Problem 5 :Recursion Tree Time Complexity Find the Worst case time Complexity of the following recursive functions
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- Given an array of integers, find the longest non-decreasing subsequences (the subsequence does not need to be consecutive). For example, A = [8,5,2,10,3,6,9,7] contains the longest subsequences [2,3,6,9] and [2,3,6,7].a. Formulate the recursive relation of the optimal solution (do not miss the base case); b. Design a bottom-up (iterative) algorithm to calculate the length of the longest non-decreasing subsequences (pseudo code); c. Analyze the complexity of your algorithm.Given an array of integers, find the longest non-decreasing subsequences (the subsequence doesnot need to be consecutive). For example, A = [8,5,2,10,3,6,9,7] contains the longest subsequences[2,3,6,9] and [2,3,6,7].(1) Formulate the recursive relation of the optimal solution (do not miss the base case);(2) Design a bottom-up (iterative) algorithm to calculate the length of the longest non-decreasingsubsequences (pseudo code);(3) Analyze the complexity of your algorithmProblem 5: Let T be a binary search tree with n elements. Write an O(n)-time algorithmthat builds a skip list S from T such that, in the worst-case, the time complexity of thesearch operation in S is O(log n).Hint: construct a deterministic skip list.
- Provide an example of a recursive function in which the amount of work on each activation is constant. Provide the recurrence equation and the initial condition that counts the number of operations executed. Specify which operations you are counting and why they are the critical ones to count to assess its execution time. Draw the recursion tree for that function and determine the Big-© by determining a formula that counts the number of nodes in the tree as a function of n.-Use course material of CMSC 451 and additional sources. Need reference also . Note dont copy from any other sites including chat gpt also . Previous ans here was also copy from chat gpt .Provide an example of a recursive function in which the amount of work on each activation is constant. Provide the recurrence equation and the initial condition that counts the number of operations executed. Specify which operations you are counting and why they are the critical ones to count to assess its execution time. Draw the recursion tree for that function and determine the Big-© by determining a formula that counts the number of nodes in the tree as a function of n.-Use course material of CMSC 451 and additional sources. Need reference also . Note dont copy from any other sitesUse 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.
- If you are given a set S of integers and a number t, prove that this issue falls into the NP class. Is there a subset of S where the total number of items is t? Note: Complexity in Data Structures and AlgorithmsDesign a nondeterministic polynomial-time algorithm for following problem: Given a graph G = (V, E), is there a spanning tree with exactly two leaves? Please give an analysis on correctness and running time of your algorithm.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)+cn
- 3 Question 4 Let G=(V,E) be undirected graph and: (u,v)∈E. (A) Build an algorithm in complexity O (V + E) that finds a spanning tree of G that includes the (u, v). (B) Build an efficient algorithm that finds all the ribs of G that are common to all the spanning trees of G.A. Construct a RECURSIVE solution for following iterative solution to find a value in a circular-linked list B. Analyze the runtime complexity (correctly explain the scenario, show how much work would be done, and represent the work using asymptotic notations, i.e. big O), of the given iterative solution and your recursive solution for the best case scenario and the worse case scenario. C. prove the correctness of your recursive solution by induction /**@param value - a value to search for@return true if the value is in the list and set the current reference to itotherwise return false and not updating the current reference*/public boolean find(T value){ if(this.cur == null) return false; //get out, nothing is in here Node<T> tmp = this.cur; //start at the current position if(tmp.data == value) return true; // found it at the starting location tmp = tmp.next; //adv. to next node, if there is one while(tmp != cur) { if(tmp.data == value){ this.cur = tmp;…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