Laboratory: Analyzing Procedures

a. Add the following line to the beginning of list-append (again, immediately after the lambda).

(display (list 'list-append front back)) (newline)

b. Determine how many times list-append is called when reversing a list of length seven using list-reverse-1.

c. Add the following line to the kernel of list-reverse-2 (immediately after the lambda).

(display (list 'kernel remaining reversed)) (newline)

d. Determine how many times the kernel is called when reversing a list of length seven using list-reverse-2.

e. Comment out the lines that you just added by prefixing them with a semicolon.

a. Replace the define for list-reverse-1 with define$, as in the following.

(define$ list-reverse-1
  (lambda (lst)
     ...))

b. Find out how many times list-append is called in reversing a list of seven elements by entering the following command in the interactions pane.

> (analyze (list-reverse-1 (list 1 2 3 4 5 6 7)) list-append)

c. Did you get the same answer as in the previous exercise? If not, why do you think you got a different result?

d. One potential issue is that we haven't told the analyst to include the recursive calls in list-append. We can do so by replacing define with define$ in the definition of list-append.

e. Once again, find out how many times list-append is called in reversing a list of seven elements by entering the following command in the interactions pane.

> (analyze (list-reverse-1 (list 1 2 3 4 5 6 7)) list-append)

f. Did you get the same answer as in exercise 1? If not, what difference do you see?

g. Replace the define in list-reverse-2 with define$.

h. Find out how many times kernel is called in reversing a list of seven elements by entering the following command in the interactions pane.

> (analyze (list-reverse-2 (list 1 2 3 4 5 6 7)) kernel)

i. Did you get the same answer as in exercise 1? If not, what difference do you see?

In the previous exercise, you considered only a single procedure in each case (list-append for list-reverse-1, list-reverse-2-kernel for list-reverse-2). Suppose we incorporate all of the other procedures. What effect does it have?

a. Find out how many total procedure calls are done in reversing a list of length seven, using list-reverse-1, with the following.

> (analyze (list-reverse-1 (list 1 2 3 4 5 6 7)))

b. How does that number of calls seem to relate to the number of calls to list-append?

c. Are there any procedures you're surprised to see?

d. Find out how many total procedure calls are done in reversing a list of length seven, using list-reverse-2, by entering the following command.

> (analyze (list-reverse-2 (list 1 2 3 4 5 6 7)))

e. How does that number of calls seem to relate to the number of calls to kernel?

f. Are there any procedures you're surprised to see?

a. Fill in the following chart to the best of your ability.

b. Predict what the entries will be for a list size of 32.

c. Check your results experimentally.

d. Write a formula for the columns, to the best of your ability.

Here is a third version of rgb-brightest.

;;; Procedure:
;;;   rgb-brightest
;;; Parameters:
;;;   colors, a list of RGB colors
;;; Purpose:
;;;   Find the brightest color in the list.
;;; Produces:
;;;   brightest, an RGB color.
;;; Preconditions:
;;;   rgb-brightness is defined.
;;;   colors contains at least one element.
;;; Postconditions:
;;;   For any color in colors,
;;;    (rgb-brightness color) <= (rgb-brightness brightest)
;;;   brightest is an element of colors.
(define rgb-brightest-3
  (lambda (colors)
    (let kernel ((brightest-so-far (car colors))
                 (remaining-colors (cdr colors))))
      (if (null? remaining-colors)
          brightest-so-far
          (kernel (rgb-brighter brightest-so-far (car remaining-colors))
                  (cdr remaining-colors)))))

a. Find out how many steps this procedure takes on lists of length 2, 4, 8, and 16 in which the elements are arranged from lightest to darkest.

b. Find out how many steps this procedure takes on lists of length 2, 4, 8, and 16 in which the elements are arranged from darkest to lightest.

c. Find out how many steps this procedure takes on lists of length 2, 4, 8, and 16 in which the elements are in no particular order.

d. Predict the number of steps this procedure will take on each kind of list, where the length is 32.

Consider rgb-brightest-4, a variant of an efficient version of rgb-brightest that has additional error checking added.

(define rgb-brightest-4
  (lambda (colors)
    (if (not (all-rgb? colors))
        (throw "rgb-brightest: expects a list of colors; received" colors))
    (if (null? (cdr colors))
        (car colors)
        (rgb-brighter (car colors)
                      (rgb-brightest-2 (cdr colors))))))
 

a. Predict the number of calls to rgb-brightest-4 in finding the brightest in a list of eight colors.

b. Check your hypothesis.

c. Predict the number of calls to all-rgb? in finding the brightest in a list of eight colors.

d. Check your hypothesis.

Rewrite rgb-brightest-4 so that it continues to check preconditions, but precondition checking does not exact such a heavy penalty.