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, Problem 4P
(a)
Program Plan Intro
To list the five inversions of the array
(b)
Program Plan Intro
To find the most inversions from the set
(c)
Program Plan Intro
To find the relation between the running time of the insertion sort and the number of inversions in the input array.
(d)
Program Plan Intro
To describe an
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CodeGiven an unsorted array A of size N that contains only positive integers, find a continuous sub-array that adds to a given number S and return the left and right index(1-based indexing) of that subarray.
In case of multiple subarrays, return the subarray indexes which come first on moving from left to right.
Note:- You have to return an ArrayList consisting of two elements left and right. In case no such subarray exists return an array consisting of element -1.
Multiplying Factors For a pair of integers (x,y) and an integer k, the multiplying factor is defined as minimum positive integer m such that LCM(x,y) * m is divisible by k. Here LCM(x,y) represents the smallest positive integer divisible by both x and y, their least common multiple. Given an array arr of n integers and an integer k, find the sum of multiplying factors over all pairs of integers (arr[i], arr[j]) and k0 <= i,j < n. Example Suppose n=3, arr =[4,5,6] and k=12. Pair Multiplying Factor Remarks (4, 3 LCM(4,4)**3=4**3 =12 4 ) LCM(5,5)**12=5** 12=60 (5, 12 LCM(6,6)**2=6**2 5) (6, 2 12 6 )
A unimodal array is an array that has a sequence of monotonically increasing integers followed by a sequence of monotonically decreasing integers, assuming all elements in the array are unique. Example: - A= {4, 5, 8, 9, 10, 11, 7, 3, 2, 1}: A is a unimodal array because there is an increasing sequence followed by a decreasing sequence and the maximum element is 11.- B= {11, 9, 8, 7, 5, 4, 3, 2, 1}: B is not a unimodal array because there is no increasing sequence It is simply a decreasing sequence and the maximum element is 11.- C= {1, 2, 3, 4, 5, 7, 8, 9, 11}: C is not a unimodal array because There is an increasing sequence, but there is no decreasing sequence and the maximum element is 11.a) Design an efficient algorithm with the lowest possible complexity to state whether a given array is unimodal or not, and explain why your algorithm is efficient. b) Analyze the complexity of your algorithm.
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Similar questions
- 1. Suppose you have these inputs: M, I, T, Q, L, H, R, E, K, P, C, A. a. Show the binary search tree for these inputs. b. Show how to store the binary search tree in an array with the node structure (key, left, right).arrow_forwardJava Insertion Sort but make it read the data 12, 11, 13, 5, 6 from a file not an array // Java program for implementation of Insertion Sort public class InsertionSort { /*Function to sort array using insertion sort*/ void sort(int arr[]) { int n = arr.length; for (int i = 1; i < n; ++i) { int key = arr[i]; int j = i - 1; /* Move elements of arr[0..i-1], that are greater than key, to one position ahead of their current position */ while (j >= 0 && arr[j] > key) { arr[j + 1] = arr[j]; j = j - 1; } arr[j + 1] = key; } } /* A utility function to print array of size n*/ static void printArray(int arr[]) { int n = arr.length; for (int i = 0; i < n; ++i) System.out.print(arr[i] + " "); System.out.println(); } // Driver method…arrow_forwardarray unordered Arr has unsorted integers. SortedArr is an integer array. SortedArr performs which task better than unsortedArr? Use the fastest algorithms. Inserting a new element II Searching for a given element III Calculating the mean of the elements (A) I alone (B) II alone (C) III alone (D) I and II alone (E) I, II, and IIIarrow_forward
- : In an array of integers, a "peak" is an element which is greater than or equalto the adjacent integers and a "valley" is an element which is less than or equal to the adjacentintegers. For example, in the array {5, 8, 6, 2, 3, 4, 6}, {8, 6} are peaks and {5, 2} are valleys. Given anarray of integers, sort the array into an alternating sequence of peaks and valleys.EXAMPLEInput: {5, 3, 1, 2, 3}Output: {5, 1, 3, 2, 3}arrow_forwardGiven a vector of integers. Find the number of elements which are out of order in the array. An element is in out of order if it is smaller than previous element and larger than next element. Note- first and last element is not to be counted as out of order.arrow_forwardLet A[0..n – 1] represent an array with length n and indices 0 to n – 1. if i > j in A[i..)]arrow_forward
- 1. Consider the algorithm for the sorting problem that sorts an array by counting,for each of its elements, the number of smaller elements and then uses thisinformation to put the element in its appropriate position in the sorted array:ALGORITHMComparisonCountingSort(A[0..n − 1])//Sorts an array by comparison counting//Input: Array A[0..n//Output: Array S[0..n− 1] of orderable values− 1] of A’s elements sorted// in nondecreasing orderfor i ← 0 to nCount− 1 do[i]←0for i ← 0 to n − 2 dofor j ← i +1 to n − 1 doif A[i] < A[j ]Count[j ]← Count[j ] + 1else Count[i]← Count[i] + 1for i ←0 to n−1 doS[Count[i]]←A[i]return Sa. Apply this algorithm to sorting the list 60, 35, 81, 98, 14, 47.b. Is this algorithm stable?c. Is it in-place?arrow_forwardSuppose an array has n elements. This _____ sorts to sort and array works as follows: Find the smallest element and place it in the first position. Then find the smallest of the remaining n-1 elements and place it in the second position. Repeat on n-2 elements, n-3 elements, ..., until the array is sorted.arrow_forward1. Consider a problem of computing the prefix average of a sequence of numbers stored in an array P consisting of P integers. We want to compute an array P such that P is the average of P for P . Consider the two algorithms related to the above problem. Answer the questions given belowarrow_forward
- Code only_Given an unsorted array A of size N that contains only positive integers, find a continuous sub-array that adds to a given number S and return the left and right index(1-based indexing) of that subarray. In case of multiple subarrays, return the subarray indexes which come first on moving from left to right. Note:- You have to return an ArrayList consisting of two elements left and right. In case no such subarray exists return an array consisting of element -1..arrow_forwardPeaks and Valleys: In an array of integers, a "peak" is an element which is greater than or equalto the adjacent integers and a "valley" is an element which is less than or equal to the adjacentintegers. For example, in the array {5, 8, 6, 2, 3, 4, 6}, {8, 6} are peaks and {5, 2} are valleys. Given anarray of integers, sort the array into an alternating sequence of peaks and valleys.EXAMPLEInput: {5, 3, 1, 2, 3}Output: {5, 1, 3, 2, 3}arrow_forwardFor the given array, simulate the working operation of Insertion Sort. Show your work at each step in c++. Make sure to show the status of the array after every insertion.[ 28, 13, 22, 7, 34, 2, 15, 18 ] iarrow_forward
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