Lecture 10

Lecture 10: Recursion vs Iteration II

Reminders and announcements:

• Project 1 due Fri Feb 9
• Homework 4 coming Thursday afternoon

Memoization

A technique to address this problem:

1. Store the results of expensive calculations.
2. Have subsequent calls retrieve stored answers instead of calculating.

Let's memoize fib.

memoized fib call graph

memoized fib call graph

Fibonacci timing summary

Measured on my old office PC (2015, Intel i7-6700K) with Python 3.8.5

Call counts

Another way to measure the cost of a recursive function is to count how many times the function is called.

Let's do this for recursive fib.

Corollary: $T(n) = 2F_{n+1}-1$.

Proof of corollary: Both sequences $T(n)$ and $2F_{n+1}-1$ have the same first two terms, and obey the same recursion relation. Induction.

Corollary: $T(n) = 2F_{n+1}-1$.

Proof of corollary: Let $S(n) = 2F_{n+1}-1$. Then $S(0)=S(1)=1$, and $$\begin{split}S(n) &= 2F_{n+1}-1 = 2(F_{n} + F_{n-1}) - 1\\ &= (2 F_n - 1) + (2 F_{n-1}-1) + 1\\ & = S(n-1) + S(n-2) + 1\end{split}$$ Therefore $S$ and $T$ have the same first two terms, and follow the same recursive definition based on the two previous terms. By induction, the set of $n$ such that $T(n) = 2F_{n+1}-1$ is all of $\mathbb{N}$.

Corollary: Every time we increase $n$ by 1, the naive recursive fib does $\approx61.8\%$ more work.

(The ratio $F_{n+1}/F_n$ approaches $\frac{1 + \sqrt{5}}{2} \approx 1.61803$.)

Recursion with backtracking

How do you solve a maze?

Recursion with backtracking

How do you solve a maze?

My guess at your approach:

• Follow a path with your eyes, choosing turns somewhat arbitrarily.
• If you reach a dead end, go back a bit and reconsider.

A formal version of this recursion with backtracking. Start with

• The maze
• A path from the start to somewhere

Add one more step to the path.

Make a recursive call to see what happens after that.

If the answer is "nothing good", consider a different next step.

Backtracking

Two stop conditions: Dead end or goal.

If goal: Great! Return the path as solution.

If dead end: Return None to tell whoever called us to try something else (to backtrack).

Depth first

This method is also called a depth first search for a path through the maze.

Here, depth first means that we always add a new step to the path before considering any other changes (e.g. going back and modifying an earlier step).

Maze coordinates

Maze API

Let's explore the Maze class from maze.py.