Fundamentals of Computer Science I: Media Computing (CS151.02 2007F)

Deep Recursion


Summary: In this laboratory, you will further explore issues of deep recursion introduced in the reading on pairs and pair structures and continued in the reading on deep recursion.

Preparation

a. Make sure that you have the reading on pairs and pair structures and the reading on deep recursion open in separate tabs and windows.

b. Make sure that you have a piece of paper and writing instrument handy.

Exercises

Exercise 1: Number Trees

Recall that a list is a data structure defined recursively as follows:

  • The empty list is a list.
  • Cons of a value and a list is a list.

In the reading on pairs and pair structures, the section entitled “Recursion with Pairs” includes a procedure that works on “number trees”, nested structures built with the pair procedure.

Write a recursive definition for number trees, trees built from only numbers and cons cells, similar to that for lists.

Exercise 2: A Number Tree Predicate

Using your recursive definition of number trees from the previous problem, write a procedure, (number-tree? val) that returns true if val is a number tree and false otherwise.

Exercise 3: Summing Number Trees

Consider again the sum-of-number-tree procedure from the reading, reproduced here.

;;; Procedure:
;;;   sum-of-number-tree
;;; Parameters:
;;;   ntree, a number tree
;;; Purpose:
;;;   Sums all the numbers in ntree.
;;; Produces:
;;;   sum, a number
;;; Preconditions:
;;;   ntree is a number tree.  That is, it consists only of numbers
;;;   and cons cells.
;;; Postconditions:
;;;   sum is the sum of all numbers in ntree.
(define sum-of-number-tree
  (lambda (ntree)
    (if (pair? ntree)
        (+ (sum-of-number-tree (car ntree))
           (sum-of-number-tree (cdr ntree)))
        ntree)))

a. Verify that sum-of-number-tree works as advertised on a single number.

b. Verify that sum-of-number-tree works as advertised on a pair of numbers.

c. Verify that it works as advertised on the first example.

> (sum-of-number-tree (cons (cons (cons 0 1)
                                  (cons 2 3))
                            (cons (cons 4 5)
                                  (cons 6 7))))

d. What do you expect sum-of-number-tree to return when given (cons 10 11) as a parameter? Verify your answer experimentally.

e. What do you expect sum-of-number-tree to return when given the empty list as a parameter? Verify your answer experimentally.

f. What do you expect sum-of-number-tree to return when given (list 1 2 3 4 5) as a parameter? Verify your answer experimentally.

Exercise 4: Preconditions

a. What preconditions should sum-of-number-tree have?

b. Use the number-tree? predicate from earlier to rewrite sum-of-number-tree so that it reports an appropriate error if its preconditions are not met.

c. Some programmers consider it inappropriate to scan a tree twice, once to make sure that it's valid and once to compute a value based on the tree. Rewrite sum-of-number-tree so that it checks for and reports errors only when it is at one of the non-pair values.

Exercise 5: Counting Cons Cells

a. Define and test a procedure named cons-cell-count that takes any Scheme value and determines how many boxes would appear in its box-and-pointer diagram. (The data structure that is represented by such a box, or the region of a computer's memory in which such a structure is stored is called a cons cell. Every time the cons procedure is used, explicitly or implicitly, in the construction of a Scheme value, a new cons cell is allocated, to store information about the car and the cdr. Thus cons-cell-count also tallies the number of times cons was invoked during the construction of its argument.)

For example, the structure in the following box-and-pointer diagram contains seven cons-cells, so when you apply cons-cell-count to that structure, it should return 7. On the other hand, the string "sample" contains no cons-cells, so the value of (cons-cell-count "sample") is 0.

In answering this question, you should consider whether each value, in turn, is a pair using the pair? predicate.

b. Use cons-cell-count to find out how many cons cells are needed to construct the list

(0 (1 (2 (3 (4)))))

See the notes at the end of the lab if you have trouble creating that list.

c. Draw a box-and-pointer diagram of this list to check the answer.

Explorations

Exploration 1: Rendering Color Trees

Recall the render-color-tree from the reading.

a. Add render-color-tree to your definitions window.

b. Create a new 200x200 image named canvas.

c. What effect do you expect the following instruction to have?

> (render-color-tree color.blue canvas 0 0 200 200)

d. Check your answer experimentally.

e. What effect do you expect the following instruction to have?

> (render-color-tree (cons color.black color.red) canvas 0 0 200 200)

f. Check your answer experimentally.

g. What effect do you expect the following instruction to have?

> (render-color-tree (cons (cons color.green color.yellow) color.orange) canvas 0 0 200 200)

h. Check your answer experimentally.

i. Create a color tree that you might be able to render in some interesting fashion.

j. Rewrite render-color-tree so that it splits the image vertically rather than horizontally.

k. Rewrite render-color-tree so that it randomly chooses between splitting the image vertically and horizontally.

l. What do you expect to have happen if you apply your new version to the following color tree?

> (define red-black (cons (cons (cons color.red (cons color.black color.red)) (cons color.black (cons color.red color.black))) (cons color.red (cons color.black color.red))))

m. Check your answer experimentally.

n. What do you expect to have happen if you apply it again?

o. Check your answer experimentally.

p. Write a version of render-color-tree that alternates between splitting the image vertically and horizontally.

q. Write a procedure that, given a list of colors and a maximum depth, randomly generates a color tree. Render your trees to see what they look like!

One potentially interesting way to experiment with color trees is to have a procedure build those trees. How? We might provide the procedure with an intended size of the tree. If the number of colors is 1, we simply return a randomly-selected color. If the number of colors is 2, we cons together two trees of size 1. Otherwise, we pick a random size for the left subtree (the size should be at least 1 and strictly less than the intended size of the whole tree). We then recursively build a tree of that size and another tree of an appropriate size and then cons them together.

a. Using this technique, write a procedure, (random-bw-tree size), that randomly builds a black and white tree of the appropriate size.

b. Using this technique, write a procedure, (random-color-tree size colors), that builds a random color tree of the desired size, randomly selecting the color from the list colors.

Create a new version of render-color-tree that has an extra parameter, vsplit?, a Boolean value. If the Boolean is true, when render-color-tree should split the area vertically (as it currently does). If the Boolean is false, render-color-tree should split the area horizontally. In both cases, the recursive calls should negate that Booelan value.

For Those with Extra Time

If you find that you have extra time, you might want to attempt one or more of the following problems.

Extra 1: Cons Cells vs. Values

As you may recall, a tree is either (a) a non-pair value or (b) the cons of two trees. In the reading, you saw a procedure that counted the number of values in a tree. In this lab, you wrote a procedure that counted the number of cons cells (pairs) in a tree. What is the relationship between the numbers returned by those two procedures?

Extra 2: Searching for Colors

a. Write a procedure, (color-tree.contains? ctree color), that determines whether color appears anywhere in ctree.

b. Rewrite the procedure to determine if a color nearby to color appears somewhere in ctree. (You might say that two colors are nearby if their red, green, and blue components all differ by less than sixteen.)

Notes

Notes on Exercise 5

If, for some reason, you are having trouble creating the list

(0 (1 (2 (3 (4)))))

try

(list 0 (list 1 (list 2 (list 3 (list 4)))))

Creative Commons License

Samuel A. Rebelsky, rebelsky@grinnell.edu

Copyright 2007 Janet Davis, Matthew Kluber, and Samuel A. Rebelsky. (Selected materials copyright by John David Stone and Henry Walker and used by permission.)

This material is based upon work partially supported by the National Science Foundation under Grant No. CCLI-0633090. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/2.5/ or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA.