Reading: Vectors

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Monday, Nov 5, 2018
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.

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 several techniques for representing collections of data:

  • We can represent the collection as a list.
  • We can represent the collection as a nested list.
  • We will soon learn how we can represent the collection persistently as a file.

Representing a collection as a list has some problems. In particular, it is relatively expensive to get a particular element and it is equally expensive to change a particular element. Why is it expensive to get an element (say, the tenth element)? In the case of a list, we need to follow the cdr of each pair through the list until we reach the element. In the case of a tree, we need to figure out how many values are in the left subtree to decide where to look. Changing an element may be even worse, because once we’ve reached the position, we need to build the structure back to a new form.

Does this mean that lists and other similar structures are inappropriate ways to represent collections? Certainly not. Rather, they work very well for some purposes (e.g., it is easy to extend a list) 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.

Indexing, a key feature of vectors

You may have noted that when we use lists to group data (e.g., the tallies for the words in a book), 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 address this problem. 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.

Mutation, another key feature of vectors

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; to update a tally). 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.

A practical detail: How DrRacket displays vectors

When showing a vector, DrRacket follows a format much like the list, but with a preceding pound sign, #. That is, the elements of the vector are separated by spaces, enclosed in parentheses, and with an extra # in the front. For instance, here’s how Scheme shows a vector containing the strings "alpha", "beta", and "gamma", in that order:

'#("alpha" "beta" "gamma")

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

Some implementations of Scheme permit us to use vector literals, in which a programmer can use a similar syntax to specify a vector when writing a Scheme program or typing commands and definitions into the Scheme interactive interface. In some such implementations, the literal begins with the hash mark. In others, the programmer must 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.

We traditionally recommend that you avoid using this notation just as we recommend that you avoid the corresponding list literal notation for lists.

Vector procedures

Standard Scheme provides the following fundamental procedures for creating vectors and selecting and replacing their elements. You’ll find that many of them correspond to similar list procedures.

vector

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

> (vector "alpha" "beta" "gamma")
'#("alpha" "beta" "gamma")
'> (vector)  ; the empty vector -- no elements!
#()
> (define beta 2)
> (vector "alpha" beta (list "gamma" 3) (vector "delta" 4) (vector "epsilon"))
'#("alpha" 2 ("gamma" 3) #("delta" 4) #("epsilon"))

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

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? (list "alpha" "beta" "gamma"))  ; a list, not a vector
#f
> (vector? "alpha beta gamma")  ; a string, not a vector
#f

vector-length

The vector-length procedure takes one argument, which must be a vector, and returns the number of elements in the vector. Unlike the length procedure for lists, which must look through the whole list to find the length, vector-length can immediately determine the length of a 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: out of bounds: 3

vector-set!

All of the previous procedures look a lot like list procedures, except that many are more efficient (e.g., vector? and vector-length take a constant number of steps; list? takes a number of steps proportional to the the length of the list and list-ref takes a number of steps proportional to the index). Now let’s see a procedure that’s much different. We can 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 supposed to be “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 the ones we use, don’t enforce this rule.)

vector->list and list->vector

The vector->list procedure 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 (vector 31 27 16))
'(31 27 16)
> (vector->list (vector))
'()
> (list->vector (list #\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.

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

Selecting random elements from vectors

You may recall that we recently defined a procedure, random-elt, that randomly selects an element from a list.

;;; Procedure:
;;;   random-elt
;;; Parameters:
;;;   lst, a non-empty list 
;;; Purpose:
;;;   Unpredictably pick an element of lst.
;;; Produces:
;;;   val, a value
;;; Preconditions:
;;;   [No additional]
;;; Postconditions:
;;;   * val is an element of lst.
;;;   * If lst contains more than one element, it is difficult to predict 
;;;     which element val is.
(define random-elt
  (lambda (lst)
    (list-ref lst (random (length lst)))))

The procedure is simple and straightforward. But it’s slow. Since we have to find the length of the list each time we look for a random element, we’ll spend time and effort stepping through the elements of the list with cdr.

Fortunately, it’s straightforward to write a similar procedure using vectors. We just change the list procedures to their corresponding vector versions.

