Consider the recurrence T(n). if n ≤ 4 T(n): m) = { T(√n])+d ifn>4 Use the recursion tree technique or repeated substitution to come up with a good guess of a tight bound on this recurrence and then prove your tight bound correct with induction or another technique.
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- Give the uppor-bound for the recurrence T(n)=2π(n/2)+n∧2, using the Recursion Tree method. You must show at least 3 levels of the tree, and give the explicit log base when using the tree height!For each of the following recurrences, verify the answer you get by applying the master method, by solving the recurrence algebraically OR applying the recursion tree method. T(N) = 2T(N-1) + 1 T(N) = 3T(N-1) + nUse a recursion tree to determine a good asymptotic upper bound on therecurrence T(n) = 3T(n/2) + n. Use the substitution method to prove your answer.
- Please explain!! Solve the recurrence: T(n)=2T(2/3 n)+n^2. first by directly adding up the work done in each iteration and then using the Master theorem. Note that this question has two parts (a) Solving IN RECURSION TREE the problem by adding up all the work done (step by step) andFor the algorthim write a recurrence for its runtime, use the recurrence tree method to solve the recurrence, and and find the tightest asymptotic upper bound on the runtime of the algorthim. Algorthim Problem: Algorithm H divides an instance of size n into 4 subproblems of size n/3 each, recursively solves each one, and then takes O(n^2) time to combine the solutions and output the answer.Use a recursion tree to determine a good asymptotic upper bound on following recurrences. Please see Appendix of your text book for using harmonic and geometric series. a) T (n) = T(n/5) + O(n)2 b) T (n) = 10T(n/2) + O(n)2 c) T (n) = 10T(n/2) + Θ (1) d) T (n) = 2T (n/2) + n/ lg n e) T (n) = 2T (n - 1) + Θ (1)
- Using a recursion tree, show the process how to solve the following recurrence in terms of the big O representation. Use the substitution method to verify your result. T(n) = T(n/2)+T(n/3)+cnFor the algorthim write a recurrence for its runtime, use the recurrence tree method to solve the recurrence, and and find the tightest asymptotic upper bound on the runtime of the algorthim. Algorrthim Problem: Algorithm V divides an instance of size n into 6 subproblems of size n/6 each, recursively solves each one, and then takes O(n) time to combine the solutions and output the answer.For any given AVL tree T of n elements, we need to design an algorithm that finds elements x in T such that a <= x <= b, where a and b are two input values. Assume that T has m elements x satisfying a <= x <= b. What is the best time complexity for such an algorithm? A. O(m) B. O(m log n) C.O(m + log n) D.O(log m + log n)
- Please explain Solve the recurrence: T(n)=2T(2/3 n)+n^2. first by directly adding up the work done in each iteration and then using the Master theorem. Note that this question has two parts (a) Solving IN RECURSION TREE the problem by adding up all the work done (step by step) and (b) using Master TheoremConsider a divide-and-conquer algorithm that calculates the sum of all elements in a set of n numbers by dividing the set into two sets of n/2 numbers each, finding the sum of each of the two subsets recursively, and then adding the result. What is the recurrence relation for the number of operations required for this algorithm? Answer is f(n) = 2 f(n/2) + 1. Please show why this is the case.The Triangle Vertex Deletion problem is defined as follows: Given: an undirected graph G = (V, E) , with IVI=n, and an integer k>= 0. Is there a set of at most k vertices in G whose deletion results in deleting all triangles in G? (a) Give a simple recursive backtracking algorithm that runs in O(3^k * ( p(n))) where p(n) is a low-degree polynomial corresponding to the time needed to determine whether a certain vertex belongs to a triangle in G. (b) Selecting a vertex that belong to two different triangles can result in a better algorithm. Using this idea, provide an improved algorithm whose running time is O((2.562^n) * p(n)) where 2.652 is the positive root of the equation x^2=x+4