Laboratory: Recursion Basics

Summary: In this laboratory, you will explore some basic concepts in recursing over lists.

Reference:

Here are the definitions from the reading. 

;;; 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))))))

;;; Procedure:
;;;   rgb-filter-out-dark
;;; Parameters:
;;;   colors, a list of RGB colors.
;;; Purpose:
;;;   Create a new list of colors, consiting only of non-dark colors.
;;; Produces:
;;;   not-dark, a list of RGB colors.
;;; Preconditions:
;;;   rgb-dark? is defined.
;;; Postconditions:
;;;   Every element of not-dark appears in colors.
;;;   If (not (rgb-dark? (list-ref colors i))) holds for some i,
;;;     then the corresponding color appears in not-dark.
(define rgb-filter-out-dark
  (lambda (colors)
    (if (null? colors)
        null
        (if (rgb-dark? (car colors))
            (rgb-filter-out-dark (cdr colors))
            (cons (car colors) (rgb-filter-out-dark (cdr colors)))))))

The latter procedure requires rgb-dark?, which follows. 

;;; Procedure:
;;;   rgb-dark?
;;; Parameters:
;;;   color, an RGB color
;;; Purpose:
;;;   Determine if the color appears dark.
;;; Produces:
;;;   dark?, a Boolean
;;; Preconditions:
;;;   [No additional]
;;; Postconditions:
;;;   If color is relatively dark, then dark? is #t.
;;;   Otherwise, dark? is #f.
(define rgb-dark?
  (lambda (color)
     (> 33 (rgb-brightness color))))

;;; Procedure:
;;;   rgb-brightness
;;; Parameters:
;;;   color, an RGB color
;;; Purpose:
;;;   Computes the brightness of color on a 0 (dark) to 100 (light) scale.
;;; Produces:
;;;   b, an integer
;;; Preconditions:
;;;   color is a valid RGB color.  That is, each component is between
;;;     0 and 255, inclusive.
;;; Postconditions:
;;;   If color1 is likely to be perceived as lighter than color2,
;;;     then (brightness color1) > (brightness color2).
(define rgb-brightness
  (lambda (color)
    (round (* 100 (/ (+ (* 0.30 (rgb-red color))
                        (* 0.59 (rgb-green color))
                        (* 0.11 (rgb-blue color)))
                      255)))))

Preparation

In this laboratory, we will not be working with images (just with colors and with lists), so you need not create an image.

a. Copy the sum and rgb-filter-out-dark procedures to your definitions pane.

b. Copy the rgb-dark? and rgb-brightness procedures to your library.

c. Load your library.

d. Create a list of a dozen or so colors (red, black, green, blue, yellow, orange, magenta, white, black, etc.). Name it my-colors.

Exercises

Exercise 1: Testing Sum

a. Read through sum so that you have a sense of its purpose.

b. Verify that sum produces the same result as in the corresponding reading.

c. What value do you expect sum to produce for the empty list? Check your answer experimentally.

d. What value do you expect sum to produce for a singleton list? Check your answer experimentally.

e. Try sum for a few other lists, too.

Exercise 2: Removing Dark Colors

a. Read through the definition of rgb-filter-out-dark to try to understand what it does.

b. Determine which colors in my-colors are dark with (map rgb-dark? my-colors).

c. Create a list of non-dark colors with (rgb-filter-out-dark my-colors).

d. Verify that all the resulting colors are not dark.

Exercise 3: Counting Values

Suppose the length procedure were not defined. We could define it by recursing through the list, counting 1 for each value in the list. In some sense, this is much like the definition of sum, except that we use the value 1 rather than the value of each element.

a. Using this idea, write a procedure, (my-length lst) that finds the length of a list (without using length).

b. Check your answer on a few examples: the empty list, the list of colors you created, a few more lists of your choice.

Exercise 4: Product

Write a procedure, (product nums), that, given a list of numbers, computes the product of those numbers. You should feel free to use sum as a template. However, you should think carefully about the base case.

