Sometimes the problem that we need an algorithm for doesn’t apply to the empty list, even in a vacuous or trivial way, and the base case for a direct recursion instead involves singleton lists – that is, lists with only one element. For instance, suppose that we want an algorithm that finds the westernmost city in one of our zip code databases. (The list must be non-empty because there is no “westernmosts city” in an empty list of cities.)
The assumption that the list is not empty is a precondition for the
meaningful use of this procedure, just as a call to Scheme’s built-in
quotient
procedure requires that the second argument, the divisor,
be non-zero. A precondition is a requirement that must be met in order
for your procedure to work correctly. You should have already formed a
habit of figuring out when such preconditions are appropriate. With the
6P technique for documenting procedures, you have likely made it a
habit to document such preconditions as you write the initial comment
for a procedure:
;;; Procedure:
;;; westernmost-city
;;; Parameters:
;;; zips, a list of lists
;;; Purpose:
;;; Find the westernmost city in zips.
;;; Produces:
;;; westernmost, a list
;;; Preconditions:
;;; * zips is nonempty.
;;; * All the entries in zips are of the form
;;; '(zip:string latitude:real longitude:real city:string state:string county:string)
;;; Postconditions:
;;; * westernmost is an element of zips (and is, by implication, an entry
;;; in the appropriate form).
;;; * For each city, c, in zips, westernmost is either at the same
;;; longitude as c or has a smaller longitude than c.
Alternately, we can make the structure of the list part of the specification of the parameters (and therefore an implicit precondition).
;;; Parameters:
;;; zips, a nonempty list of lists.
Whether you specify the precondition in the parameters or the preconditions section is often a matter of personal taste. What is most important is that you specify it somewhere.
Now that we’ve documented the procedure, let’s think about how to implement it. If a list of cities contains only one element, the answer is trivial – its only element is its westernmost. Otherwise, we can take the list apart into its car and its cdr, invoke the procedure recursively to find the westernmost of the cdr, and then try to figure out which comes first. How do we figure whether or not one city is to the west of another? As the documentation suggests, we compare their longitudes.
Let’s use a helper procedure to compare the two entries and return the
westernmost. (This is not a recursive helper procedure. Rather, like
the built-in max
, it is a relatively straightforward procedure that
simplifies our recursive definitions.)
;;; Procedure:
;;; westernmost
;;; Parameters:
;;; city1, a list in the standard "zip code format"
;;; city2, a list in the standard "zip code format"
;;; Purpose:
;;; Determine the westernmost of city1 and city2
;;; Produces:
;;; city-west, a list in the standard "zip code format"
;;; Preconditions:
;;; city1 and city2 are in standard "zip code format".
;;; Postconditions:
;;; * city-west is either city1 or city2.
;;; * the longitude of city-west is less than or equal to
;;; the longitude of city1.
;;; * the longitude of city-west is less than or equal to
;;; the longitude of city2.
(define westernmost
(lambda (city1 city2)
(if (< (caddr city1) (caddr city2))
city1
city2)))
We can test whether the given list has a single element by checking
whether its cdr is an empty list. The value of the expression (null? (cdr
zips))
is #t
if cities
has a single element and #f
if cities
has two or more elements. (It gives an error if cities
has zero
elements.)
Here, then, is the procedure definition:
(define westernmost-city
(lambda (zips)
(if (null? (cdr zips))
(car zips)
(westernmost (car zips) (westernmost-city (cdr zips))))))
If someone who uses this procedure happens to violate its precondition, applying the procedure to the empty list, the Scheme interpreter notices the error and prints out a diagnostic message:
> (westernmost-city null)
cdr: expects argument of type <pair>; given ()
If you think back to the tail-recursive version of difference
, you may
note another time that we had a special singleton case. When we compute
v
_{1} - v
_{2} - v
_{3} - v
_{k},
the base case is not “we have nothing to subtract”, but rather “we
have nothing to subtract from v
_{1}”.
