Can someone help solve this recurrence relation? [closed]

Answer 1

Let's look at the first recurrence, T(n) = 2T(n/2) + 1. The n/2 is our clue here: each nested term's parameter is half that of its parent. Therefore, if we start with n = 2^k then we will have k terms in our expansion, each adding 1 to the total, before we hit our base case, T(0). Hence, assuming T(0) = 1, we can say T(2^k) = k + 1. Now, since n = 2^k we must have k = log_2(n). Therefore T(n) = log_2(n) + 1.

We can apply the same trick to your second recurrence, T(n) = T(n^0.5) + 1. If we start with n = 2^2^k we will have k terms in our expansion, each adding 1 to the total. Assuming T(0) = 1, we must have T(2^2^k) = k + 1. Since n = 2^2^k we must have k = log_2(log_2(n)), hence T(n) = log_2(log_2(n)) + 1.