Discussion 9: Scheme, Scheme Lists

Attendance

Your TA will come around during discussion to check you in.

If you miss discussion for a good reason (such as sickness or a scheduling conflict), email cs61a@berkeley.edu within one week to receive attendance credit.

Scheme

You can use scheme.cs61a.org to try things out.

An if expression has 3 subexpressions:

  • The first one, called the predicate, is always evaluated to choose between the next two.
  • The second one, called the consequent, is evaluated if the predicate's value is true.
  • The third one, called the alternative, is evaluated if the predicate's value is false.

The value of the second or third expression (whichever gets evaluated) is the value of the whole if expression.

Only #f is a false value in Scheme. All other values are true values, including #t.

The logical special forms and and or can have any number of sub-expressions. Use and to check if all its sub-expressions are true values. Use or to check if any of them are.

An Example:

scm> (define y (+ 1 2))  ; y is now 3
y
scm> (define z (or (> y 4) (> y 5) (< y 5)))  ; z is true because y < 5
z
scm> (+ 7 (if (and (> y 1) z) y 1))  ; the if expression evaluates to 3
10

The expect form is built into scheme.cs61a.org and checks to see if its first sub expression evaluates to its second one. It is used for testing.

Q1: Perfect Fit

Definition: A perfect square is k*k for some integer k.

Implement fit, which takes non-negative integers total and n. It returns whether there are n different positive perfect squares that sum to total.

Use the (or _ _) special form to combine two recursive calls: one that uses k*k in the sum and one that does not.
This is a tree recursion problem. Your implementation should try different ways of summing up n different perfect squares to see if the sum is total. For the example (fit 10 2), the call to (f 10 2 1) returns whether there are 2 perfect squares that sum to 10 and are each 1*1 or greater. To figure that out, you'll need to make two recursive calls:
  • (f 10 2 2) which returns whether there are 2 perfect squares that sum to 10 and are each 2*2 or greater (nope)
  • (f 9 1 2) which returns whether there is 1 perfect square that sums to 9 and is 2*2 or greater (yep: 3*3)

Once your implementation makes these two recursive calls, it needs to combine them into one result.

The first recursive call should subtract k*k from total and subtract 1 from n; the other recursive call should leaves total and n unchanged.

Q2: Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest integer that evenly divides two positive integers.

Write the procedure gcd, which computes the GCD of numbers a and b using Euclid's algorithm, which recursively uses the fact that the GCD of two values is either of the following:

  • the smaller value if it evenly divides the larger value, or
  • the greatest common divisor of the smaller value and the remainder of the larger value divided by the smaller value

In other words, if a is greater than b and a is not divisible by b, then

gcd(a, b) = gcd(b, a % b)  # please note that this is Python syntax

You may find the provided procedures min and max helpful. You can also use the built-in modulo and zero? procedures.

>scm> (modulo 10 4)
>2
>scm> (zero? (- 3 3))
>#t
>scm> (zero? 3)
>#f

Scheme Lists & Quotation

Scheme lists are linked lists. Lightning review:

  • nil and () are the same thing: the empty list.
  • (cons first rest) constructs a linked list with first as its first element. and rest as the rest of the list, which should always be a list.
  • (car s) returns the first element of the list s.
  • (cdr s) returns the rest of the list s.
  • (list ...) takes n arguments and returns a list of length n with those arguments as elements.
  • (append ...) takes n lists as arguments and returns a list of all of the elements of those lists.
  • (draw s) draws the linked list structure of a list s. It only works on code.cs61a.org/scheme. Try it now with something like (draw (cons 1 nil)).

Quoting an expression leaves it unevaluated. Examples:

  • 'four and (quote four) both evaluate to the symbol four.
  • '(2 3 4) and (quote (2 3 4)) both evaluate to a list containing three elements: 2, 3, and 4.
  • '(2 3 four) and (quote (2 3 four)) evaluate to a list containing 2, 3, and the symbol four.

Here's an important difference between list and quotation:

scm> (list 2 (+ 3 4))
(2 7)
scm> '(2 (+ 3 4))
(2 (+ 3 4))

Q3: Remove

Implement a procedure remove that takes in a list of numbers s and a number x. It returns a list with all instances of x removed from s. You may assume that s only contains of numbers and will not have nested lists.

Q4: Pair Up

Implement pair-up, which takes a list s. It returns a list of lists that together contain all of the elements of s in order. Each list in the result should have 2 elements. The last one can have up to 3.

Look at the examples to make sure you understand what this procedure does.

pair-up takes a list (of numbers) and returns a list of lists, so when (length s) is less than or equal to 3, return a list containing the list s. For example, (pair-up (list 2 3 4)) should return ((2 3 4)).

Use (cons _ (pair-up _)) to create the result, where the first argument to cons is a list with two elements: the (car s) and the (car (cdr s)). The argument to pair-up is everything after the first two elements.

Q5: List Insert

Write a Scheme function that, when given an element, a list, and an index, inserts the element into the list at that index. You can assume that the index is in bounds for the list.

Use Ok to unlock and test your code:

python3 ok -q list-insert -u
python3 ok -q list-insert

Q6: Increasing Rope

Definition: A rope in Scheme is a non-empty list containing only numbers except for the last element, which may either be a number or a rope.

Implement up, a Scheme procedure that takes a positive integer n. It returns a rope containing the digits of n that is the shortest rope in which each pair of adjacent numbers in the same list are in increasing order.

Reminder: the quotient procedure performs floor division, like // in Python. The remainder procedure is like % in Python.

Compare first to (car result) to decide whether to cons the value first onto the result or whether to form a new list that contains first and result as elements.
To correctly call helper from within up, build a rope that only contains the last digit of n: (remainder n 10).