Pairs and pair structures

Exercise 1

Draw box-and-pointer diagrams for each of the following lists:

((x) y z)
(x (y z))
((a) b (c ()))

Exercise 2

Have DrScheme evaluate each of the following expressions. In each case, explain why Scheme does or does not use the dot notation when displaying the value.

(cons 'a "Walker")
(cons 'a '())
(cons '() 'a)
(cons '() (cons '() '()))

Exercise 3

Draw a box-and-pointer representation of the value of each expression in the previous exercise.

Exercise 4

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 last box-and-pointer diagram shown above 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.

Use cons-cell-count to find out how many cons cells are needed to construct the list (0 (1 (2 (3 (4))))). Draw a box-and-pointer diagram of this list to check the answer.

Exercise 5

Define and test a procedure called pair-structure-max that takes any Scheme value as its argument, and returns the greatest exact positive integer contained in that value (considered as a pair structure), or 0 if the structure contains no exact positive integers.

In other words, if pair-structure-max is given an exact positive integer, it should return that argument unchanged; if it is given a pair, it should apply itself recursively to the pair's car and cdr, and return whichever result is larger; and otherwise it should return 0.

> (pair-structure-max 17)
17
> (pair-structure-max '((3 . 1) . (4 . (5 . 2))))
5
> (pair-structure-max '())
0
> (pair-structure-max "foo")
0
> (pair-structure-max '(((-3 . "bar") . 17.9) . (baz . #f)))
0