Write a recursive method which searches for a key in an ascending list (binary search). Show that this function takes O(log n) time. Time Complexity
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- A certain recursive algorithm takes an input list of n elements. Divides the list into Vn sub-lists, each with yn elements. Recursively solves each of these yn smaller sub- instances. Then spends an additional 0(n) time to combine the solutions of these sub- instances to obtain the solution of the main instance. As a base case, if the size of the input list is at most a specified positive constant, then the algorithm solves such a small instance directly in 0(1) time. a) Express the recurrence relation that governs T(n), the time complexity of this algorithm. b) Derive the solution to this recurrence relation: T(n) = 0(?). Mention which methods you used to derive your solution.Consider the following implementation of the recursive binary searchalgorithm that passes arrays in its recursive calls. Give a recurrence for the worst-case running time of the above binary search implementation.Solve the recurrence to derive a tight bound on the running time of the above algorithm.Write for the following problem a recursive algorithm whose worst-case timecomplexity is not worse than Θ(n ln n). Given a list of n distinct positiveintegers, partition the list into two sublists, each of size n/2, such that thedifference between the sums of the integers in the two sublists is maximized.You may assume that n is a multiple of 2.
- Given an unsorted array, A, of integers and an integer k, write a recursivejava code for rearranging the elements in A so that all elements less than or equal to k come before any elements larger than k. What is the running time of your algorithm on an array of n values.Give a recursive (decrease-by-one) algorithm for finding the position of the smallest element in an array of n real numbers. and Determine the running time complexity of this algorithm.Suppose a recursive algorithm performs 2 recursive calls. Assume the first recursive call isof size at most 70% the original input size, and the second call is of size at most 25% of theoriginal input size. In addition, the algorithm performs O(n) additional work after makingthese recursive calls. What is the big-Oh run time of this algorithm?
- Use induction to develop a recursive algorithm for finding theaverage of n real numbers in an array A[1..n]. Analyze its running time.RecursionWe have learned the binary search algorithm this semester, but we implementbinary search using a while loop. In this exercise, we are going to see binarysearch implemented by recursion and trace the recursion.Tracing the Recursion. Observe the recursive solution provided below and answer the followingquestions:1. Which line(s) of this program define(s) the base case of the binary() method?2. Which line(s) of this program include recursive call(s)?3. Trace the recursion below. You must show the tracing step by step (write themdown); otherwise – little to no credit!4. At what step of your recursion tracing did you hit the base case?5. What is the final output of this code?Your main task is to write a recursive function sierpinski() that plots a Sierpinski triangle of order n to standard drawing. Think recursively: sierpinski() should draw one filled equilateral triangle (pointed downwards) and then call itself recursively three times (with an appropriate stopping condition). It should draw 1 filled triangle for n = 1; 4 filled triangles for n = 2; and 13 filled triangles for n = 3; and so forth. Sierpinski.java When writing your program, exercise modular design by organizing it into four functions, as specified in the following API: public class Sierpinski { // Height of an equilateral triangle with the specified side length. public static double height(double length) // Draws a filled equilateral triangle with the specified side length // whose bottom vertex is (x, y). public static void filledTriangle(double x, double y, double length) // Draws a Sierpinski triangle of order n, such that the largest filled //…
- Recursively define the set P of strings (composed of lowercase letters) that have an even number of each vowel (i.e. a word with 2 a's and 4 e's is valid but not 1 a and 5 e's). Use set A to represent all alphabet characters, set V to represent all vowels.Determine a recurrence relation for the divide-and-conquer sum-computation algorithm. The problem is computing the sum of n numbers. This algorithm divides the problem into two instances of the same problem: to compute the sum of the first ⌊n/2⌋ numbers and compute the sum of the remaining ⌊n/2⌋ numbers. Once each of these two sums is computed by applying the same method recursively, we can add their values to get the sum in question. A-)T(n)=T(n/2)+1 B-)T(n)=T(n/2)+2 C-)T(n)=2T(n/2)+1 D-)T(n)=2T(n/2)+2Plz solve correctly and don't use chat gpt. Give a recursive definition for the set of all strings of 0’s and 1’s that have more 0’s than 1’s.
