Class 23: Recursion Basics

Held: Wednesday, 1 March 2017

We begin our exploration of *recursion, the most general form of repetition available in Scheme. You can use recursion to both build and iterate over different kinds of values.*

Preliminaries

Overview

• Background
• Key ideas in recursion
• Sample recursive procedures
• Expressing recursion in Scheme

Related Pages

• Eboard 23 (Source)

• Reading: Recursion Basics

• Lab: Recursion Basics

Updates

News / Etc.

• New partners!

Quiz topics

• Anonymous procedures
• Local bindings (let let*)
• image-compute
• Heterogeneous lists and car, cons, and cdr

Rotating reminders

• Visit mentor sessions! We have mentor sessions on Thursday evenings from 7:00-8:00 p.m. and 8:00-9:00 p.m. in the CS Commons.
• Visit review sessions! I run review sessions on Thursdays at 9am in this room.

About the exam

• We will quickly look through the exam.

Upcoming Work

• Exam 2 prologue due Friday.
• Exam 2 due next Tuesday.
• No lab writeup!
• Revisit: Recursion basics

Extra credit (Academic/Artistic)

• CS Extras, Thursday at 4:15 pm, Sam’s research and PM’s research

Extra credit (Peer)

• P.U. ‘n club, Saturday, 4:00 p.m., Younker lounge

Extra credit (Misc)

Good things to do

• Bagel party in Math commons, Thursday, 7-9 pm. Lox status still unclear.

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.
  • basic values and operations: the things we can work with
  • sequencing choosing the order in which we do things
• We’ve already seen how to do all of these things.
• 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?
• We know a few ways to repeat actions:
  • Using lists: map
  • Using images: image-variant, image-transform!, and image-compute-pixels!.
• Today we begin to consider more general forms of repetition.

Some Challenges

• You may recall that one of the issues in writing algorithms is that we are often limited to a few basic operations. Let’s explore how we might accomplish some more challenging tasks with only a few basic operations and a powerful helper function.
• We will develop some functions to answer questions about lists under the following assumptions:
  • You get a collection of values
  • You may assess only the first item in the collection (assuming there is such an item)
  • You may pass the rest of the collection to the person next to you and ask one question of that person
  • You have answer to give if there are no items in the collection
• The silly model I have:
  • You have an executive assistant. You can ask your executive assistant to do most of the work, but not all of it.
  • In particular, if you have a non-empty list, you have to remove the car of the list and you may not look at the cdr. (You have to ask your assistant to deal with the cdr.)
• Here are some possibilities
  • Counting the number of values we have
  • Determining whether a collection contains a value
  • Finding the alphabetically first thing in a collection
  • Finding all values in a collection that meet some criteria
  • (Maybe) summing some values
• After I’ve done a few, I’ll have you write a few
• If we do this right, you’ll see some interesting patterns

Recursion

• In Scheme, the most common mechanism for repetition is recursion.
• To do something that involves repeated actions, you
  • Do one action, computing a result
  • Do the remaining actions, computing a result
  • 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 it’s 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

Remember that I tend to think in abstractions first. But we’ll look at some concrete forms, too.

Here’s the form of a typical recursive procedure:

(define PROC
  (lambda (VAL)
    (if (BASE-CASE-TEST)
        (COMPUTE-BASE-CASE VAL)
        (COMBINE (EXTRACT-DATA VAL)
                 (PROC (SIMPLIFY 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 (EXTRACT-DATA (car LST))
                 (PROC (cdr LST))))))

Sometimes it’s useful to grab the recursive result first, particularly if you’re going to use it in multiple ways.

(define PROC
  (lambda (LST)
    (if (null? LST)
        NULL-CASE
        (let ((recursive-result (PROC (cdr LST))))
          (if (TEST)
              (COMBINE-1 (EXTRACT-DATA-1 (car LST)) recursive-result)
              (COMBINE-2 (EXTRACT_DATA-2 (car LST)) recursive-result))))))

Lab

• Start the lab on recursion
• We will continue this lab in the next class session.
• Be prepared to reflect and to ask questions.