Class 41: Tail Recursion (1)

Held: Monday, 14 April 2003

Summary: Today we revisit recursion and consider a more efficient way to write recursive procedures. This technique is called tail recursion.

Related Pages:

• EBoard.
• Reading on Tail Recursion.
• Lab on Tail Recursion.

Notes:

• Are there questions on the project?
• What did you think about meeting in Saints' Rest?

Overview:

• Kinds of recursion.
• Why do tail recursion.
• Generating lists tail-recursively.
• Lab: Tail Recursion.

Two Kinds of Recursion

• We've seen (at least) two kinds of shallow recursion in Scheme.
• Recursion in which you do something with the recursive result.
  • E.g., definition of add-to-all, factorial, traditional right-associative sum.
• Recursion in which you simply return the recursive result, unchanged.
  • E.g., member?, the left-associative sum.
• The latter kind of recursion is called tail recursion and, in Scheme, is likely to be faster. Why?
• Let's look at the two versions of sum (in class).
  • Version 1, right associative, not tail recursive.

    (define sumr
      (lambda (values)
        (if (null? values) 0
            (+ (car values) (sumr (cdr values))))))
    

      + Version 2, left associative, tail recursive.
    (define suml
      (lambda (values)
        (letrec ((suml-helper
                  (lambda (remaining-values partial-sum)
                    (if (null? remaining-values) partial-sum
                        (suml-helper (cdr remaining-values)
                                     (+ partial-sum
                                        (car remaining-values)))))))
        (suml-helper values 0))))
    

• Let's sum the list (1 2 3 4) and see if one seems to be easier to deal with.
  • Right associative

    (sumr (1 2 3 4))
    --> (+ 1 (sumr (2 3 4)))
    --> (+ 1 (+ 2 (sumr (3 4))))
    --> (+ 1 (+ 2 (+ 3 (sumr (4)))))
    --> (+ 1 (+ 2 (+ 3 (+ 4 (sumr ())))))
    --> (+ 1 (+ 2 (+ 3 (+ 4 0))))
    --> (+ 1 (+ 2 (+ 3 4)))
    --> (+ 1 (+ 2 7))
    --> (+ 1 9)
    --> 10
    

  • Left associative

    (suml (1 2 3 4))
    --> (suml-helper (1 2 3 4) 0)
    --> (suml-helper (2 3 4) 1)
    --> (suml-helper (3 4) 3)
    --> (suml-helper (4) 6)
    --> (suml-helper () 10)
    --> 10
    

• Which would you prefer to evaluate by hand?
• In the right-associative case, Scheme needs to remember the things to do once the recursion reaches the base case.
  • Remembering that stuff takes extra memory and time.
• Hence, tail-recursive procedures are normally quicker than non-tail-recursive procedures.
• We can also use named lets for tail-recursive procedures.

Making Procedures Tail Recursive

• The normal strategy for making procedures tail recursive is the one we used for suml:
  • Make a helper procedure with one additional parameter.
  • The additional parameter accumulates partial solutions. Hence, we often call it an accumulator.
• You can often transform a non-tail-recursive (but recursive) procedure in to a tail-recursive procedure:
  • Build that helper with an extra parameter.
  • The accumulator normally starts out with the value from the old base case (the one in the non-tail-recursive procedure).
  • The operation used to update the accumulator is similar to (but not necessarily identical to) the one used in the standard recursive call.

Tail Recursion and List Generation

• While tail recursion is generally a good thing, it's a bad idea to spend lots of extra computational effort on building the accumulated partial result.
  • The problem appears most frequently in procedures that build lists.
• Consider the problem of adding two to each value in a list.
• Suppose our accumulator contains all the values we've added two to.
• Then at each step, we'll need to put the next value at the end of the accumulator.
  • How do you add to the end of a list?
  • How much computational effort does it involve?
• You'll find that your procedure spends a lot of time walking through the accumulated result.
• Can we do something better? Yes. Accumulate the reverse of the desired result.
  • Updating at each step is each; just cons to the front.
  • We do need to spend some extra effort at the end to reverse that result.
• Alternately, we can reverse the parameter before calling the helper.
• See the reading for more details.

Lab

• If there's time, start the lab on list recursion.
• If not, we'll start it tomorrow.