We didn’t note the need for a singleton base case until we tried to write a tail-recursive version, but the need was there. Of course, that means that we might consider rewriting the non-tail-recursive version, but that version gave us the wrong answer, anyway.
If you consider the examples you’ve studied over the past few days, you will see that there is a common form for most of the procedures. The form goes something like this:
(define recursive-proc
(lambda (val)
(if (base-case-check? val)
(base-case-computation val)
(combine (partof val)
(recursive-proc (simplify val))))))
For example, for the westernmost-city
procedure,
westernmost-city
.zips
, our list of cities.(null? (cdr zips))
, which checks whether zips
has only one element.car
, which extracts the one drawing left in zips
.car
, which extracts the first entry in zips
.cdr
, which drops the first element, thereby giving us a simpler (well, less long) list.westernmost
.Similarly, consider the first complete version of sum
.
;;; Procedure:
;;; sum
;;; Parameters:
;;; numbers, a list of numbers.
;;; Purpose:
;;; Find the sum of the elements of a given list of numbers
;;; Produces:
;;; total, a number.
;;; Preconditions:
;;; All the elements of numbers must be numbers.
;;; Postcondition:
;;; total is the result of adding together all of the elements of numbers.
;;; If all the values in numbers are exact, total is exact.
;;; If any values in numbers are inexact, total is inexact.
(define sum
(lambda (numbers)
(if (null? numbers)
0
(+ (car numbers) (sum (cdr numbers))))))
In the sum
procedure,
sum
.numbers
, a list of numbers.(null? numbers)
, which checks if we have no numbers.0
. (This computation does not quite match the form above, since we don’t apply the 0 to numbers
. As this example suggests, sometimes the base case does not involve the parameter.)car
, which extracts the first value in numbers
.cdr
, which drops the the first element.When you write your own recursive procedures, it’s often useful to
start with the general structure and then to fill in the pieces. When
you are recursing over lists (as you have in our first explorations of
recursion), partof is almost always car
and simplify is almost
always cdr
. There’s a bit more to the base-case-test, since we’ve
used both (null? ___)
and (null? (cdr? ___))
. We may also find
other techniques.
However, when you work with other kinds of information (as you will do soon), you’ll have different techniques for extracting a piece of information, for simplifying the argument, and for deciding when you’re done.
Note, also, that examples like filtering suggest a similar, but more complex structure for recursive procedures.
(define recursive-proc
(lambda (val)
(cond
[(one-base-case-test?)
(one-base-case-computation val)]
[(another-base-case-test?)
(another-base-case-computation val)]
...
[(special-recursive-case-test-1?)
(combine-1 (partof-1 val)
(recursive-proc (simplify-1 val)))]
[(special-recursive-case-test-2?)
(combine-2 (partof-2 val)
(recursive-proc (simplify-2 val)))]
...
[else
(combine (partof val)
(recursive-proc (simplify val)))])))
However, in practice you will find that you rarely have more than two base-case tests (mostly one) and rarely more than two recursive cases.
When we write tail-recursive procedures, we simply use the result of the recursive call, and don’t combine it with anything. Here’s a simple tail recursive pattern.
(define procedure
(lambda (val)
(procedure-helper initial-result initial-remaining)))
(define procedure-helper
(lambda (so-far remaining)
(if (base-case-test? remaining)
(final-computation so-far)
(procedure-helper (combine (part-of remaining) so-far)
(simplify remaining)))))
and
and or
Of course, these common forms are not the only way to define recursive
procedures. In particular, when we define a predicate that uses direct
recursion on a given list, the definition is usually a little simpler
if we use and
- and or
-expressions rather than if
-expressions. For
instance, consider a predicate all-numbers?
that takes a given
list of values and determines whether all of them are numbers. As usual,
we consider the cases of the empty list and non-empty lists separately:
Since the empty list has no elements, it is (as mathematicians
say) “vacuously true” that all of its elements are numbers – there is
certainly no counterexample that one could use to refute the assertion. So
all-numbers?
should return #t
when given the empty list.
