Section 2.2 describes a method for solving recurrence relations which is based on analyzin...

Section 2.2 describes a method for solving recurrence relations which is based on analyzing the recursion tree and deriving a formula for the work done at each level. Another (closely related) method is to expand out the recurrence a few times, until a pattern emerges. For instance, let’s start with the familiar T(n) = 2T(n/2) + O(n). Think of O(n) as being ≤ cn for some constant c, so: T (n) ≤ 2T (n/2) + cn. By repeatedly applying this rule, we can bound T (n) in terms of T(n/2), then T(n/4), then T(n/8), and so on, at each step getting closer to the value of T(∙) we do know, namely T(1) = O(1).

A pattern is emerging... the general term is

Plugging in k = log2 n, we get T(n)< n T(1) + cnlog2 n = O (n log n).

(a) Do the same thing for the recurrence T (n) = 3T (n/2) + O (n). What is the general kth term in this case? And what value of k should be plugged in to get the answer?

(b) Now try the recurrence T(n) = T(n − 1) + O(1), a case which is not covered by the master theorem. Can you solve this too?