THIS IS THE QUESTION ABOUT THE D&L OF ALGORITHMS

Asymptotic Notation, Comparing Orders and Order Formula

QUESTION 1: Using the Ratio Limit Theorem , prove the following.

O(108) O( ln n) O(n) O( n ln n) O( n^2) O( n^3) O( 2^n) O( 3^n) .

QUESTION 2: Obtain a formula for the order of .

Recurrence Relations for Complexity Analysis

QUESTION 3: A very 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, at least in the worst case. 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= 3^k , for some nonnegative integer k.

(a) Give a recurrence relation for the worst-case complexity W(n) of TriMergeSort for an input list of size n.

(b) Solve the recurrence formula you have given in (a) to obtain an explicit formula for the worst-case complexity W(n) of TriMergeSort.

(c) Which is more efficient in the worst case, MergeSort or TriMergeSort? Discuss.

QUESTION 4: Consider the sorting algorithm Insertion Sort for sorting a list L[0:n 1]. Derive a recurrence relation for the worst-case complexity W(n) and solve.

Solving Recurrence Relations

QUESTION 5: Solve the following recurrence relations:

a. t(n) = 3t (n-1) + n, n 1, init.cond. t(0) = 0.

b. t(n) = 4t (n-1) + 5, n 1, init.cond. t(0) = 2.

Mathematical induction

QUESTION 6: a. Prove by induction that

=

b. Briefly describe how the preceding formula can be employed to design an algorithm for computing fib(n) using only at most 8log2n multiplications.

Please help me solve for all the six questions. Thanks.

S(n) = (logi)