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 2.3, Problem 6E
Program Plan Intro
To observe that the while loop of lines 5-7 of the insertion sort uses the linear search to sort the sub-array and also state whether the binary search can be used instead of linear search to improve the running time complexity.
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Although merge sort runs in (n lg n) worst-case time and insertion sort runs
in (n²) worst-case time, the constant factors in insertion sort can make it faster
in practice for small problem sizes on many machines. Thus, it makes sense to
coarsen the leaves of the recursion by using insertion sort within merge sort when
Chapter 2 Getting Started
subproblems become sufficiently small. Consider a modification to merge sort in
which n/k sublists of length k are sorted using insertion sort and then merged
using the standard merging mechanism, where k is a value to be determined.
a. Show that insertion sort can sort the n/k sublists, each of length k, in (nk)
worst-case time.
b. Show how to merge the sublists in (n lg(n/k)) worst-case time.
c. Given that the modified algorithm runs in (nk + n lg(n/k)) worst-case time,
what is the largest value of k as a function of n for which the modified algorithm
has the same running time as standard merge sort, in terms of -notation?
d. How should we…
Write 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.
Observe that the while loop of lines 6–8 of the Insertion-Sort procedure given in class uses a linear search to scan (backward) through the sorted subarray A[1..j − 1]. Can we use a binary searh instead to im- prove the overall worst-case running time of insertion sort to Θ(n lg n)?
Chapter 2 Solutions
Introduction to Algorithms
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- QuickSort is run for an array A in a manner that PARTITION consistently produces a 5:1 split for the (sub)arrays to be sorted (recursively) next. In this case, the recurrence equation for QuickSort's runtime is what? Group of answer choices T(n) <= T(5n/10) + T(n/10) + Theta(n) T(n) <= T(5/n) + T(1/n) + Theta(n) T(n) <= T(5n/6) + T(n/6) + Theta(n) T(n) <- T(6n/5) + T(6n) + Theta(n)arrow_forwardOn his way back from detention, Harry runs into his friend Hermione. He is upset because Professor Snape discovered that his sorting spell failed. Instead of sorting the papers correctly, each paper was within k slots of the proper position. Hermione immediately suggests that insertion sort would have easily fixed the problem. In this problem, we show that Hermione is correct (as usual). As before, A[1 ..n] in an array of n distinct elements. Show that sorting a list in which each paper is within k slots of its proper position takes N (n log k) comparisons. Hint: Use the decision-tree technique.arrow_forwardWrite 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_forward
- Consider the following variant of Mergesort. If the array has more thantwo elements, it recursively sorts the first two-third and the last one-third. Then, it mergesthe first two-third with the last one-third. If the array has at most two elements, then ittrivially sorts the array. For completeness, here is the pseudocode for sorting the arrayA[l..r]. Write a recurrence relation for the worst-case running time of Newsort. Solve the recurrencerelation to obtain aΘ-bound on the running time of the algorithm, in terms of the lengthnof the array. You must use the tree method to solve the recurrence. You may also ignorefloors and ceilings in your calculations.arrow_forward1)If we consider Recurrences and Running Time, what is the running time T(n) for size of an array n? BINARY-SEARCH (A, lo, hi, x) { if (lo > hi) return FALSE mid = (lo + hi)/2 if x = A[mid]Assignment 1 return TRUE if (x < A[mid]) BINARY-SEARCH (A, lo, mid-1, x) if (x > A[mid]) BINARY-SEARCH (A, mid+1, hi, x) } 2)6. Based on T(n) that you found on previous question, solve recurrence with iteration method.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
- 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_forwardCreate a merge-sort multithread.A is an array, and p and q are array indexes used to sort the sub-array A[p..r]. B is a subarray that will be filled by the sort. A p-merge-sort(A,p,r,B,s) call sorts items from A[p..r] and places them in B[s..s+r-p].arrow_forwardProblem 2 In the INSERTIONSORT implementation we showed in class we compared the element we were placing with each element in the sorted subarray until we found the correct position. Since the subarray is sorted we can use binary search to find the correct final position. (Reminder: BINARYSEARCH performs O(log n) comparison in the worst case) (a) How does this change impact the total number of com- parisons performed in the worst case? Explain your answer.arrow_forward
- create a plan for implementation Instead of the binary-search-based linearithmic technique, TwoSumFaster employs a linear algorithm to count the pairings that sum to zero once the array is sorted. Then, using a similar concept, create a quadratic method for the 3-sum issue.arrow_forwardEstablish Proposition H. Bottom-up mergesort sorts an array of length N using at most 6N lg N array accesses and between 12 N lg N and N lg N comparisons.arrow_forwardCreate a bottom-up mergesort that takes advantage of array order by doing the following each time it has to identify two arrays to merge: locate a sorted subarray (by incrementing a pointer until it finds an entry in the array that is smaller than its predecessor), then locate the next, and finally merge them. Examine the algorithm's running time in terms of array size and the number of maximal rising sequences in the array.arrow_forward
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