Recurrences arise naturally in analyzing the complexity of recursive algorithms and in probabilistic analysis. We introduce some basic techniques for solving recurrences. A recurrence is a recursive relation for a complexity function T(n). Here are two examples: F(n) = F(n − 1) + F(n − 2) (1) and Looks famil-T(n) = n + 2T(n/2). (2) iar? The reader may recognize the first as the recurrence for Fibonacci numbers, and the second as the complexity of the Merge Sort, described in Lecture 1. These recurrences have 1 the following “separable form”: T(n) = G(n, T(n1),..., T(nk)) (3) where G(x0, x1,..., xk) is a function or expression in k + 1 variables and n1,...,nk are all strictly less than n. Each ni is a function of n. E.g., in (1), we have k = 2 and n1 = n − 1, n2 = n − 2. But in (2), we have k = 1 and n1 = n/2. What does it mean to “solve ” recurrences such as equations (1) and (2)? The Fibonacci recurrence and the Mergesort recurrence has the following well-known solutions:

An unpleasant feature of modern life is that people often tell you certain things are impossible that you wonder, or even strongly suspect, should indeed be possible. I want to talk (of course) about a mathematical instance of this idea. Ironically, mathematics is one of the very few fields in which it is possible to prove – really prove – that something is impossible. However, in my experience impossibility proofs in mathematics often depend sensitively on the precise set of hypotheses. For instance, most students of mathematics know that it is not possible to trisect a 60 degree angle. What this means of course is that it is not possible using a compass and a straightedge. However, if you are allowed to take your straightedge and mark two points on it, then it becomes possible to trisect a 60 degree angle, and in fact any angle. (Keyword: neusis constructions.) The moral here is that, when you are told something is impossible in mathematics, you are actually being told that it is impossible under certain precise conditions. This does not necessarily mean you should give up. Often there are other reasonable conditions under which the impossible becomes possible. Sometimes these

1.1. “Induction is fundamentally discrete... ” 1 1.2....is dead wrong! 2