;;; Procedure:
;;;   random-vector-elt
;;; Parameters:
;;;   vec, a non-empty vector
;;; Purpose:
;;;   Unpredictably pick an element of vec.
;;; Produces:
;;;   val, a value
;;; Preconditions:
;;;   [No additional]
;;; Postconditions:
;;;   * val is an element of vec.
;;;   * If vec contains more than one element, it is difficult to predict 
;;;     which element val is.
(define random-vector-elt
  (lambda (vec)
    (vector-ref vec (random (vector-length vec)))))

Let’s check it.

> (define words (vector "alpha" "beta" "gamma" "delta" "epsilon"))
> (random-vector-elt words)
"beta"
> (random-vector-elt words)
"epsilon"
> (random-vector-elt words)
"alpha"
> (random-vector-elt words)
"alpha"
> (random-vector-elt words)
"beta"

We’ll see in the lab just how much difference this makes.

Implementing number vectors

We frequently store only one type in a collection. For example, just as we might restrict a list or pair structure to contain only numbers, we might restrict the numbers in a vector to store only integers. Those integers might, for example, represent tallies of letters or words.

Say we had such a vector of numbers–how could we increment the tally in one of the positions? After some reflection, it seems We need three steps, one to get the current value in the vector, another to increment that value, and the last step to make that value the new entry in the given position. We put these steps together as follows.

;;; Procedure:
;;;   number-vector-increment-at!
;;; Parameters:
;;;   vec, a vector
;;;   index, an integer
;;; Purpose:
;;;   Increment the value at a vector position
;;; Produces:
;;;   [Nothing; called for side effect.]
;;; Preconditions:
;;;   (vector-ref vec index) is a number
;;; Postconditions:
;;;   Let val be (vector-ref vec index) before the procedure call. After the
;;    call (vector-ref vec index) produces val+1.
(define number-vector-increment-at!
  (lambda (vec index)
    (vector-set! vec 
                 index 
                 (increment (vector-ref vec index)))))

The fact that vectors allow mutation makes the increment straightforward. What if we wanted to increment every value in a vector? With lists we might think about using basic list recursion to pass over the list and add one to each item. (And then we might think better of it and use map). There is no analog for car/cdr that we might use to process lists, neither is there an analog for map.

Fortunately, indexing is fast in vectors, so we can use numeric recursion to iterate over all the positions in a vector, applying our increment function along the way. Thus, we might track pos, the current position to modify in the vector, starting at zero and ending when we reach the length of the vector. We can encapsulate this repeated mutation with a named let

;;; Procedure:
;;;   number-vector-increment!
;;; Parameters:
;;;   vec, a vector
;;; Purpose:
;;;   Increment the value at all vector positions
;;; Produces:
;;;   [Nothing; called for side effect.]
;;; Preconditions:
;;;   (vector-ref vec index) for 0 <= index < (vector-length vec) is a number.
;;;   number-vect-increment-at! is defined
;;; Postconditions:
;;;   Let val be (vector-ref vec index) before the procedure call. After the
;;    call (vector-ref vec index) produces val+1.
(define number-vector-increment!
  (lambda (vec)
    (let ([len (vector-length vec)]) ; unchanging value, tells recursion to stop
      (let kernel! ([pos 0])  ; Start the recursion at the first position
        (when (< pos len) ; When the position is valid,
          (number-vector-increment-at! vec pos) ; increment the number at pos
          (kernel! (+ 1 pos))))))) ; and process the rest of the vector

There are various ways to mutate all vector elements; the lab will suggest some alternatives. Unfortunately, we do not always have a special helper function like number-vector-increment-at! that allows us to write such streamlined code. Instead we must combine the indexing and mutation steps directly in the recursion.

As an example, suppose we wished to convert the tallies to percentages by dividing each number by the sum of all the numbers in the vector. Assuming you have a means of totalling these numbers (a procedure you will write in the lab), we still need to iterate over all vector positions, just as we did in number-vector-increment! only this time we use the position variable pos directly to index the vector with vector-ref, rather than with a helper. Putting this together, we might write the following procedure.

