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 6.5, Problem 9E
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Let M(n) be the minimum number of comparisons needed to sort an array A with exactly n elements in Merge sort. For example, M(1) = 0, M(2) = 1, and M(4) = 4. If n is an even number, clearly explain why M(n) = 2M(n/2) + n/2.
Q4. Consider Following 4 lists lists:
L1
45
12
06
34
02
18
09
L2
35
22
05
14
08
22
30
Apply Heap sort, Quick sort and Merge sort algorithms (step by step and complete) on all
of above lists individually. Also compute the time complexity for each algorithm.
solution should have O(l1.length + l2.length) time complexity, since this is what you will be asked to accomplish in an interview.
Given two singly linked lists sorted in non-decreasing order, your task is to merge them. In other words, return a singly linked list, also sorted in non-decreasing order, that contains the elements from both original lists.
Chapter 6 Solutions
Introduction to Algorithms
Ch. 6.1 - Prob. 1ECh. 6.1 - Prob. 2ECh. 6.1 - Prob. 3ECh. 6.1 - Prob. 4ECh. 6.1 - Prob. 5ECh. 6.1 - Prob. 6ECh. 6.1 - Prob. 7ECh. 6.2 - Prob. 1ECh. 6.2 - Prob. 2ECh. 6.2 - Prob. 3E
Ch. 6.2 - Prob. 4ECh. 6.2 - Prob. 5ECh. 6.2 - Prob. 6ECh. 6.3 - Prob. 1ECh. 6.3 - Prob. 2ECh. 6.3 - Prob. 3ECh. 6.4 - Prob. 1ECh. 6.4 - Prob. 2ECh. 6.4 - Prob. 3ECh. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.5 - Prob. 1ECh. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Prob. 4ECh. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - Prob. 8ECh. 6.5 - Prob. 9ECh. 6 - Prob. 1PCh. 6 - Prob. 2PCh. 6 - Prob. 3P
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- It takes O(log n) time to execute Max-heapify. True False We are given two arrays, A = (a1, az, ..., am) and B = (bı, b2, ..., bm) such that a; for all i e {1,2, .., n}. Insertion-sort makes the same number of comparisons on input A as it does on input B. 2b; %3D True False The average running time of the Randomized Selection is O(n) if the pivot is chosen uniformly at random. True Falsearrow_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_forwardAlthough 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…arrow_forward
- For each question, an algorithm will be described that operates on N elements, and your answer should include: (a) a big-O expression that describes the total number of operations in the worst case (for ex- ample, O(N³)) (b) a description of how to achieve the same effect as the algorithm described, but achieved with a better big-O time bound (for example, "use mergesort instead of insertion sort") (c) the big-O time bound for your improved approach. Your improved algorithm does not need to be provably the best possible, but it should have a different and better big-O bound. (It may not be as simple as substituting one named algorithm for another; consider what is redundant about the work done by the existing algorithm.) You don't need to use pseudocode to describe your algorithms - the style used in the problem descriptions is also sufficient for your solutions. You can use pseudocode if you like. Do not write real code. If you wish to use an algorithm described in class, you can name…arrow_forwardGiven k sorted lists and the total number of elements in all lists is n, please design an algorithm to merge k sorted lists into one sorted list in O(n lg k) time. Argue your algorithm is correct and analyze the time complexity of your algorithm.arrow_forwardThe selection sort algorithm could be modified to stop when the unsorted section of the list contains only one number, because that one number must be in the correct position. Show that this modification would have no effect on the number of comparisons required to sort an n-element list.arrow_forward
- 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_forwardMergeSort algorithm has running time in terms of n O(n) O(lg(n)) O(n*lg(n)) O(n^2) O(n^2*lg(n)) O(n^3) otherarrow_forwardExercise 2. We have seen that Dijkstra's algorithm can be implemented in two ways: Variant (a) uses an array to store the dist[] values of the unknown nodes, and Variant (b) uses a MIN-HEAP to store these values. (a) Suppose in your application m <3n. Which variant gives a faster runtime? Justify your answer. (b) Suppose in your application m 2 n²/3. Which variant gives a faster runtime? Justify your answer.arrow_forward
- Suppose you are given an array A[1..n] with a max-heap structure. If you run HEAPSORT on the array, obviously you will find the smallest and second smallest elements in Theta(n Ig n) running time. Explain using plain English that you can do better and find those two elements with an IN PLACE algorithm that uses at most n-2 comparisons. Response Feedback: If you run insertion sort in the leaves (of which there are about n/2) that could be (n/2)^2 comparisons in the worst case to sort them.arrow_forwardMerge sort is an efficient sorting algorithm with a time complexity of O(n log n). This means that as the number of elements (chocolates or students) increases significantly, the efficiency of merge sort remains relatively stable compared to other sorting algorithms. Merge sort achieves this efficiency by recursively dividing the input array into smaller sub-arrays, sorting them individually, and then merging them back together. The efficiency of merge sort is primarily determined by its time complexity, which is , where n is the number of elements in the array. This time complexity indicates that the time taken by merge sort grows logarithmically with the size of the input array. Therefore, even as the number of chocolates or students increases significantly, merge sort maintains its relatively efficient performance. Regarding the distribution of a given set of x to y using iterative and recursive functions, the complexity analysis depends on the specific implementation of each…arrow_forwardA sequential search of a sorted list can halt when the target is less than a givenelement in the list. Define a modified version of this algorithm and state thecomputational complexity, using big-O notation, of its best-, worst-, and average-case performances. Using this template in python: def sequential_search(input_list, target):target_index = -1#TODO: Your work here# Return target_index. If not found, -1return target_indexif __name__ == "__main__":my_list = [1, 2, 3, 4, 5]print(sequential_search(my_list, 3)) # Correct Output: 2print(sequential_search(my_list, 0)) # Correct Output: -1arrow_forward
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