For a non-empty list, we separate the car and the cdr. If the list
is to count as all numbers, the car must clearly be a number, and in
addition the cdr must be a list of only numbers. We can use a recursive
call to determine whether the cdr is all numbers, and we can combine the
expressions that test the car and cdr conditions with and
to make sure
that they are both satisfied.
Thus, all-numbers?
should return #t
when the given list either is empty or has a number as its first element and all numbers after that. This yields the following definition:
;;; Procedure:
;;; all-numbers?
;;; Parameters:
;;; values, a list of Scheme values
;;; Purpose:
;;; Determine whether all of the elements of a list are numbers.
;;; Produces:
;;; allnum?, a Boolean.
;;; Preconditions:
;;; [No additional]
;;; Postconditions:
;;; allnum? is #t if all of the elements of values are numbers.
;;; allnum? is #f if at least one element is not a number.
(define all-numbers?
(lambda (values)
(or (null? values)
(and (number? (car values))
(all-numbers? (cdr values))))))
When values
is the empty list, all-numbers?
applies the first test
in the or
-expression, finds that it succeeds, and stops, returning
#t
. In any other case, the first test fails, so all-numbers?
proceeds
to evaluate the first test in the and
-expression. If the first element
of values
is not a number, the test fails, so all-numbers?
stops,
returning #f
. However, if the first element of numbers
is a number,
the test succeeds, so all-numbers?
goes on to the recursive procedure
call, which checks whether all of the remaining elements are numbers, and
returns the result of this recursive call, however that result turns out.
Here’s a template for that solution.
(define all-____?
(lambda (lst)
(or (null? lst)
(and (____? (car lst))
(all-____? (cdr lst))))))
We can use a similar technique to ask if any value in a list is a number. In this case, if there are no values in the list, we know that no values are numbers. Otherwise, we check if the first value is a number. If it is, then some value must be a number.
The complicated part is getting the base case right (particularly if we
want to avoid using if
). The standard technique is to require that
the list not be null (using not
and and
). If the list is null,
(not (null? lst))
returns #f
. And, since (and #f ...)
is #f
,
we get false back for the empty list, just as we wanted.
;;; Procedure:
;;; any-numbers?
;;; Parameters:
;;; numbers, a list of Scheme values
;;; Purpose:
;;; Determine whether any of the elements of a list is a number.
;;; Produces:
;;; anynum?, a Boolean.
;;; Preconditions:
;;; [No additional]
;;; Postconditions:
;;; anynum? is #t if at least one of the elements of values is a number.
;;; anynum? is #f if all of the elements are not numbers.
(define any-numbers?
(lambda (values)
(and (not (null? values))
(or (number? (car values))
(any-numbers? (cdr values))))))
And, once again, we can generalize.
(define any-____?
(lambda (lst)
(and (not (null? lst))
(or (____? (car lst))
(any-____? (cdr lst))))))
Recall the following procedure from the reading on recursion with helper procedures which finds the value in a list of real numbers with the largest absolute value.
(define furthest-from-zero
(lambda (numbers)
(if (null? (cdr numbers))
(car numbers)
(further-from-zero (car numbers)
(furthest-from-zero (cdr numbers))))))
Using the common form for recursive procedures given above applied to furtest-from-zero
, fill in the blanks:
Recall the alternate approach from the reading on recursion with helper procedures.
(define furthest-from-zero
(lambda (numbers)
(furthest-from-zero-helper (car numbers) (cdr numbers))))
(define furthest-from-zero-helper
(lambda (furthest-so-far numbers-remaining)
(if (null? numbers-remaining)
furthest-so-far
(furthest-from-zero-helper
(further-from-zero furthest-so-far (car numbers-remaining))
(cdr remaining-numbers)))))
Using the common form for tail recursive procedures given above applied to this version of furthest-from-zero
, fill in the blanks: