1. a) [10pts] Give pseudocode of a recursive version InsertionSortRec of Insertion Sort, where the input list is an array b) [10pts] Repeat for linked list. 2. Using the Ratio Limit Theorem, prove the following. O(10)O(lnn)O(n)O(nlnn)O(n2)O(n3)O(2n)O(4n). 3. [20pts] Suppose L[0:n1] is a sorted list of distinct integers (negative integers allowed). Design (describe at a high level, then give pseudocode) and analyze (best- and worst-case complexities) an algorithm similar to Binary Search that finds a fixed point, i.e., an index i such that L[i]=i or returns 1 if no such index exists. 4. a) [15pts] Prove by induction that (fib(n)fib(n+1))=(0111)n(01) b) [5pts] Briefly describe how the formula in part a) can be employed to design an algorithm for computing fib(n) using O(logn) multiplications. 5. A natural question to ask related to MergeSort is whether we can do better by dividing the list into more than two parts. It turns out we can't do any better. In this exercise you will verify this fact mathematically. TriMergeSort, given below, is a variant of MergeSort, where the list is split into three equal parts instead of two. For convenience, assume that n=3k, for some nonnegative integer k. a) [9pts] Give a recurrence relation for the worst-case complexity W(n) of TriMergeSort for an input list of size n. b) [9pts] Solve the recurrence formula you have given in (a) to obtain an explicit formula for the worst-case complexity W(n) of TriMergeSort. c) [2pts] Which is more efficient in the worst case, MergeSort or TriMergeSort? Discuss