Fundamentals of Computer Science I (CS151.01 2006F)

Vectors

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Summary: Vectors are data structures that are very similar to lists in that they arrange data in linear fashion. Vectors differ from lists in two significant ways: Unlike lists, vectors are indexed and vectors are mutable.

Contents:

Introduction: Deficiencies of Lists

As you've seen in many of the procedures and programs we've written so far, there are many problems in which we have to deal with collections of information. We have now learned three techniques for representing collections of data:

All of these ways to represent collections have some similar deficiencies. In particular, it is relatively expensive to extract a particular element of a list (or file) and it is equally expensive to change a particular element.

Does this mean that lists, files, and trees are inappropriate ways to represent collections? Certainly not. Rather, they work very well for some purposes (e.g., it is easy to add an element to a list; files persist between invocations of Scheme) and less well for other purposes (e.g., extracting and changing).

To resolve these deficiencies, Scheme provides an alternate mechanism for representing collections, the vector.

A Key Feature of Vectors: Indexing

You may have noted that when we use lists to group data (e.g., the information on a course), we need to use list-ref or repeated calls to cdr to get later elements of the list. Unfortunately, list-ref works by cdr'ing down the list. Hence, it takes about five steps to get to the fifth element of the list and about one hundred steps to get to the one hundredth element of a list. Similarly, to get to the fifth element of a file, we'll need to read the preceding elements and to get to the hundredth element, we'll also need to read through the preceding elements. It would be nicer if we could access any element of the group of data in the same amount of time (preferably a small amount of time).

Vectors contain a fixed number of elements and provide indexed access (also called random access) to those elements, in the sense that each element, regardless of its position in the vector, can be recovered in the same amount of time. In this respect, a vector differs from a list or a file: The initial element of a list is immediately accessible, but subsequent elements are increasingly difficult and time-consuming to access.

Key Feature 2: Mutating

You may have also noted that we occasionally want to change an element of a group of data (e.g., to change a student's grade in the structure we use to represent that student). When we use lists, we essentially need to build a new list to change one element. When we use files, we often have to build a new file, copying both preceding and subsequent values.

Vectors are mutable data structures: It is possible to replace an element of a vector with a different value, just as one can take out the contents of a container and put in something else instead. It's still the same vector after the replacement, just as the container retains its identity no matter how often its contents are changed.

The particular values that a vector contains at some particular moment constitute its state. One could summarize the preceding paragraph by saying that the state of a vector can change and that state changes do not affect the underlying identity of the vector.

Showing Vectors

When showing a vector, Scheme shows each of its elements, enclosed in parentheses, with an extra character, #, in front of the left parenthesis. For instance, here's how Scheme shows a vector containing the symbols alpha, beta, and gamma, in that order:

#(alpha beta gamma)

The mesh (also called pound, sharp, or hash) character distinguishes the vector from the list containing the same elements.

We can use the same syntax to specify a vector when writing a Scheme program or typing commands and definitions into the Scheme interactive interface, except that we have to place a single quotation mark before the mesh so that Scheme will not try to evaluate the vector as if it were some exotic kind of procedure call. (DrScheme will not make this mistake even if you forget the single quotation mark, but not all implementations of Scheme are so generous.)

I recommend that you avoid using this vector literal notation (that is, with the quotation mark), just as I recommend that you avoid the corresponding list literal notation for lists.

DrScheme's Variations for Showing Vectors

In the interaction window, when DrScheme shows a vector as the value of some top-level expression that the user supplied, it does something more ambitious, but potentially rather confusing: It inserts a numeral between the mesh character and the left parenthesis to indicate the number of elements in the vector, thus:

#3(alpha beta gamma)

However, you can instruct DrScheme to use the straight mesh-and-parentheses representation, with no numeral, by giving the following command at the beginning of your program or interactive session:

(print-vector-length #f)

That is, Don't print the lengths of vectors.

Vector Procedures

Standard Scheme provides the following fundamental procedures for creating vectors and selecting and replacing their elements:

vector

The constructor vector takes any number of arguments and assembles them into a vector, which it returns.

