Sorting:
- recursive definition:
  sorting an empty list is the empty list
  else insert the first item into the sorted rest of the list
- needs helper function to insert
  inserting into an empty list is the singleton list
  compare item to first in the list -- what do we do?


Iteration and recursion: Computational processes

- Last time, the question of why
(define (fib n)
  (cond
    [(= n 0) 1]
    [(= n 1) 1]
    [else (+ (fib (- n 1)) (fib (- n 2)))]))
is so much slower than
(define (! n)
  (cond
    [(= n 0) 1]
    [else (* n (! (- n 1)))]))
was asked...

Let's answer that

Do a couple of levels of substitution...
Fibonacci numbers:
  two recursive calls per level ("tree", exponential growth)
  repeated computation

Factorial:
  one recursive call per level
  no repeated computation
  list of n computations must be remembered, then computed
    after unrolling the function calls.

How could we fix the Fibonacci numbers?
  memoization: remember the value of a function after you compute it
  different algorithm

For that matter, can we "fix" the factorial?

"Iteration" versus "Recursion"

Iterative factorials
Iterative exponentiation
Iterative Fibonaccis

Iteration: constant space, keep track of "current" value
Recursion: space grows, need all values

Can we do even better?
Exponentiation...