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 1.2, Problem 2E
Program Plan Intro
To calculate the values of
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Suppose we are comparing implementations of insertion sort and merge sort on the same machine.
For inputs of size n, insertion sort runs in 8n2 steps, while merge sort runs in 32nlogn steps.
For which values of n does insertion sort beat merge sort?
Group of answer choices:
1. n < 1024
2. n > 1024
3. n < 16
4. n > 16
5. Merge sort is always better since O(nlogn) is better than O(n^2)
Q: Suppose we are comparing implementations of insertion sort and merge sort on the same machine. For inputs of size n, insertion sort runs in 8n2 steps, while merge sort runs in (64n log(n)) steps. Which one is batter choice and why ?
Merge 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…
Chapter 1 Solutions
Introduction to Algorithms
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- Given two sorted arrays nums1 and nums2 of size m and n respectively, return the median of the two sorted arrays. The overall run time complexity should be O(log (m+n)). Example 1: Input: nums1 = [1,3], nums2 = [2] Output: 2.00000 Explanation: merged array = [1,2,3] and median is 2. Example 2: Input: nums1 = [1,2], nums2 = [3,4] Output: 2.50000 Explanation: merged array = [1,2,3,4] and median is (2 + 3) / 2 = 2.5. Constraints: nums1.length == m nums2.length == n 0 <= m <= 1000 0 <= n <= 1000 1 <= m + n <= 2000 -106 <= nums1[i], nums2[i] <= 106 Write the whole code in python language Attach the code outputs also and explain the implementation.arrow_forwardGiven two sorted arrays nums1 and nums2 of size m and n respectively, return the median of the two sorted arrays. The overall run time complexity should be O(log (m+n)). Example 1: Input: nums1 = [1,3], nums2 = [2] Output: 2.00000 Explanation: merged array = [1,2,3] and median is 2. Example 2: Input: nums1 = [1,2], nums2 = [3,4] Output: 2.50000 Explanation: merged array = [1,2,3,4] and median is (2 + 3) / 2 = 2.5. Constraints: nums1.length == m nums2.length == n 0 <= m <= 1000 0 <= n <= 1000 1 <= m + n <= 2000 -106 <= nums1[i], nums2[i] <= 106arrow_forwardConsider a hybrid sorting algorithm that combines Mergesort with Insertion Sort.It uses Mergesort until the number of elements in the input becomes smaller than or equal to 8, after which it switches to Insertion Sort. What is the number of key comparisons performed by this hybrid sorting algorithm in the best case when running on an input array of size n? Briefly justify your answer. You could assume n = 2k, k is more than 3arrow_forward
- Given two sorted arrays nums1 and nums2 of size m and n respectively, return the median of the two sorted arrays. The overall run time complexity should be o(log (m+n)). Example 1: Input: nums1 [1,3], nums2 = [2] Output: 2.00000 Explanation: merged array = [1,2,3] and median is 2. Example 2: Input: nums1 [1,2], nums2 = [3,4] Output: 2.50000 Explanation: merged array = [1,2,3,4] and median is (2 + 3) / 2 = 2.5. Constraints: nums1.length == m • nums2.length == n 0 <= m <= 1000 0 <= n <= 1000 1 <= m + n <= 2000 -10° <= nums1[i], nums2[i] <= 10°arrow_forwardSorting a data set is an important sub-problem in data science. Given the size n of a data set, which statements are correct? a) Bubble Sort has worst-case run-time complexity O(n). b) Bubble Sort has worst-case run-time complexity O(n log N). c) Bubble Sort has worst-case run-time complexity O(n²). d) Merge Sort has worst-case run-time complexity O(n). e) Merge Sort has worst-case run-time complexity 0(n log N). f) Merge Sort has worst-case run-time complexity 0(n²). g) Quick Sort has worst-case run-time complexity O(n). h) Quick Sort has worst-case run-time complexity O(n log N). i) Quick Sort has worst-case run-time complexity O(n?).arrow_forwardOne disadvantage of using a For-loop in sorting algorithms is that the body of theFor-loop is repeatedly executed even if it is no longer necessary. One example isBubble Sort:Input: Array A[1 . . . n]Output: Array A sortedfor i ←1 to (n −1) dofor j ←1 to (n −i) doif A[j] > A[j + 1] thenA[j] ↔A[j + 1]return AIn the array [2, 1, 3, 7, 9] we know that when 2 and 1 are interchanged, early on in theexecution of the algorithm, the array is now sorted, and all subsequent iterations ofthe outer loop are unnecessary. *Write a version of Bubble Sort that avoids unnecessary comparisons*arrow_forward
- ) Which one is the computationally best-case scenario for the insertion sort algorithm. InsertionSort(a[ ]){ int N = size; for (int i = 0; i < N; i++) for (int j = i; j > 0; j--) if (less(a[j], a[j-1])) // compare swap(a, j, j-1); // exchange else break;} A) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 B) 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 C) 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3 D) 10, 10, 1, 1, 2, 2, 3, 3, 4, 4 E) 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0arrow_forwardSorting a data set is an important sub-problem in data science. Given the size n of a data set, whichstatements are correct?a) Bubble Sort has worst-case run-time complexity O(n).b) Bubble Sort has worst-case run-time complexity O(n log N).c) Bubble Sort has worst-case run-time complexity O(n2).d) Merge Sort has worst-case run-time complexity O(n).e) Merge Sort has worst-case run-time complexity O(n log N).f) Merge Sort has worst-case run-time complexity O(n2).g) Quick Sort has worst-case run-time complexity O(n).h) Quick Sort has worst-case run-time complexity O(n log N).i) Quick Sort has worst-case run-time complexity O(n2).arrow_forwardQuestion 2: Implement sequential search and binary search algorithms on your computer. Note down run time for each algorithm on arrays of size (1, 250, 500, 750, 1000, 1250, 1500, 5000). For both algorithms, store the values 0 throughn 1 in order in the array, and use a variety of random search values in the range 0 to n 1 on each size n. Graph the resulting times. When is sequential search faster than binary search for a sorted array? Solution: Execution Time for Input Size(N) Search Element Execution Time for Sequential Search(n) Binary Search(logn)arrow_forward
- 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.arrow_forwardprocedure bubble-sort(A) for i= 1 to A.length−1 do //1+n+n-1 for j=A.length down to i+ 1 do //n-1+1 if A[j]< A[j−1] then; exchange A[j] with A[j−1] end if end for end for end procedure Derive Θ(g(n)),O(g(n)), and Ω(g(n)). State the constants for these running times.arrow_forwardGiven below is the Randomized Quick Sort Algorithm, where p and r represent lower and upper bounds of array A respectively. RANDOMIZED-PARTITION (A, p, r) 1 i = RANDOM(p,r) 2 exchange A[r] with A[i] 3 return PARTITION (A, p, r) RANDOMIZED-QUICKSORT if parrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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