> (print-vector-length #f)
> (vector 'alpha 'beta 'gamma))
#(alpha beta gamma)
> (vector)  ; the empty vector -- no elements!
#()
> (vector 'alpha "beta" '(gamma 3) '#(delta 4) (vector 'epsilon))
#(alpha "beta" (gamma 3) #(delta 4) #(epsilon))

As the last example shows, Scheme vectors can be heterogeneous, containing elements of various types, just like Scheme lists.

make-vector

The make-vector procedure takes two arguments, a natural number k and a Scheme value obj, and returns a k-element vector in which each position is occupied by obj.

> (make-vector 12 'foo)
#(foo foo foo foo foo foo foo foo foo foo foo foo)
> (make-vector 4 0)
#(0 0 0 0)
> (make-vector 0 4)  ; the empty vector, again
#()

The second argument is optional; if you omit it, the value that initially occupies each of the positions in the array is left unspecified. Various implementations of Scheme have different ways of filling them up, so you should omit the second argument of make-vector only when you intend to replace the contents of the vector right away.

vector?

The type predicate vector? takes any Scheme value as argument and determines whether it is a vector.

> (vector? (vector 'alpha 'beta 'gamma))
#t
> (vector? '#(alpha beta gamma)) ; discouraged notation
#t
> (vector? (list 'alpha 'beta 'gamma))  ; a list, not a vector
#f
> (vector? "alpha beta gamma")  ; a string, not a vector
#f
> (vector? '#(#f))  ; a one-element vector (the element is #f)
#t

vector-length

The vector-length procedure takes one argument, which must be a vector, and returns the number of elements in the vector.

> (vector-length (vector 3 1 4 1 5 9))
6
> (vector-length (vector 'alpha 'beta 'gamma))
3
> (vector-length (vector))
0 

vector-ref

The selector vector-ref takes two arguments -- a vector vec and a natural number k (which must be less than the length of vec). It returns the element of vec that is preceded by exactly k other elements. (In other words, if k is 0, you get the element that begins the vector; if k is 1, you get the element after that; and so on.)

> (vector-ref (vector 3 1 4 1 5 9) 4)
5
> (vector-ref (vector 'alpha 'beta 'gamma) 0)
alpha
> (vector-ref (vector 'alpha 'beta 'gamma) 3)
vector-ref: index 3 out of range [0, 2] for vector: #(alpha beta gamma)

vector-set!

All of the previous procedures look a lot like list procedures, except that many are more efficient (e.g., vector? takes a constant number of steps; list? takes a number of steps proportional to the the length of the list). Now let's see a procedure that's much different. We can actually use procedures to change vectors.

The mutator vector-set! takes three arguments -- a vector vec, a natural number k (which must be less than the length of vec), and a Scheme value obj -- and replaces the element of vec that is currently in the position indicated by k with obj. This changes the state of the vector irreversibly; there is no way to find out what used to be in that position after it has been replaced.

It is a Scheme convention to place an exclamation point meaning Proceed with caution! at the end of the name of any procedure that makes such an irreversible change in the state of an object.

The value returned by vector-set! is unspecified; one calls vector-set! only for its side effect on the state of its first argument.

> (define sample-vector (vector 'alpha 'beta 'gamma 'delta 'epsilon))
> sample-vector 
#(alpha beta gamma delta epsilon)
> (vector-set! sample-vector 2 'zeta)
> sample-vector  ; same vector, now with changed contents
#(alpha beta zeta delta epsilon)
> (vector-set! sample-vector 0 "foo")
> sample-vector  ; changed contents again
#("foo" beta zeta delta epsilon)
> (vector-set! sample-vector 2 -38.72)
> sample-vector  ; and again
#("foo" beta -38.72 delta epsilon)

Vectors introduced into a Scheme program by means of the mesh-and-parentheses notation are immutable -- applying vector-set! to such a vector is an error, and the contents of such vectors are therefore constant. (Warning! Some implementations of Scheme, including DrScheme, don't enforce this rule.)

vector->list and list->vector

The vector->list takes any vector as argument and returns a list containing the same elements in the same order; the list->vector procedure performs the converse operation.

