#lang pl 10
#|
Skeleton code for the queens problem.
Reminder: in Schlac all values are one-argument functions, so you'll
never get any kind of a "type error". You can only get such errors
when you use one of the special conversion-to-racket with a bad
value if you're lucky -- and a meaningless result if you're not.
This means that you need to be extremely careful about the types:
they're only comments, but make sure that you use the right types.
Use the test cases as much as possible to make your life easier. If
you get stuck, you can try to implement the problem in the course
language, then translate the Racket code back to Schlac. See the
homework text for details.
|#
;; a convenient rewrite for local bindings
(rewrite (with [x E1] E2) => ???)
;; tests
(test (->nat (with [x (* 2 4)] (+ x x))) => '16)
(test (->nat (with [x 3] (with [x (* x 3)] (+ x x)))) => '18)
;; ==================== Numeric utilities ====================
;; diff : Nat Nat -> Nat
;; computes the difference between two numbers: |x-y|
(define diff
(lambda (x y)
;; reminder about how our `-` behaves: if y>x, then (- x y) => 0
???))
;; tests
(test (->nat (diff 0 0)) => '0)
(test (->nat (diff 3 3)) => '0)
(test (->nat (diff 2 4)) => '2)
(test (->nat (diff 4 2)) => '2)
;; = : Nat Nat -> Bool
;; comparison for natural numbers
???
;; tests
(test (->bool (= 0 0)) => '#t)
(test (->bool (= 1 1)) => '#t)
(test (->bool (= 2 1)) => '#f)
(test (->bool (= (* (* 2 3) 2) (* 3 4))) => '#t)
;; ==================== List utilities ====================
;; append : (Listof A) (Listof A) -> (Listof A)
;; appends two lists
???
;; tests
(test (->listof ->nat (append null null)) => '())
(test (->listof ->nat (append null l123)) => '(1 2 3))
(test (->listof ->nat (append l123 null)) => '(1 2 3))
(test (->listof ->nat (append l123 l123)) => '(1 2 3 1 2 3))
;; append* : (Listof (Listof A)) -> (Listof A)
;; appends a list of lists
(define/rec append*
(lambda (lists)
???))
;; tests
(test (->listof ->nat (append* null)) => '())
(test (->listof ->nat (append* (cons null null))) => '())
(test (->listof ->nat (append* (cons l123 (cons l123 null))))
=> '(1 2 3 1 2 3))
;; map : (A -> B) (Listof A) -> (Listof B)
;; maps a function over a list, returning a list of result values
???
;; tests
(test (->listof ->nat (map add1 null)) => '())
(test (->listof ->nat (map (+ 2) l123)) => '(3 4 5))
;; the `map' examples below are just for fun
(test (->listof ->bool
(with [tf (cons #t (cons #f null))]
(append* (map (lambda (x) (map (and x) tf)) tf))))
=> '(#t #f #f #f))
;; this is doing the same, using the fact that `not' returns a
;; function with swapped arguments
(test (->listof ->bool
(with [tf (cons #t (cons #f null))]
(append* (map (not map tf) (map and tf)))))
=> '(#t #f #f #f))
;; (map and tf) = (not map tf and), so abstract the (not map tf) part
(test (->listof ->bool
(with [nmtf (not map (cons #t (cons #f null)))]
(append* (map nmtf (nmtf and)))))
=> '(#t #f #f #f))
;; finally, do this for both `and' and `or'
(test (->listof ->bool
(with [nmtf (not map (cons #t (cons #f null)))]
(with [ao (cons and (cons or null))]
(append* (map nmtf (append* (map nmtf ao)))))))
=> '(#t #f #f #f #t #t #t #f))
;; filter : (A -> Bool) (Listof A) -> (Listof A)
;; filters a list according to a given predicate
???
;; tests
(define =3 (= 3))
(test (->listof ->nat (filter =3 null)) => '())
(test (->listof ->nat (filter =3 l123)) => '(3))
(test (->listof ->nat (filter (lambda (x) (= 1 (diff x 2))) l123))
=> '(1 3))
;; ==================== Main code ====================
;; From this point we start with the actual solution. A configuration
;; of a board is a list of numbers: each number represents the column
;; of a queen at that row. For example (0 1 0) represents three rows,
;; where the queen of the first row stands on the first column, the
;; second row queen is at the second column, and the third is on the
;; first column too.