Exercise 5: Counting Special Values

What if we don't want to count every value in a list? For example, what if we only want to count the dark values in a list of colors. In this case, we still recurse through the list, but we sometimes count 1 (when the color is dark) and sometimes count 0 (when the color is not dark).

a. Using this idea, write a procedure, (rgb-tally-dark colors), that, given a list of colors, counts how many are dark.

b. Test your procedure.

Exercise 6: Summing Components

In the past, we've found it useful to find the average of two colors. Let's consider how we might find the average of a list of colors. First, we would need to find the number of colors in the list. That's easy, we just use the length procedure. Next, we need to find the sum of each component. That's a bit harder, but let's suppose we can do it. We next divide each sum by the length, and get the average of that component. Finally, we put it all together with rgb-new.

That is, we might write 

(define rgb-list-average
  (lambda (colors)
    (let ((count (length colors)))
      (rgb-new (/ (sum-red colors) count)
  	       (/ (sum-green colors) count)
  	       (/ (sum-blue colors) count)))))

Of course, for this to work, we need to write sum-red, sum-blue, and sum-green. For now, we'll write one of the three. (One we've written that one, the other two should be obvious.)

a. Write a procedure, (sum-red colors), that computes the sum of the red components in a list of colors. You should use direct recursion in your definition sum-red. You may want to base your definition on the definition of sum.

b. Test your procedure on a list of a single color.

c. Test your procedure on the my-colors list you wrote earlier.

d. It is possible to write sum-red without using direct recursion. How? An appropriate combination of map and sum. Try doing so. (If you can't find a solution, look at the notes on this problem.)

Exercise 7: Filtering Out Reds

Write a procedure, (filter-reds colors), that filters out all elements of colors with a red component of at least 128.

For Those With Extra Time

Extra 1: Filtering Out Spots

Write a procedure, (spots-bound spots left top right bottom), that filters out all elements of spots which do not appear in the rectangle bounded at the left by left, at the top by top, at the right by right, and at the bottom by bottom.

For example, if left is 5, top is 10, right is 8, and bottom is 20, the procedure should remove any spots whose column is less than 5 or more than 8 or whose row is less than 10 or more than 20.

Extra 2: The Darkest Color

Write a procedure, (rgb-darkest colors), that, given a nonempty list of colors, finds the darkest of those colors.

Extra 3: Closest to Zero

Write a procedure, (closest-to-zero values), that, given a list of real numbers (including both positive and negative numbers), returns the value closest to zero in the list.

Hint: Think about how, given two numbers, you determine which is closer to zero.

Hint: Think about how this problem is similar to a problem or problems we've solved before.

Extra 4: Distances Between Spots

Write a procedure, (spot-distances spots), that, given a list of spots, returns a list of the distance between neighboring spots in the list. For example,

For example, 

> (define some-spots
    (list (spot-new 0 0 "red")
          (spot-new 10 0 "orange")
          (spot-new 10 8 "yellow")
          (spot-new 6 8 "green")
          (spot-new 6 5 "blue")
          (spot-new 9 9 "indigo")
          (spot-new 10 10 "violet")))
> (spot-distances some-spots)
(10 8 4 3 5 1.4142135)

Notice that spot-distances returns a list whose length is one less than the length of its input list.

Extra 5: Averaging Colors

Write a procedure, (rgb-averages colors), that, given a list of colors, computes a new list of colors, by averaging subsequent pairs of colors. For example, if the input list is the standard seven rainbow colors (red, orange, yellow, green, blue, indigo, and violet), the output list will consist of a red-orange average, an orange-yellow average, a yellow-green average, a green-blue average, a blue-indigo average, and an indigo-violet average.

Once again, the length of the result list is one less than the length of the input list.

Extra 6: Rendering Spots, Revisited

Write (image-render-spots! image spots) without using map or foreach!.

Notes

Notes on Problem 6: Summing Components

We can use map to extract the red component of each color. 

(map rgb-red colors)

That gives us a list of numbers, which we can sum with sum. 

(sum (map rgb-red colors))

Putting it all together in a procedure, we get 

(define sum-red
  (lambda (colors)
    (sum (map rgb-red colors))))

Jerod Weinman