# Tail Recursion

Summary: In this laboratory, you will reflect on the appropriate and inappropriate uses of tail recursion.

Contents:

## Exercises

### Exercise 0: Preparation

a. Review the reading on tail recursion.

b. Start DrScheme.

### Exercise 1: Identifying Tail-Recursive Procedures

Identify three (or more) tail-recursive procedures you've already written.

### Exercise 2: Does Tail Recursion Really Make a Difference?

In DrScheme, you can find out how long it takes to evaluate an expression by using `(time expression)`, which prints out the time it takes to evaluate and then returns the value computed. Please scan the reading on using `time` before continuing this exercise.

a. Try the two versions of `factorial` on some large numbers. Does one seem to be faster than the other? You can find the two versions of factorial in the notes on this problem.

b. Try the three versions of `add-to-all` on some lists of varying sizes (you'll probably need at least a hundred values in each list, and possibly more) until you can determine a difference in running times. Note that you should make sure to build the list before you start timing.

### Exercise 3: Selecting Values

Here's an attempt to write a tail-recursive version of the `select` procedure that selects all the values in a list for which a predicate holds.

```;;; Procedure:
;;;   select
;;; Parameters:
;;;   pred?, a unary predicate
;;;   lst, a list of values
;;; Purpose:
;;;   Selects all values in lst for which pred? holds.
;;; Produces:
;;;   selected, a list
;;; Preconditions:
;;;   pred? can be applied to any element of lst.
;;; Postconditions:
;;;   every value in selected appears in lst.
;;;   (pred? val) holds for every val in selected.
;;;   (pred? val) holds for every val in lst not in selected.
;;;   values in selected appear in the same order in which they appear in lst.
(define select
(lambda (pred? lst)
(let kernel ((lst lst)
(selected null))
(cond
; If nothing's left in the original list, use selected
((null? lst) selected)
; If the predicate holds for the first value in lst,
; select it and continue
((pred? (car lst)) (kernel (cdr lst) (cons (car lst) selected)))
; Otherwise, skip it and continue
(else (kernel (cdr lst) selected))))))
```

a. What do you expect the results of the following two expression to be?

```(select odd? (list 1 2 3 4 5 6 7))
(select even? (list 1 2 3 4 5 6 7))
```

c. Correct any errors in `select` that you observed in a or b.

### Exercise 4: Finding the Longest String on a List

Write a tail-recursive `longest-string-on-list` procedure which, given a list of strings as a parameter, returns the longest string in the list.

Note that you can use `string-length` to find the length of a string.

### Exercise 5: Finding the Index

Define a tail-recursive procedure ```(index val lst)``` that returns the index of val in lst. That is, it should return zero-based location of `val` in `lst`. If the item is not in the list, the procedure should return -1. Test your procedure on:

```(index 3 (list 1 2 3 4 5 6)) => 2
(index 'so (list 'do 're 'mi 'fa 'so 'la 'ti 'do)) => 4
(index "a" (list "b" "c" "d")) => -1
(index 'cat null) => -1
```

### Exercise 6: Iota, Once Again

Recall that `(iota n)` is to be the list of all nonnegative integers less than n in increasing order. Define and test a tail-recursive version of `iota`.

### Exercise 7: Reverse

Write your own tail-recursive version of `reverse`.

## For Those Who Finish Early

If you finish early, you may want to consider the following problem.

Here's a non-tail-recursive version of `append`.

```(define append
(lambda (list-one list-two)
(if (null? list-one) list-two
(cons (car list-one)
(append (cdr list-one) list-two)))))
```

a. Write your own tail-recursive version.

b. Determine experimentally which of the three versions (built-in, given above, tail-recursive) is fastest.

## Notes on Problem 2

Here are the two versions of `factorial`.

```;;; Procedures:
;;;   factorial
;;;   factorial-tr
;;; Parameters:
;;;   n, a non-negative integer
;;; Purpose:
;;;   Computes n! = 1*2*3*...*n
;;; Produces:
;;;   result, a positive integer
;;; Preconditions:
;;;   n is a non-negative integer [unchecked]
;;; Postconditions:
;;;   result = 1*2*3*...*n

(define factorial
(lambda (n)
(if (<= n 0) 1
(* n (factorial (- n 1))))))

(define factorial-tr
(lambda (n)
(letrec ((kernel (lambda (m acc)
(if (<= m 0) acc
(kernel (- m 1) (* acc m))))))
(kernel n 1))))
```

Here are the three versions of `add-to-all`.

```;;; Procedures:
;;; Parameters:
;;;   value, a number
;;;   values, a list of numbers
;;; Purpose:
;;;   Creates a new list by adding value to each member of values.
;;; Produces:
;;;   new-values, a list of numbers
;;; Preconditiosn:
;;;   value is a number [unchecked]
;;;   values is a list of numbers [unchecked]
;;; Postconditions:
;;;   (list-ref new-values i) equals (+ value (list-ref values i))
;;;     for all reasonable values of i.

(lambda (value values)
(if (null? values) null
(cons (+ value (car values))

(lambda (value values)
(let kernel  ((values-remaining values)
(values-processed null))
(if (null? values-remaining) values-processed
(kernel (cdr values-remaining)
(append values-processed
(list (+ value (car values-remaining)))))))))

(lambda (value values)
(let kernel  ((values-remaining values)
(values-processed null))
; If we've run out of values to process,
(if (null? values-remaining)
(reverse values-processed)
(kernel (cdr values-remaining)
(cons (+ value (car values-remaining))
values-processed))))))
```

## History

Monday, 30 October 2000 [Samuel A. Rebelsky]

Sunday, 8 April 2001 [Samuel A. Rebelsky]

Thursday, 7 November 2002 [Samuel A. Rebelsky]

• Added problem on `select`.
• Moved `append` to if you have time.

Monday, 11 November 2002 [Samuel A. Rebelsky]

Sunday, 11 April 2003 [Samuel A. Rebelsky]

• A few more minor updates.
• Validated HTML.

Disclaimer: I usually create these pages on the fly, which means that I rarely proofread them and they may contain bad grammar and incorrect details. It also means that I tend to update them regularly (see the history for more details). Feel free to contact me with any suggestions for changes.

This document was generated by Siteweaver on Tue May 6 09:29:21 2003.
The source to the document was last modified on Mon Apr 14 21:26:50 2003.
This document may be found at `http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2003S/Labs/tail-recursion.html`.

Samuel A. Rebelsky, rebelsky@grinnell.edu