] by first finding the smallestelement of A[1:n] and exchanging it with the element in A[1]. Then find thesmallest element of A[2:n], and exchange it with A[2]. Then find the smallestelement of A[3:n], and exchange it with A[3]. Continue in this manner for thefirst n-1 elements of A. Write pseudocode for this algorithm, which is knownas selection sort. What loop invariant does this algorithm maintain? Why does itneed to run for only the first n-1 elements, rather than for all n elements? Give theworst-case running time of selection sort in big theta notation. Is the best-case runningtime any better?
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Introduction to Algorithms
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- Implement a quicksort with a sample size of 2k 1. Sort the sample first, then have the recursive procedure partition on the sample's median and shift the two halves of the rest of the sample to each subarray so that they may be utilised in the subarrays without having to be sorted again. This algorithm is known as samplesort. Put into practise a quicksort based on a 2k sample. The sample should be sorted first, after which you should set up the recursive procedure to split the sample based on its median and to shift the two halves of the remaining sample to each subarray so they can be utilised in the subarrays without needing to be sorted again. The name of this algorithm is samplesort.arrow_forwardYou want to design an algorithm, called minMax(A,p,r), that takes an array of integers and indexes of the first and last elements, and returns the minimum and maximum values in that range. Now write the pseudo code of a divide and conquer (and therefore, recursive) algorithm with the same time complexity (Θ(n)). You can assume that p ≤ r. Also, in your code, you can return two numbers by returning a pair, e.g. “return (a, b)”, and can save the output in a similar way, e.g. “(a, b) = minMax(parameters)”. (Short answer please)arrow_forwarda. Write a version of the sequential search algorithm that can be used to search a sorted list. (1, 2) b. Consider the following list: 2, 20, 38, 41, 49, 56, 62, 70, 88, 95, 100, 135, 145 Using a sequential search on ordered lists, that you designed in (a), how many comparisons are required to determine whether the following items are in the list? (Recall that comparisons mean item comparisons, not index comparisons.) (1, 2) 2 57 88 70 135 Write a program to test the function you designed. Note: Have the function,seqOrdSearch, return -1 if the item is not found in the list. (return the index of the item if found).arrow_forward
- Given an unsorted array A of integers of any size, n ≥ 3, and an integer value x, write an algorithm as a pseudo code (not a program!) that would find out if there exist EXACTLY3 occurrences in the array with value x. What is the time complexity of your algorithm, in terms of Big-O? What is the space complexity of your algorithm, in terms of Big-O? What if , the given array A is sorted. Will time complexity change from the case that A was unsorted? • If yes; give a new algorithm that achieves this better complexity (indicate the time complexity as of that algorithm). • If no, explain why such new constraints/conditions cannot lead to a better time complexity.arrow_forwarddesign an algorithm to find all the common elements in two sorted lists of numbers. For example, for the lists 2, 5, 5, 5 and 2, 2, 3, 5, 5, 7, the output should be 2, 5, 5.What is the maximum number of comparisons your algorithm makes if the lengths of the two given lists are m and n, respectively?arrow_forwardWrite a program that sorts an array of random or sorted numbers using Radix sort algorithms, fill the array with a nearly ordered list. Construct your nearly ordered list by reversing elements 19 and 20 in the sorted random list. Count the number of comparisons and moves necessary to order this list. Run the program three times, once with an array of 100 items, once with an array of 500 items, and once with an array of 1000 items. For the first execution only (100 elements), Print the unsorted data followed by the sort data 10x10 matrixes (10 rows of 10 numbers each).arrow_forward
- in java , Consider the array of integers A={ 49, 26, 93, 17, 77, 31, 44, 55, 23}. Now, Perform the following sorting operations on ‘A’: Using Insertion sort, show how array would look like after each step. Using quick sort, show how array will be partitioned (first partition only) choose 23 the pivot element.arrow_forwardQ1: Suppose you are given an array A of n Your task is to sort n numbers stored in array A by reading the first element of A and placing it on its original position (position after sorting). Then read the second element of A, and place it on its original position. Continue in this manner for the first n-1 elements of A. What type of sorting is this? Write the algorithm and also mention the name of this sorting algorithm. What loop invariant does this algorithm maintain? Give the best-case and worst-case running times of this sorting algorithm.arrow_forwardCreate an algorithm that demonstrates the Fibonacci search procedure. The number of data elements n is chosen in such a way that: I Fk+1 > (n+1); and ii) Fk + m = (n +1) for some m 0, where Fk+1 and Fk are two consecutive Fibonacci numbers.arrow_forward
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