771 lines
29 KiB
Scheme
771 lines
29 KiB
Scheme
;;; MATRIX -- Obtained from Andrew Wright.
|
|
|
|
(import (scheme base)
|
|
(scheme read)
|
|
(scheme write))
|
|
|
|
(define div quotient)
|
|
(define mod modulo)
|
|
|
|
; Chez-Scheme compatibility stuff:
|
|
|
|
(define (chez-box x) (cons x '()))
|
|
(define (chez-unbox x) (car x))
|
|
(define (chez-set-box! x y) (set-car! x y))
|
|
|
|
; Test that a matrix with entries in {+1, -1} is maximal among the matricies
|
|
; obtainable by
|
|
; re-ordering the rows
|
|
; re-ordering the columns
|
|
; negating any subset of the columns
|
|
; negating any subset of the rows
|
|
; Where we compare two matricies by lexicographically comparing the first row,
|
|
; then the next to last, etc., and we compare a row by lexicographically
|
|
; comparing the first entry, the second entry, etc., and we compare two
|
|
; entries by +1 > -1.
|
|
; Note, this scheme obeys the useful fact that if (append mat1 mat2) is
|
|
; maximal, then so is mat1. Thus, we can build up maximal matricies
|
|
; row by row.
|
|
;
|
|
; Once you have chosen the row re-ordering so that you know which row goes
|
|
; last, the set of columns to negate is fixed (since the last row must be
|
|
; all +1's).
|
|
;
|
|
; Note, the column ordering is really totally determined as follows:
|
|
; all columns for which the second row is +1 must come before all
|
|
; columns for which the second row is -1.
|
|
; among columns for which the second row is +1, all columns for which
|
|
; the third row is +1 come before those for which the third is
|
|
; -1, and similarly for columns in which the second row is -1.
|
|
; etc
|
|
; Thus, each succeeding row sorts columns withing refinings equivalence
|
|
; classes.
|
|
;
|
|
; Maximal? assumes that mat has atleast one row, and that the first row
|
|
; is all +1's.
|
|
(define maximal?
|
|
(lambda (mat)
|
|
(let pick-first-row
|
|
((first-row-perm
|
|
(gen-perms mat)))
|
|
(if first-row-perm
|
|
(and (zunda first-row-perm mat)
|
|
(pick-first-row (first-row-perm 'brother)))
|
|
#t))))
|
|
|
|
(define zunda
|
|
(lambda (first-row-perm mat)
|
|
(let* ((first-row
|
|
(first-row-perm 'now))
|
|
(number-of-cols
|
|
(length first-row))
|
|
(make-row->func
|
|
(lambda (if-equal if-different)
|
|
(lambda (row)
|
|
(let ((vec
|
|
(make-vector number-of-cols)))
|
|
(do ((i 0 (+ i 1))
|
|
(first first-row
|
|
(cdr first))
|
|
(row row
|
|
(cdr row)))
|
|
((= i number-of-cols))
|
|
(vector-set! vec
|
|
i
|
|
(if (= (car first) (car row))
|
|
if-equal
|
|
if-different)))
|
|
(lambda (i)
|
|
(vector-ref vec i))))))
|
|
(mat
|
|
(cdr mat)))
|
|
(zebra (first-row-perm 'child)
|
|
(make-row->func 1 -1)
|
|
(make-row->func -1 1)
|
|
mat
|
|
number-of-cols))))
|
|
|
|
|
|
(define zebra
|
|
(lambda (row-perm row->func+ row->func- mat number-of-cols)
|
|
(let _-*-
|
|
((row-perm
|
|
row-perm)
|
|
(mat
|
|
mat)
|
|
(partitions
|
