2.3 Lists, Iteration, and Recursion
Racket is a dialect of the language Lisp, whose name originally stood for “LISt Processor.” The built-in list datatype remains a prominent feature of the language.
The list function takes any number of values and returns a list containing the values:
| > (list "red" "green" "blue") |
'("red" "green" "blue") |
| > (list 1 2 3 4 5) |
'(1 2 3 4 5) |
A list usually prints with ', but the printed form of a list depends on its content. See Pairs and Lists for more information.
As you can see, a list result prints in the REPL as a quote ' and then a pair of parentheses wrapped around the printed form of the list elements. There’s an opportunity for confusion here, because parentheses are used for both expressions, such as (list "red" "green" "blue"), and printed results, such as '("red" "green" "blue"). In addition to the quote, parentheses for results are printed in blue in the documentation and in DrRacket, whereas parentheses for expressions are brown.
Many predefined functions operate on lists. Here are a few examples:
| > (length (list "hop" "skip" "jump")) ; count the elements |
3 |
| > (list-ref (list "hop" "skip" "jump") 0) ; extract by position |
"hop" |
| > (list-ref (list "hop" "skip" "jump") 1) |
"skip" |
| > (append (list "hop" "skip") (list "jump")) ; combine lists |
'("hop" "skip" "jump") |
| > (reverse (list "hop" "skip" "jump")) ; reverse order |
'("jump" "skip" "hop") |
| > (member "fall" (list "hop" "skip" "jump")) ; check for an element |
#f |
2.3.1 Predefined List Loops
In addition to simple operations like append, Racket includes functions that iterate over the elements of a list. These iteration functions play a role similar to for in Java, Racket, and other languages. The body of a Racket iteration is packaged into a function to be applied to each element, so the lambda form becomes particularly handy in combination with iteration functions.
Different list-iteration functions combine iteration results in different ways. The map function uses the per-element results to create a new list:
| > (map sqrt (list 1 4 9 16)) | |||
'(1 2 3 4) | |||
| |||
'("peanuts!" "popcorn!" "crackerjack!") |
The andmap and ormap functions combine the results by anding or oring:
| > (andmap string? (list "a" "b" "c")) |
#t |
| > (andmap string? (list "a" "b" 6)) |
#f |
| > (ormap number? (list "a" "b" 6)) |
#t |
The filter function keeps elements for which the body result is true, and discards elements for which it is #f:
| > (filter string? (list "a" "b" 6)) |
'("a" "b") |
| > (filter positive? (list 1 -2 6 7 0)) |
'(1 6 7) |
The map, andmap, ormap, and filter functions can all handle multiple lists, instead of just a single list. The lists must all have the same length, and the given function must accept one argument for each list:
| |||
'("peanut" "pop" "cracker") |
The foldl function generalizes some iteration functions. It uses the per-element function to both process an element and combine it with the “current” value, so the per-element function takes an extra first argument. Also, a starting “current” value must be provided before the lists:
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14 |
Despite its generality, foldl is not as popular as the other functions. One reason is that map, ormap, andmap, and filter cover the most common kinds of list loops.
Racket provides a general list comprehension form for/list, which builds a list by iterating through sequences. List comprehensions and related iteration forms are described in see Iterations and Comprehensions.
2.3.2 List Iteration from Scratch
Although map and other iteration functions predefined, they are not primitive in any interesting sense. You can write equivalent iterations using a handful of list primitives.
Since a Racket list is a linked list, the two core operations on a non-empty list are
Examples: |
| > (first (list 1 2 3)) |
1 |
| > (rest (list 1 2 3)) |
'(2 3) |
To create a new node for a linked list—
| > empty |
'() |
| > (cons "head" empty) |
'("head") |
| > (cons "dead" (cons "head" empty)) |
'("dead" "head") |
To process a list, you need to be able to distinguish empty lists from non-empty lists, because first and rest work only on non-empty lists. The empty? function detects empty lists, and cons? detects non-empty lists:
| > (empty? empty) |
#t |
| > (empty? (cons "head" empty)) |
#f |
| > (cons? empty) |
#f |
| > (cons? (cons "head" empty)) |
#t |
With these pieces, you can write your own versions of the length function, map function, and more.
Examples: | |||||
| |||||
| > (my-length empty) | |||||
0 | |||||
| > (my-length (list "a" "b" "c")) | |||||
3 |
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If the derivation of the above definitions is mysterious to you, consider reading How to Design Programs. If you are merely suspicious of the use of recursive calls instead of a looping construct, then read on.