> (vector->list '#(31 27 16))
(31 27 16)
> (vector->list (vector))
()
> (list->vector '(#\a #\b #\c))
#(#\a #\b #\c)
> (list->vector (list 31 27 16))
#(31 27 16)

vector-fill!

The vector-fill! procedure takes two arguments, the first of which must be a vector. It changes the state of that vector, replacing each of the elements it formerly contained with the second argument.

> (define sample-vector (vector 'rho 'sigma 'tau 'upsilon))
> sample-vector  ; original vector
#(rho sigma tau upsilon)
> (vector-fill! sample-vector 'kappa)
> sample-vector  ; same vector, now with changed contents
#(kappa kappa kappa kappa)

The vector-fill! procedure is invoked only for its side effect and returns an unspecified value.

Implementing Standard Vector Procedures

Some older implementations of Scheme may lack the list->vector, vector->list, and vector-fill! procedures, but it is straightforward to define them in terms of the others.

list->vector

;;; Procedure:
;;;   list->vector
;;; Parameters:
;;;   lst, a list.
;;; Purpose:
;;;   Convert the list to a vector.
;;; Produces:
;;;   vec, a vector
;;; Preconditions:
;;;   lst is a list
;;; Postconditions:
;;;   vec is a vector.
;;;   The length of vec equals the length of lst.
;;;   The ith element of vec equals the ith element of lst for
;;;     all "reasonable" i.

In most implementations, we'll need to recurse over the list, adding elements to a corresponding vector. We'll also need to build that vector, first. Since we only want to build one vector, we should use the husk-and-kernel technique (which we've previously used for error checking). The husk creates the vector and tells the kernel and then calls the kernel.

However, we may need other parameters for the kernel, so it's time to think a little bit about the kernel. Since we want to copy all the elements of the list to the vector, we probably need to repeat some basic step, and the only way we know how to do repetition is recursion. To keep track of what we're copying, we'll probably need a counter that we pass to the helper. I'll call that counter pos, since it keeps track of a position in the list or vector.

What should we copy from at each step? We could copy the element at position pos of the list into position pos of the vector. However, that's somewhat inefficient because it requires us to step through to the ith element of the list.

Since we no longer need a value from the list once we've copied it, we can cdr through the list as we step through the vector. Now, our goal is to copy the initial element of the list into the appropriate position of the vector.

When do we stop? When we run out of elements to copy.

So, here's how to set things up.

(define list->vector
  (lambda (lst)
    (list->vector-kernel! lst ; Copy the whole list
                          0   ; Starting at position 0
                          (make-vector (length lst) null)
                          ; Into a new vector of the appropriate length
                          )))

The kernel is a little bit more complicated. We need to keep track of where we are in the vector. We may also want to carefully specify what this kernel is supposed to do. (Such specification is not always necessary for helpers, but I think it clarifies things in this case.)

;;; Procedure:
;;;   list->vector-kernel!
;;; Parameters:
;;;   lst, a list to copy
;;;   pos, a position in the vector
;;;   vec, a vector
;;; Purpose
;;;   Copy values from lst into positions pos, pos+1, ...
;;; Produces:
;;;   vec, the same vector but with different contents.
;;; Preconditions:
;;;   The length of vec = pos + the length of list. [Unverified]
;;;   pos >= 0. [Unverified]
;;; Postconditions:
;;;   Element pos of vec now contains element 0 of list.
;;;   Element pos+1 of vec now contains element 1 of lst.
;;;   Element pos+2 of vec now contains element 2 of lst.
;;;   ...
;;;   The last element of vec now contains the last element of lst.
(define list->vector-kernel!
  (lambda (lst pos vec)
    ; We don't bother to verify the preconditions because this
    ; procedure should only be called by something that has already
    ; verified the preconditions (explicitly or implicitly).

    ; If there's nothing left to copy, stop.
    (if (null? lst) vec
        ; Otherwise, copy the initial element of the list and
        ; then copy the remaining elements
        (begin
          (vector-set! vec pos (car lst))
          (list->vector-kernel! (cdr lst) (+ pos 1) vec)))))

Is this the only way to write the list->vector procedure? No. More advanced students might know about the apply procedure which makes life significantly easier.