;; threaten? : Nat Nat Nat -> Bool
;; determines whether a queen at column x and a queen at column y are
;; threatening each other when they are n rows apart
(define threaten?
(lambda (x y n)
??? ; hint: use (diff x y)
))
;; tests
;; if both are at column 1 (ie, the second column), then they always
;; threaten each other
(test (->bool (threaten? 1 1 1)) => '#t)
(test (->bool (threaten? 1 1 2)) => '#t)
(test (->bool (threaten? 1 1 3)) => '#t)
;; if the first is at 1 and the second at 3, then they threaten each
;; other if they are two rows apart
(test (->bool (threaten? 1 3 1)) => '#f)
(test (->bool (threaten? 1 3 2)) => '#t)
(test (->bool (threaten? 1 3 3)) => '#f)
;; same when the columns are swapped
(test (->bool (threaten? 3 1 1)) => '#f)
(test (->bool (threaten? 3 1 2)) => '#t)
(test (->bool (threaten? 3 1 3)) => '#f)
;; safe? : Nat Nat (Listof Nat) -> Bool
;; determines whether it's safe to put a queen at column col when it's
;; n rows before a list of column positions cols; in other words,
;; (safe col 1 cols) determines if you can put a queen at column col
;; just before the configuration specified by cols; (this is a helper
;; for `configurations' below, note that the cols argument is
;; essentially a part of a whole-board configuration)
(define/rec safe?
(lambda (col n cols)
(or (null? cols)
??? ; uses `threaten?', of course
)))
;; tests
(test (->bool (safe? 5 1 l123)) => '#t)
;; configurations : Nat (Listof Nat) -> (Listof (Listof Nat))
;; this is the core of the solution: finds all valid n-row
;; configurations where the queen columns are all taken from the given
;; list of input cols
(define/rec configurations
(lambda (n cols)
(if (zero? n)
(cons null null)
(append* (map (lambda (rest)
(map ???
(filter ??? ; uses `safe?'
cols)))
(configurations (sub1 n) cols))))))
;; No need to test this, since the main `queens' function is a simple
;; call to this function. But to clarify, if you call it with
;; (configurations 2 (list 1 2 4)), then the result will be a list of
;; "board configurations" of queen placements: each such configuration
;; is a list of 2 column numbers from the (list 1 2 4), such that no
;; queen threatens another queen. In this case, (list 2 4) is a valid
;; result, (list 1) is invalid (need two rows not one), and (list 1 2)
;; is not (because it represents two diagonally placed queens).
;; range : Nat Nat -> (Listof Nat)
;; returns the list of numbers starting from the first argument, up to
;; (but not including) the second argument
(define/rec range
(lambda (from to)
???))
;; tests
(test (->listof ->nat (range 0 0)) => '())
(test (->listof ->nat (range 0 5)) => '(0 1 2 3 4))
(test (->listof ->nat (range 2 5)) => '(2 3 4))
;; queens : Nat -> (Listof Nat)
;; finally, the solution is simple -- find all the configurations of
;; `size' rows, where each row has a queen at some column 0 to `size'
;; and return the first one, or return an empty list if there are no
;; solutions
(define queens
(lambda (size)
???))
;; tests
;; single solution for a 1x1 board:
(test (->listof ->nat (queens 1)) => '(0))
;; no solution for 2x2 or 3x3 boards
(test (->listof ->nat (queens 2)) => '())
(test (->listof ->nat (queens 3)) => '())
;; and finally test a few solution (note that these tests depend on
;; the specific algorithm since there are many correct solutions, so
;; they are *not* good tests)
(test (->listof ->nat (queens 4)) => '(2 0 3 1))
(test (->listof ->nat (queens 5)) => '(3 1 4 2 0))
;; 8 : Nat
;; define this so you can run the real problem conveniently
(define 8 (+ 4 4))
;; tests
(test (->nat 8) => '8)
;; Finally, this is what you want to try:
;; (->listof ->nat (queens 8))
;; You can also try to see how long it takes to find all solutions by
;; making `queens' return the whole list, and use it like this:
;; (->listof (->listof ->nat) (queens-all 4))