A certain recursive algorithm takes an input list of n elements. Divides the list into yn sub-lists, each with /n 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 a scenario in which you are presented with a data set of length K. Write a simple recursive algorithm to choose all possible pairs of elements in the set. Assess the computational complexity and explain your calculations.

Another recursive algorithm is applied to some data A = (a1, ..., am) where m = 2* (i.e. 2, 4, 8,16 ...) where x is an integer ≥ 1. The running time T is characterised using the following recurrence equations: T(1) = c when the size of A is 1 T(m) = 2T (2) + c otherwise Determine the running time complexity of this algorithm.

Give a recursive algorithm which takes as input a sequence of numbers and returns the minimum (i.e., smallest) member in the sequence. Your algorithm should not use an iterative loop. FindMin (al, a2,..., an) If (length (alan) = 1), Return (_A_) m: FindMin (al an-1) // The recursive call If (an S m) Else End-if Return (_B__) Return (_C_)

write a computer program that produces the desired output from the given input. Input: Elements in a finite set SOutput: Elements in ℘(S)Algorithm: Use recursion.

The n-th harmonic is the sum of the reciprocals of the first n natural numbers given by: 17 1.1 1 1 H1 =Σ =1+=+=+ - 2 3 4 k=1 k +. + n (i) Write a recursive algorithm of this function returning, H, with nε N. (ii) Give a recurrence relation for the number of divisions the recursive algorithm calculates for an input n & N. iii) Solve the recurrence relation and give the O(n) of the algorithm.

Given three sequences of length m, n, and p each, you are to design and analyze an algorithm to find the longest common subsequence (LCSS) for the three sequence. Is it possible to use dynamic programming to find the LCSS between the three sequences? If yes, provide recursive solution to the above problem.

Given the following recursively defined set S: Basis: 0 € S and 7 € S Recursive rule: if x ES and y = S, then: • x+yЄS x-yes Prove that every element in S is divisible by 7.

A recursive algorithm is applied to some data A = (a1,..., am) where m≥ 2. The running time T is characterised using the following recurrence equations: T(2) = c when the size of A is 2 T(m) = T(m-1) + 2c otherwise Determine the running time complexity of this algorithm.

With the Binary Search algorithm, the algorithm can be developed by the loop-based form as well as a recursive form. Which of the following is not true? O If the item is not found, the loop version returns when the range bounds reach, but the recursive version finishes when the recursive depth is more than half the initial search range. The search range starts with the whole array, and only the recursive version can work on a subrange passed through arguments. O Both forms of the program divide the search range repeatedly in half. O If the item is found, the loop version returns from the entire method, whereas the recursive version returns from one level of recursion.