|
(list (miota number-of-cols))))
|
|
(or (not row-perm)
|
|
(and
|
|
(zulu (car mat)
|
|
(row->func+ (row-perm 'now))
|
|
partitions
|
|
(lambda (new-partitions)
|
|
(_-*- (row-perm 'child)
|
|
(cdr mat)
|
|
new-partitions)))
|
|
(zulu (car mat)
|
|
(row->func- (row-perm 'now))
|
|
partitions
|
|
(lambda (new-partitions)
|
|
(_-*- (row-perm 'child)
|
|
(cdr mat)
|
|
new-partitions)))
|
|
(let ((new-row-perm
|
|
(row-perm 'brother)))
|
|
(or (not new-row-perm)
|
|
(_-*- new-row-perm
|
|
mat
|
|
partitions))))))))
|
|
|
|
|
|
(define zulu
|
|
(let ((cons-if-not-null
|
|
(lambda (lhs rhs)
|
|
(if (null? lhs)
|
|
rhs
|
|
(cons lhs rhs)))))
|
|
(lambda (old-row new-row-func partitions equal-cont)
|
|
(let _-*-
|
|
((p-in
|
|
partitions)
|
|
(old-row
|
|
old-row)
|
|
(rev-p-out
|
|
'()))
|
|
(let _-split-
|
|
((partition
|
|
(car p-in))
|
|
(old-row
|
|
old-row)
|
|
(plus
|
|
'())
|
|
(minus
|
|
'()))
|
|
(if (null? partition)
|
|
(let _-minus-
|
|
((old-row
|
|
old-row)
|
|
(m
|
|
minus))
|
|
(if (null? m)
|
|
(let ((rev-p-out
|
|
(cons-if-not-null
|
|
minus
|
|
(cons-if-not-null
|
|
plus
|
|
rev-p-out)))
|
|
(p-in
|
|
(cdr p-in)))
|
|
(if (null? p-in)
|
|
(equal-cont (reverse rev-p-out))
|
|
(_-*- p-in old-row rev-p-out)))
|
|
(or (= 1 (car old-row))
|
|
(_-minus- (cdr old-row)
|
|
(cdr m)))))
|
|
(let ((next
|
|
(car partition)))
|
|
(case (new-row-func next)
|
|
((1)
|
|
(and (= 1 (car old-row))
|
|
(_-split- (cdr partition)
|
|
(cdr old-row)
|
|
(cons next plus)
|
|
minus)))
|
|
((-1)
|
|
(_-split- (cdr partition)
|
|
old-row
|
|
plus
|
|
(cons next minus)))))))))))
|
|
|
|
(define all?
|
|
(lambda (ok? lst)
|
|
(let _-*-
|
|
((lst
|
|
lst))
|
|
(or (null? lst)
|
|
(and (ok? (car lst))
|
|
(_-*- (cdr lst)))))))
|
|
|
|
(define gen-perms
|
|
(lambda (objects)
|
|
(let _-*-
|
|
((zulu-future
|
|
objects)
|
|
(past
|
|
'()))
|
|
(if (null? zulu-future)
|
|
#f
|
|
(lambda (msg)
|
|
(case msg
|
|
((now)
|
|
(car zulu-future))
|
|
((brother)
|
|
(_-*- (cdr zulu-future)
|
|
(cons (car zulu-future)
|
|
past)))
|
|
((child)
|
|
(gen-perms
|
|
(fold past cons (cdr zulu-future))))
|
|
((puke)
|
|
(cons (car zulu-future)
|
|
(fold past cons (cdr zulu-future))))
|
|
(else
|
|
(error 'gen-perms "Bad msg: ~a" msg))))))))
|
|
|
|
(define fold
|
|
(lambda (lst folder state)
|
|
(let _-*-
|
|
((lst
|
|
lst)
|
|
(state
|
|
state))
|
|
(if (null? lst)
|
|
state
|
|
(_-*- (cdr lst)
|
|
(folder (car lst)
|
|
state))))))
|
|
|
|
(define miota
|
|
(lambda (len)
|
|
(let _-*-
|
|
((i 0))
|
|
(if (= i len)
|
|
'()
|
|
(cons i
|
|
(_-*- (+ i 1)))))))
|
|
|
|
(define proc->vector
|
|
(lambda (size proc)
|
|
(let ((res
|
|
(make-vector size)))
|
|
(do ((i 0
|
|
(+ i 1)))
|
|
((= i size))
|
|
(vector-set! res
|
|
i
|
|
(proc i)))
|
|
res)))
|
|
|
|
; Given a prime number P, return a procedure which, given a `maker' procedure,
|
|
; calls it on the operations for the field Z/PZ.