2.3.3 Tail Recursion
Both the my-length and my-map functions run in O(n) time for a list of length n. This is easy to see by imagining how (my-length (list "a" "b" "c")) must evaluate:
| (my-length (list "a" "b" "c")) |
| = (+ 1 (my-length (list "b" "c"))) |
| = (+ 1 (+ 1 (my-length (list "c")))) |
| = (+ 1 (+ 1 (+ 1 (my-length (list))))) |
| = (+ 1 (+ 1 (+ 1 0))) |
| = (+ 1 (+ 1 1)) |
| = (+ 1 2) |
| = 3 |
For a list with n elements, evaluation will stack up n (+ 1 ...) additions, and then finally add them up when the list is exhausted.
You can avoid piling up additions by adding along the way. To accumulate a length this way, we need a function that takes both a list and the length of the list seen so far; the code below uses a local function iter that accumulates the length in an argument len:
| (define (my-length lst) |
| ; local function iter: |
| (define (iter lst len) |
| (cond |
| [(empty? lst) len] |
| [else (iter (rest lst) (+ len 1))])) |
| ; body of my-length calls iter: |
| (iter lst 0)) |
Now evaluation looks like this:
| (my-length (list "a" "b" "c")) |
| = (iter (list "a" "b" "c") 0) |
| = (iter (list "b" "c") 1) |
| = (iter (list "c") 2) |
| = (iter (list) 3) |
| 3 |
The revised my-length runs in constant space, just as the evaluation steps above suggest. That is, when the result of a function call, like (iter (list "b" "c") 1), is exactly the result of some other function call, like (iter (list "c") 2), then the first one doesn’t have to wait around for the second one, because that takes up space for no good reason.
This evaluation behavior is sometimes called tail-call optimization, but it’s not merely an “optimization” in Racket; it’s a guarantee about the way the code will run. More precisely, an expression in tail position with respect to another expression does not take extra computation space over the other expression.
In the case of my-map, O(n) space complexity is reasonable, since it has to generate a result of size O(n). Nevertheless, you can reduce the constant factor by accumulating the result list. The only catch is that the accumulated list will be backwards, so you’ll have to reverse it at the very end:
Attempting to reduce a constant factor like this is usually not worthwhile, as discussed below.
| (define (my-map f lst) |
| (define (iter lst backward-result) |
| (cond |
| [(empty? lst) (reverse backward-result)] |
| [else (iter (rest lst) |
| (cons (f (first lst)) |
| backward-result))])) |
| (iter lst empty)) |
It turns out that if you write
| (define (my-map f lst) |
| (for/list ([i lst]) |
| (f i))) |
then the for/list form in the function both is expanded to essentially the same code as the iter local definition and use. The difference is merely syntactic convenience.
2.3.4 Recursion versus Iteration
The my-length and my-map examples demonstrate that iteration is just a special case of recursion. In many languages, it’s important to try to fit as many computations as possible into iteration form. Otherwise, performance will be bad, and moderately large inputs can lead to stack overflow. Similarly, in Racket, it is sometimes important to make sure that tail recursion is used to avoid O(n) space consumption when the computation is easily performed in constant space.
At the same time, recursion does not lead to particularly bad performance in Racket, and there is no such thing as stack overflow; you can run out of memory if a computation involves too much context, but exhausting memory typically requires orders of magnitude deeper recursion than would trigger a stack overflow in other languages. These considerations, combined with the fact that tail-recursive programs automatically run the same as a loop, lead Racket programmers to embrace recursive forms rather than avoid them.
Suppose, for example, that you want to remove consecutive duplicates from a list. While such a function can be written as a loop that remembers the previous element for each iteration, a Racket programmer would more likely just write the following:
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In general, this function consumes O(n) space for an input
list of length n, but that’s fine, since it produces an
O(n) result. If the input list happens to be mostly consecutive
duplicates, then the resulting list can be much smaller than
O(n)—
| (remove-dups (list "a" "b" "b" "b" "b" "b")) |
| = (cons "a" (remove-dups (list "b" "b" "b" "b" "b"))) |
| = (cons "a" (remove-dups (list "b" "b" "b" "b"))) |
| = (cons "a" (remove-dups (list "b" "b" "b"))) |
| = (cons "a" (remove-dups (list "b" "b"))) |
| = (cons "a" (remove-dups (list "b"))) |
| = (cons "a" (list "b")) |
| = (list "a" "b") |