;;; Procedure:
;;;   number-vector-scale!
;;; Parameters:
;;;   vec, a vector
;;;   divisor, a number
;;; Purpose:
;;;   Scale all the elements in the vector by dividing by the given
;;;   divisor.
;;; Produces:
;;;   [Nothing; called for side effect.]
;;; Preconditions:
;;;   (vector-ref vec index) for 0 <= index < (vector-length vec) is a number.
;;; Postconditions:
;;;   Let val be (vector-ref vec index) before the procedure call. After the
;;    call (vector-ref vec index) produces val/divisor.
(define number-vector-scale!
  (lambda (vec divisor)
    (let ([len (vector-length vec)]) ; unchanging and tells recursion to stop
      (let kernel! ([pos 0])  ; Start the recursion at the first position
        (when (< pos len)     ; When the position is valid,
          (vector-set! vec    ; Set the new value in the vector
                       pos    ; at the current position
                       (/ (vector-ref vec pos) divisor)) ; to the quotient
          (kernel! (+ 1 pos))))))) ; and process the rest of the vector

Of course, we need not change the vector as we iterate over it. Perhaps we just want to find the largest value in a vector. We still need to iterate over all the positions, except we might now use the standard recursive pattern that requires us to use a combination step to get a complete answer from the partial (recursive) answer.

;;; Procedure:
;;;   number-vector-largest
;;; Parameters:
;;;   vec, a vector
;;; Purpose:
;;;   Find the largest number in a vector
;;; Produces:
;;;   largest, a number
;;; Preconditions:
;;;   (vector-ref vec index) for 0 <= index < (vector-length vec) is a number.
;;; Postconditions:
;;;   (vector-ref vec index) <= largest for 0 <= index < (vector-length vec) 
;;;   largest is a value in vec, i.e., there exists an integer index such that
;;;      0 <= index < (vector-length vec)  and
;;;      (vector-ref vec index) = largest
(define number-vector-largest
  (lambda (vec)
    (let ([last (- (vector-length vec) 1)]) ; last position to test
      (let kernel ([pos 0])  ; Start the recursion at the first position
        (if (= pos last)
            ; We are at the last position, so return the number
            (vector-ref vec pos)
            ; Otherwise return the maximum of the current position and
            ; the largest number in the rest of the vector
            (max (vector-ref vec pos)
                 (kernel (+ 1 pos))))))))

Patterns of recursion over vectors

Each time we learn a new structure, we learn techniques for recursion 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. (We also typically precompute a stopping point so that we don’t have to recompute it for each pass through the kernel.)

The test for the base case is then to check whether the current position has reached the stopping point and the “simplify” step is to add 1 to the position. As usual, the “combine” step is problem dependent.

(define vector-proc
  (lambda (vec other)
    (let ([len (vector-length vec)])
      (let kernel ([pos 0])
        (if (= pos len)
            (base-case vec other)
            (combine (vector-ref vec pos)
                     (kernel (+ pos 1))))))))

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)
    (let kernel ([pos (- (vector-length vec) 1)])
      (if (< pos 0)
          (base-case vec other)
          (combine (vector-ref vec pos)
                   (kernel (- pos 1)))))))

Because vectors are mutable, we often use an imperative pattern with (perhaps multiple) operations that have side-effects.

(define vector-proc
  (lambda (vec other)
    (let ([len (vector-length vec)])
      (let kernel! ([pos 0])
        (cond
          [(= pos len)
           (base-case! vec other)
           ...]
          [else
           (operation! vec pos)
           ...
           (kernel! (+ pos 1))])))))

Summary of important vector procedures

(list->vector lst)
Convert a list to a vector.

(vector val_1 ... val_n)
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.

Self checks

Check 1: Creating simple vectors

a. In DrRacket’s Interactions pane, type in a vector literal that denotes a vector containing just the two elements 3.14159 and 2.71828. How does DrRacket display the value of this vector?

b. Create a vector that contains the same two values by using the vector procedure.

c. Create a vector that contains the same two values by using the make-vector and vector-set! procedures.

Check 2: Processing numeric vectors

a. Make a copy of number-vector-increment-at! from above.

b. Try using number-vector-increment-at! on a vector from the previous check.

> (define v1 (vector 3.14159 2.71828))
> (number-vector-increment-at! v1 1)
> v1

c. Use number-vector-increment! on the vector to verify it behaves as intended.

> (define v2 (make-vector 2 3.14159))
> (vector-set! v2 12.71828))
> (number-vector-increment! v2)
> v2

d. Checks 2.b and 2.c relied on the vectors you created in 1.b and 1.c, respectively. What do you suppose would happen if we tried these operations on the vector you created in check 1.a?

> (number-vector-increment-at! '#(3.14159 2.71828) 1)

e. Verify your prediction. Why do you think this happens?