(b) Set up a recurrence desginating a lower bound on the number of key comparisons. (c) Set up a recurrence designating an upper bound on the number of key comparisions.
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Q: recurrence relation Un
A: Answer:- an = ( 2 - n + n²) (1)^n
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A: The relationship is mentioned below for the recurrence relation:
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A: Answer: I have given answered in the handwritten format.
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A: Recurrence relation for above are: i) an = an-1 + 3.2 ; n >= 1 a0 = 5
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Q: primary index is a non-dense O True
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Q: the Master Theorem to show that T(n) = Θ(log n).
A: the recurrence relation is true the Master Theorem to show that T(n) = Θ(log n).
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- Consider the following divide and conquer algorithm for searching for a value K in a sorted list L[1.n): 1 1. If the list is empty, then the element is not found. If the list contains only one element, then simply compare it to the value K and act accordingly. 2. Otherwise, search recursively by comparing K with the element located at [n/4|. If it is equal, then youre done. If its smaller then continue with the sublist to the left of n/4|. 3. If, on the other hand, its bigger than [n/4], then compare it with the element located at [3n/4] and do exactly as you did in (2) i.e. you either found it at 3n/4] or you continue with one of the sublists that are on the left or right of [3n/4]. (a) Write the pseudocode of the above describe search algorithm. (b) Set up a recurrence desginating a lower bound on the number of key comparisons. (c) Set up a recurrence designating an upper bound on the number of key comparisions. (d) Solve the lower bound recurrence for n = (e) Solve the upper bound…Write an algorithm that sorts a list of n items by dividing it into three sublistsof about n/3 items, sorting each sublist recursively and merging the threesorted sublists. Analyze your algorithm, and give the results under orderWrite an algorithm that searches a sorted list of n items by dividing it into three sublists of almost n/3 items. This algorithm finds the sublist that might contain the given item and divides it into three smaller sublists of almost equal size. The algorithm repeats this process until it finds the item or concludes that the item is not in the list. Analyze your algorithm and give the results using order notation.
- AssuAssume we want to analyze empirically 4 variants of the Quicksort algorithm by varying the selection of the pivot and the recursive call as follows: ● Try the following values when selecting a pivot: - Pick the last element as pivot - Pick a random element as pivot ● Do not make a recursive call to QuickSort when the list size falls below a given threshold, and use Insertion Sort to complete the sorting process instead. Try the following values for the threshold size: - Log2(N) - Sqrt(N) Therefore, you are asked: a. (12 points) Write the java code for the 4 Quicksort implementations. b. (5 points) Write a driver program that allows you to measure the running time in milliseconds of the 4 implementations for N = 10000, 20000, 40000, 80000 and 160000. For each data size N, generate a random list of N random integers ranging from 1 to 107 and use the same list to measure the running time of the 4 implementations. Present the results in the following table:me we want to analyze…Write an algorithm that sorts a list of n items by dividing it into three sublists of about n/3 items, sorting each sublist recursively and merging the three sorted sublists. Analyze your algorithm, and give the results under order notation.A binary search only works if the values in the list are sorted. A bubble sort is a simple way to sort entries. The basic idea is to compare two adjacent entries in a list-call them entry[j] and entry[j+1]. If entry[j] is larger, then swap the entries. If this is repeated until the last two entries are compared, the largest element in the list will now be last. The smallest entry will ultimately get swapped, or "bubbled" up to the top. The algorithm could be described in C as: last = num; while (last > 0) { pairs = last – 1: for (j = 0; j entry (j+1] { temp = entry[il: entryli] = entrylj+1]; entrylj+1] = temp; last = i: } } Here, num is the number of entries in the list. Write an assembly language program to implement a bubble sort algorithm, and test it using a list of 8 elements. Each element should be a halfword in length. Please show your code works with the Keil tools or VisUAL, by grabbing a screen shot with your name somewhere on the screen.
- 1. Consider the following search algorithm for sorted lists: Starting from the first element, the algorithm jumps K elements at once until finding or passing the element we are searching for. If it passes that element, then starts jumping back previous items one by one. If there are not enough elements for the last jump, then it jumps to the last element. Example: Let K=3 – we are searching for 5 1 2 3 4 5 7 8 9. 10 a. Write a pseudocode or draw a flowchart. b. Develop a Python program in a function form with parameters K and searchFor c. Try the algorithm for (i) K=4, searchFor=13 & (ii) K=5, searchFor=44 considering the following list: [2,7,8,9,11,13,16,21,28,29,30,34,37,39,41,44,47,48]| d. Considering the list in (c), which K-searchFor combination would lead to the highest number of jumps?Implement the following two sorting algorithms in a program called p3.py. Write two separate functions for these algorithms. Both functions must take a list of integers as the input parameter.1) Bogosort: first shuffle the list argument (i.e., randomize the positions of every element) and then check to see if the result is in sorted order. If it is, the algorithm terminates successfully and returns True, but if it is not then the process must be repeated.2) Bozosort: choose two elements in the list at random, swap them, and then check if the result is in sorted order. If it is, the algorithm terminates successfully and returns True, but if it is not then the process must be repeated.Write a main() function and call both sorting functions using the same list as their arguments. The list can be of any size (try a small list first). Does any of your algorithms terminate? If yes, count the number of iterations it uses to sort the list. Does it always use the same number of repetitions? If…design an algorithm to find all the common elements in two sorted lists of numbers. For example, for the lists 2, 5, 5, 5 and 2, 2, 3, 5, 5, 7, the output should be 2, 5, 5.What is the maximum number of comparisons your algorithm makes if the lengths of the two given lists are m and n, respectively?
- Suppose you have the following sorted list [3, 5, 6, 8, 11, 12, 14, 15, 17, 18] and are using the recursive binary search algorithm. Which group of numbers you will compare to in order to find the key 8. 11, 5, 3, 8 12, 6, 11, 8 3, 5, 6, 8 18, 12, 6, 8Consider sorting n numbers stored in array A by first finding the smallest element of A and exchanging it with the element in A[1]. Then find the second smallest element of A, and exchange it with A[2]. Continue in this manner for the first n -1 elements of A. write pseudocode for this algorithm , which is known as selection sort. What loop invariant does this algorithm maintain? Why does it need to run for only the first n – 1 elements, rather than for all n elements? Give the best-case and worst-case running times of selection sort in Θ-notation.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…