Analysis of recursive algorithm. Consider the pseudocode of the following three algorithms for computing 2n, where n is a non-negative integer.

Alg1 (n)

if (n == 0) return 1;

return 2 * Alg1(n - 1);

end

Alg2 (n)

if (n == 0) return 1;

return Alg2(n - 1) + Alg2(n - 1);

end

Alg3 (n)

if (n == 0) return 1;

m = floor (n / 2);

p = Alg3(m);

p = p * p;

if (n % 2 == 1) // n is an odd number

return 2 * p;

else // n is an even number;

return p;

end

end

a. (6 points) Trace the three algorithms using two small examples (n = 2 and n = 3) to nd out the outputs of the three algorithms, respectively.

b. (4 points) Prove the correctness of Alg1 using induction

c. (12 points) Let A(n), B(n), and C(n) be the running time of the three algorithms, respectively, as a function of n. Write down the recurrences for A(n), B(n), and C(n).

d. (18 points) Analyze and compare the running time of the three algorithms above. You can use either recursion tree or master method to solve the recurrences.