Problem 1: Solve the recurrence relation T(n) T(n 1) and give the analysis of the algorithm that it describes, given that T(1) = 2.

Problem 2: Analyze an algorithm whose behavior is described by the recurrence relation T(n) 4T(n/2) + 2n3.

Problem 3: Suppose you have the following two algorithms.

Algorithm A solves problems of size n by dividing them into five subproblems of size n/2, recursively solving each subproblem, and then combining the solutions using a linear number of operations. Algorithm B, which solves the same set of problems as algorithm A, solves problems of size n by dividing them into four subproblems of size n/2, recursively solving each subproblem, and then combining the solutions using at least n and at most n2 operations to do so. Analyze these two algorithms, giving as complete a big-Oh, big-Omega, or big-Theta expression or set of expressions as possible. Show your work and explain any assumptions you make. State which algorithm, other things being equal, you would prefer to use, and why.