Consider the following three sorting algorithms: Insertion Sort, Heapsort, and Quicksort. Here are three statements. A. Iam an incredibly fast sorting algorithm that runs in O(n log n) time on average, though my worst case run time is O(n²). B. I run very quickly for small values of n, but unfortunately am really slow when n is large. My average run time is O(n²). C. I am an in-place sorting algorithm whose average case and worst case running time is O(n log n). For each statement (A, B, C), determine the correct sorting algorithm.

2. Given three different sorting algorithms to solve a sorting problem and their runtimes are expressed as given below: T₁(N) = O(N²) T₂(N) = (N²) T3(N) = Q(N²) Which algorithm should be preferred and which one should not be. Give your answer by using the definitions of the three symbols used.

Consider the following algorithm. What does this algorithm compute? What is its basic operation? How many times is the basic operation executed? What is the efficiency class of this algorithm? Suggest an improvement of the given algorithm, and indicate its efficiency class. If you cannot do it, try to prove that, in fact, it cannot be done.

Consider the Sort-and-Count algorithm explained in section 5.3 of our text: "Counting Inversions"Suppose that the initial list is: 92   71   36   91   27   48   14   34   81   26   24   65   78   51   37   22 Sort-and-Count makes two recursive calls. The first recursive call inputs the first half of the initial list: 92   71   36   91   27   48   14   34 and returns the sorted version of the first half, as well as the number of inversions found in the first half (22). The second recursive call inputs the second half of the initial list: 81   26   24   65   78   51   37   22 and returns the sorted version of the second half, as well as the number of inversions found in the second half (19). Sort-and-Count then calls Merge-and-Count. To Merge-and-Count, Sort-and-Count passes the sorted versions of the two halves of the original list: 14   27   34   36   48   71   91   92, and 22   24   26   37   51   65   78   81 Merge-and-Count begins merging the two half-lists together, while counting…

You are given an array A[1, ., n] in which the elements A[2] to A[n] are already sorted, i.e., A[2, .., n] is a sorted array. Write down an efficient algorithm with proper pseudocode for moving A[1] in the correct place, so that the full array A[1, ..., n] is sorted.

Assume that we are given n pairs of items as input, where the first item is a number and the second item is one of three colors (red, blue, or yellow). Further assume that the items are sorted by number. Give an O(n) algorithm to sort the items by color (all reds before all blues before all yellows) such that the numbers for identical colors stay sorted.   For example: (1,blue), (3,red), (4,blue), (6,yellow), (9,red) should become (3,red), (9,red), (1,blue), (4,blue), (6,yellow).

Suppose that each row of an n × n array A consists of 1's and 0's such that in any row of A all the 1's come before any O's in that row. Assuming that A is already in memory, describe an algorithm running in O(n) time for finding the row that contains the most 1's. Write the algorithm following the style of your course-notes.

Write an efficient algorithm for the following problem (either pseudocode or java), and describe your reasoning. Determine the Time complexity and if you cannot find any polynomial time algorithm, then give a backtracking algorithm. Problem will be on repeat numbers input will be ARRAY[1,2, ... , n] number of positive numbes. Output will any one number that is repeated more than n/3 times. for example, if you have [1,1,2,2,2,1], since n = 6 and n/3 = 2, both 1 and 2 showed up more than n/3 times. the output will be either 1 or 2 (just one of the values are required)   if you have [1,1,3,4,5,6,7,8,9], since 1 is only repeated twice, and n/3 = 3, output will be "none"

You want to design an algorithm, called minMax(A,p,r), that takes an array of integers and indexes of the first and last elements, and returns the minimum and maximum values in that range. Now write the pseudo code of a divide and conquer (and therefore, recursive) algorithm with the same time complexity (Θ(n)). You can assume that p ≤ r. Also, in your code, you can return two numbers by returning a pair, e.g. “return (a, b)”, and can save the output in a similar way, e.g. “(a, b) = minMax(parameters)”. (Short answer please)

1. a.An algorithm can be specified in various ways. Identify and explain four ways in which an algorithm can be specified. b. (b). Two algorithms A, B sort the same problem. When you go through each algorithm and break them down into their primitive operations, each can be represented as follows:A = 2n7 + 100n4 + 26n + 50 B = 7n4 + 22n2 + nlogn + 200For very large values of n, which of the algorithms A or B will runin the shortest time to solve the problem and why? (c). Two Computer Science students, Priscilla and Julius, are discussing how to compare two algorithms for solving a given problem. Priscilla suggests that they should use the execution times of the algorithms as criterion; but Julius insists that they should use the number of statements the algorithms execute as criterion. (i). Discuss the reasons why both criteria they are considering are not good for comparing algorithms.(ii). Recommend an ideal solution/criterion that they should rather use for comparing algorithms