Class 14: Repetition with Recursion

Held Wednesday, February 14, 2001

Summary

Today we begin to consider one of the most powerful tools in Scheme, recursion. Recursion allows you to repeat operations.

Notes

• Are there questions on homework 2?
• Happy valentines day!
• We'll spend a few days on recursion, so don't worry if you don't get it all today.

Overview

• Repetition
• Recursion
• Recursion in Scheme
• Examples: Sum

Repetition

• You may recall that when we first considered algorithms we identified a number of key aspects of algortihms:
  • variables: the ability to name things;
  • conditions: the ability to choose between things;
  • procedures: the ability to name (and parameterize) collections of steps;
  • repetition: the ability to do something multiple times;
  • input and output: the ability to get and report values.
• We've already seen how to do almost all of these things, except for repetition.
• Examples of repetition from baking:
  • Stir the mix 50 times
  • Knead the bread dough until it feels like your earlobe
  • Bake until golden-brown.
• Examples of repetition from mathematics:
  • Sum these values
  • Find the smallest of these values
• Examples of repetition from everyday life:
  • Naively find a name in the phone book
  • Do I have a CD by Van Morrison?

Recursion

• In Scheme, the most common mechanism for repetition is recursion.
• To do something that involves repeated actions, you
  • Do one action
  • Repeat the rest
  • Combine the results if necessary.
• For example, to stir your cake mix 50 times, you stir it one time and then stir it 49 more times.
• More generally, to stir a cake mix n times, you stir it one time and then n-1 more times.
• Similarly, to knead dough until its the right consistency, you knead it a little, check the consistency, and, if it's not the right consistency, knead it until its the right consistency.
• In the case of mathematics, to sum a list we might add the first value to the sum of the remaining values (or add the last value to the sum of the initial values).
• There are a few key aspects to recursive design:
  • You need to know when you're done (and what to do when you're done). This aspect of recursive design is called the base case.
  • You need to know what to do when you're not done. Here, you should do a little, try again, and then perhaps combine the results. This aspect of recursive design is called the recursive case.
  • You need to be sure that you're getting closer to the base case (otherwise you'll never stop).

Recursion in Scheme

• Here's a typical recursive procedure:

  (define proc
    (lambda (val)
      (if (base-case-test)
          (base-case val)
          (combine (partof val)
                   (proc (update val))))))
  

• When the value you're working with is a list and your base case is the null list, the form is somewhat simpler:

  (define proc
    (lambda (lst)
      (if (null? lst)
          null-case
          (combine (onestep (car val))
                   (proc (cdr val))))))
  

• For example, consider sum (see the reading).