Lecture 13

Lecture 13: Recursion vs Iteration

Course bulletins:

• Please complete anonymous feedback survey (link in Blackboard announcement)
• Project 1 grades and solutions posted
• Project 2 will be posted by Friday; due Feb 26

Loose end

Let's implement the paper folding sequence recursively.

Cool fact

If you use the infinite paper folding sequence as the binary digits of a real number (starting at $\frac{1}{2}$ place and moving right), you get the paper folding constant.

It is irrational. In 2007 it was shown¹ that this constant is furthermore transcendental, i.e. cannot be expressed in terms of square roots, cube roots, or any solutions of polynomials with rational coefficients.

1  Adamczewski and Bugeaud, On the complexity of algebraic numbers I: Exapansions in integer bases, Annals of Mathematics 165 (2007) 547-565.

Stack overflow

Recursive functions are limited by a maximum call stack size.

Python imposes a limit to prevent the memory area used to store the call stack from running out (a stack overflow), which would abruptly stop the interpreter.

Iterative solutions

Let's write iterative versions of factorial, Fibonacci, and paper folding. (Or as many as time allows.)

Timing comparison

How do iterative and recursive versions compare on speed?

I made a module decs contains a decorator called timed that prints the time a function takes to return.

Question

Why is recursive fact() somewhat competitive, but fib() is dreadfully slow?

Decorator decs.count_calls will keep track of number of function calls.

fact call graph

fib call graph

fib call graph

Memoization

fib computes the same terms over and over again.

Instead, let's store all previously computed results, and use the stored ones whenever possible.

This is called memoization. It only works for pure functions, i.e. those which always produce the same return value for any given argument values.

math.sin(...) is pure; time.time() is not.

memoized fib call graph

memoized fib call graph

Fibonacci timing summary

Measured on a 4.00Ghz Intel i7-6700K CPU (2015 release date) with Python 3.8.5

Memoization summary

Recursive functions with multiple self-calls often benefit from memoization.

Memoized version is conceptually similar to an iterative solution.

Memoization does not alleviate recursion depth limits.