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 7.4, Problem 6E
Program Plan Intro
To show that the probability of getting at worst an
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Run mergesort for huge random arrays and determine empirically, as a function of N (the total of the two subarray sizes for a given merge), the average length of the other subarray when the first subarray exhausts.
(a) Carry out the Select algorithm on the following set, using k = 19 (return the element of rank 19). Show your steps! Ensure that you run the partition algorithm in such a way that you maintain theelements in their original relative order.56; 78; 34; 19; 67; 32; 13; 12; 90; 92; 50; 51; 30; 1; 99; 58; 43; 42; 24; 65; 21; 25; 68; 69; 101
(b) Carry out the Randomized Select algorithm from class on the set of elements from part (a), usingk = 19. Show your steps! When selecting a random pivot, suppose you always chose the last element inthe array. Ensure that you run the partition algorithm in such a way that you maintain the elements intheir original relative order
Run mergesort for large random arrays, and make an empirical determination of the average length of the other subarray when the first subarray exhausts, as a function of N (the sum of the two subarray sizes for a given merge).
Chapter 7 Solutions
Introduction to Algorithms
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- Consider merging two sorted subarrays of array A = (8 10 12 23 5 17 30) by calling merge(A 1 4 7). How many key comparisons will be made?arrow_forwardRandomized 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.arrow_forwardImplement a quicksort based on partitioning on themedian of a random sample of five items from the subarray. Put the items of the sampleat the appropriate ends of the array so that only the median participates in partitioning.Run doubling tests to determine the effectiveness of the change, in comparison bothto the standard algorithm and to median-of-3 partitioning (see the previous exercise).Extra credit : Devise a median-of-5 algorithm that uses fewer than seven compares onany input.arrow_forward
- Suppose we have the following sorted set of positive integers: 1 2 4 4 6 7 9 9 15 18 22. Also assume that we use linear search (starting from the left as normal) and searching for element 10. How many comparisons are needed at least then before we come to the conclusion that the element is not in the set? 10 true 8 true 9 truearrow_forwardWe are given a collection of n key-value pairs (k,m) in which k is a positive integer number in arange [0,N-1] for small enough N. Assume that we may have multiple values for the same keys.a) if m is a positive number in a range [0,M-1] for some small enough integer M, design alookup table for storing these items such that the time complexity of search, insert and deleteare constant. b) if m is an arbitrary integer number, design a lookup table for storing these itemssuch that the time complexity of search is log n.arrow_forwardProve 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_forward
- 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.arrow_forwardGiven an array A[1…n] of numeric values (can be positive, zero, and negative) determine the subarray A[i…j] (1≤ i ≤ j ≤ n) whose sum of elements is maximum over all subvectors. Below is a brute-force algorithm. Analyze its best case, worst case and average case time complexity in terms of a polynomial of n and the asymptotic notation of ɵ. You need to show the steps of your analysis.arrow_forwardIt is straightforward and efficient to compute the union of two sets using Boolean values. We may create a new union set by Oring the matching items of the two BitArrays since the union of two sets is a combination of the members of both sets. At other words, if the value in the corresponding place of either BitArray is True, a member is added to the new set.Computing the intersection of two sets is analogous to computing the union, except that the And operator is used instead of the Or operator Using the same technique we used to detect the difference, we can determine if one set is a subset of another. For example, if:setA(index) && !(setB(index))evaluates to False then setA is not a subset of setB.The BitArray Set ImplementationWrite The code for a CSet class based on a BitArray.arrow_forward
- Multiple Choice: Suppose we want to apply Radix Sort to sort 100,000 4-letter words, with each letter taken from the English alphabet (26 letters, all lower cases). Assuming that the running time for sorting n elements within range 1..k using Counting Sort is 2n+2k, whichof the following strategies will lead to the lowest running time? Show the justification.(1) treat letters at each of the four positions as a digit or(2) treat 2-letter subwords at positions 1-2 as a digit and 2-letter subwords at positions 3-4 as another digit or(3) treat all 4 letters as a digit.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_forwardcreate an implementation strategy To count the pairings that total to zero once the array is sorted, TwoSumFaster uses a linear method as opposed to the binary-search-based linearithmic approach. Then, develop a quadratic technique for the 3-sum problem using a related idea.arrow_forward
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