Exercise 3. Consider the following algorithms A and B for the problem of computing 2n (mod 317) (This is the modular exponentiation problem that we have discussed in class. Algorithm A is the one that we have seen in class, and algorithm B is a variant of it. Algorithm A. mod_exp_A (n) { }} if (n== 0) return 1; else { }} t = mod_exp_A (n/2); if (n is even) return t*t (mod 317); if (n is odd) return t*t*2 (mod 317); Algorithm B. mod_exp_B (n) { if (n== 0) return 1; else { if (n is even) return mod_exp_B (n/2) * mod_exp_B (n/2) (mod 317); if (n is odd) return mod_exp_B (n/2) * mod_exp_B (n/2) *2 (mod 317); 2

Using recursion, write a Python function def before(k,A) which takes an integer k and an array A of integers as inputs and returns a new array consisting of all the integers in A which come before the last  occurrence of k in A, in the same order they are in A. For example, if A is [1,2,3,6,7,2,3,4] then before(3,A) will return [1,2,3,6,7,2]. If k does not occur in A, the function should return None.

Consider a recursive function, called f, that computes powers of 3 using only the + operator. Assume n > = 0. int f(int n) { if (n == 0) return 1; return f(n-1) + f(n-1) + f(n-1); } Give an optimized version of f, called g, where we save the result of the recursive call to a temporary variable t, then return t+t+t. i got int g(int n) { if (n == 0) return 1; int t = g(n - 1); return t+t+t; } so now Write a recurrence relation for T(n), the number addition operations performed by g(n) in terms of n.

Give a recursive definition for the set of all strings of a’s and b’s that begins with an a and ends in a b. Say, S = { ab, aab, abb, aaab, aabb, abbb, abab..} Let S be the set of all strings of a’s and b’s that begins with a and ends in a b. The recursive definition is as follows – Base:... Recursion: If u ∈ S, then... Restriction: There are no elements of S other than those obtained from the base and recursion of S.