2.5. Recurrence Relations¶
2.5.1. Recurrence Relations¶
The running time for a recursive algorithm is most easily expressed by a recursive expression because the total time for the recursive algorithm includes the time to run the recursive call(s). A recurrence relation defines a function by means of an expression that includes one or more (smaller) instances of itself. A classic example is the recursive definition for the factorial function:
Another standard example of a recurrence is the Fibonacci sequence:
\[\begin{split}\mbox{Fib}(n) = \mbox{Fib}(n-1) + \mbox{Fib}(n-2)\ \mbox{for}\ n>2; \quad\mbox{Fib}(1) = \mbox{Fib}(2) = 1.\end{split}\]
From this definition, the first seven numbers of the Fibonacci sequence are
Notice that this definition contains two parts: the general definition for \(\mbox{Fib}(n)\) and the base cases for \(\mbox{Fib}(1)\) and \(\mbox{Fib}(2)\). Likewise, the definition for factorial contains a recursive part and base cases.
Recurrence relations are often used to model the cost of recursive
functions.
For example, the number of multiplications required by a recursive
version of the factorial function for an input of size
\(n\) will be zero when \(n = 0\) or \(n = 1\) (the base
cases), and it will be one plus the cost of calling fact on a
value of \(n-1\).
This can be defined using the following recurrence:
As with summations, we typically wish to replace the recurrence relation with a closed-form solution. One approach is to expand the recurrence by replacing any occurrences of \(\mathbf{T}\) on the right-hand side with its definition.
A slightly more complicated recurrence is
Again, we will use expansion to help us find a closed form solution.