(define list->vector
  (lambda (ls)
    (apply vector ls)))

In other words: Call the vector procedure, giving it the elements of ls as its arguments.

vector->list

Now let's consider how to go in the opposite direction. Once again, we'll probably need to recurse to step through positions and need to keep track of the position with an extra variable. Should we create a list first and then populate it? No. We don't typically mutate lists. So, we'll update the list on the fly, adding elements and extending the list at each step. Let's start with the helper. The helper will build a list from a subrange of the vector.

;;; Procedure:
;;;   vector->list-kernel
;;; Parameters:
;;;   vec, a vector
;;;   start, an integer
;;;   finish, an integer
;;; Purpose:
;;;   Builds a list that contains elements start ... finish-1 of vec
;;; Produces:
;;;   lst, a new list.
;;; Preconditions:
;;;   start >= 0
;;;   finish >= start
;;;   finish <= length of vec
;;; Postconditions:
;;;   The element at position i of lst is the element at position
;;;     i+start of vec for all reasonable i.
;;;   Does not change vec.
(define vector->list-kernel
  (lambda (vec start finish)
    ; We stop at the finish.  
    (if (= start finish)
        ; There are no more elements to put into a list, so use
        ; the empty list.
        null
        ; Otherwise, add the element at position start to a list of
        ; the remaining elements.
        (cons (vector-ref vec start)
              (vector->list-kernel vec (+ 1 start) finish)))))

Now, we just have to set things up for this kernel. We start at position 0. We end just before the length of the vector. So, here goes ...

;;; Procedure:
;;;   vector->list
;;; Parameters:
;;;   vec, a vector
;;; Purpose:
;;;   Builds a list that contains the elements of vec in the same order.
;;; Produces:
;;;   lst, a new list.
;;; Preconditions:
;;;   [None]
;;; Postconditions:
;;;   The element at position i of lst is the element at position
;;;     i of vec for all reasonable i.
;;;   Does not change vec.
(define vector->list
  (lambda (vec)
    (vector->list-kernel vec 0 (vector-length vec))))

Patterns of Recursion Over Vectors

Each time we learn a new structure, we learn techniques for recursing over that structure. As the previous examples suggested, recursion over vectors is relatively straightforward, but usually requires that we have a helper procedure that includes additional parameters - the current position in the vector and the length of the vector. (Why do we include the length? So that we don't have to recompute it each time.)

The test for the base case is then to check whether the current position equals the length and the simplify step is to add 1 to the position.

(define vector-proc
  (lambda (vec other)
    (vector-proc-helper vec other 0 (vector-length vec))))
(define vector-proc-helper
  (lambda (vec other pos len)
    (if (= pos len)
        (base-case vec other)
        (combine (vector-ref vec pos)
                 (vector-proc-helper vec other (+ post 1) len)))))

At times, it's better to start at the end of the vector and work backwards. In this strategy, we get the base case when the position reaches 0 and we simplify by subtracting 1.

(define vector-proc
  (lambda (vec other)
    (vector-proc-helper vec other (- (vector-length vec) 1))))
(define vector-proc-helper
  (lambda (vec other pos)
    (if (< pos 0)
        (base-case vec other)
        (combine (vector-ref vec pos)
                 (vector-proc-helper vec other (- pos 1))))))

Summary of Important Vector Procedures

(list->vector lst)
Convert a list to a vector.
(print-vector-length status)
Tell DrScheme to print the length of vectors (or not to print the length of vectors).
(vector val0 ... valn)
Create a vector with the given elements.
(vector? val)
Determine if val is a vector.
(vector-fill vec val)
Fill vec with multiple copies of val.
(vector-ref vec pos)
Extract an element from vec.
(vector-set! vec pos val)
Set element pos of vec to val.
(vector->list vec)
Convert a vector to a list.

 

History

 

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.

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The source to the document was last modified on Sun Oct 8 16:47:43 2006.
This document may be found at http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2006F/Readings/vectors.html.

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Samuel A. Rebelsky, rebelsky@grinnell.edu

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