Recurrence Equation Simplification

Question

Given a pseudocode like this:

int algorithm(int[] a, i, j)
    if i > j
        return 0

    int n = j - i + 1
    int d = n / 4

    algorithm(a, i, i+3d-1)
    algorithm(a, i+d, j)

    return 1 

The main cost is the cost given by the two recursive calls. The dimension of this two recursive calls should be in my opinion: 3n/4 for the first one (ignoring the 1) and n/4 for the second one. This means:

T(n) = T(3n/4) + T(n/4) + O(1)

• Question 1: is this correct?

• Question 2: is T(n) = 2T(3n/4) + O(1) correct?

Answer 1

In this code, the variables i and j are redundant because only their difference is used. So we can substitute j-i+1 with n everywhere. We can also ignore the argument a for clarity. The code becomes

int algorithm(n)
    if 1>n
         return 0
    int d= n/4
    algorithm(3d)
    algorithm(n-1-d)
    return 1 

Next transformation: substitute d . We can also discard the returned value !

algorithm(n)
    if 1>n
         return
    algorithm(3(n/4))
    algorithm(n-1-n/4)