|
|
(define make-modular
|
|
(lambda (modulus)
|
|
(let* ((reduce
|
|
(lambda (x)
|
|
(mod x modulus)))
|
|
(coef-zero?
|
|
(lambda (x)
|
|
(zero? (reduce x))))
|
|
(coef-+
|
|
(lambda (x y)
|
|
(reduce (+ x y))))
|
|
(coef-negate
|
|
(lambda (x)
|
|
(reduce (- x))))
|
|
(coef-*
|
|
(lambda (x y)
|
|
(reduce (* x y))))
|
|
(coef-recip
|
|
(let ((inverses
|
|
(proc->vector (- modulus 1)
|
|
(lambda (i)
|
|
(extended-gcd (+ i 1)
|
|
modulus
|
|
(lambda (gcd inverse ignore)
|
|
inverse))))))
|
|
; Coef-recip.
|
|
(lambda (x)
|
|
(let ((x
|
|
(reduce x)))
|
|
(vector-ref inverses (- x 1)))))))
|
|
(lambda (maker)
|
|
(maker 0 ; coef-zero
|
|
1 ; coef-one
|
|
coef-zero?
|
|
coef-+
|
|
coef-negate
|
|
coef-*
|
|
coef-recip)))))
|
|
|
|
; Extended Euclidean algorithm.
|
|
; (extended-gcd a b cont) computes the gcd of a and b, and expresses it
|
|
; as a linear combination of a and b. It returns calling cont via
|
|
; (cont gcd a-coef b-coef)
|
|
; where gcd is the GCD and is equal to a-coef * a + b-coef * b.
|
|
(define extended-gcd
|
|
(let ((n->sgn/abs
|
|
(lambda (x cont)
|
|
(if (>= x 0)
|
|
(cont 1 x)
|
|
(cons -1 (- x))))))
|
|
(lambda (a b cont)
|
|
(n->sgn/abs a
|
|
(lambda (p-a p)
|
|
(n->sgn/abs b
|
|
(lambda (q-b q)
|
|
(let _-*-
|
|
((p
|
|
p)
|
|
(p-a
|
|
p-a)
|
|
(p-b
|
|
0)
|
|
(q
|
|
q)
|
|
(q-a
|
|
0)
|
|
(q-b
|
|
q-b))
|
|
(if (zero? q)
|
|
(cont p p-a p-b)
|
|
(let ((mult
|
|
(div p q)))
|
|
(_-*- q
|
|
q-a
|
|
q-b
|
|
(- p (* mult q))
|
|
(- p-a (* mult q-a))
|
|
(- p-b (* mult q-b)))))))))))))
|
|
|
|
; Given elements and operations on the base field, return a procedure which
|
|
; computes the row-reduced version of a matrix over that field. The result
|
|
; is a list of rows where the first non-zero entry in each row is a 1 (in
|
|
; the coefficient field) and occurs to the right of all the leading non-zero
|
|
; entries of previous rows. In particular, the number of rows is the rank
|
|
; of the original matrix, and they have the same row-space.
|
|
; The items related to the base field which are needed are:
|
|
; coef-zero additive identity
|
|
; coef-one multiplicative identity
|
|
; coef-zero? test for additive identity
|
|
; coef-+ addition (two args)
|
|
; coef-negate additive inverse
|
|
; coef-* multiplication (two args)
|
|
; coef-recip multiplicative inverse
|
|
; Note, matricies are stored as lists of rows (i.e., lists of lists).
|
|
(define make-row-reduce
|
|
(lambda (coef-zero coef-one coef-zero? coef-+ coef-negate coef-* coef-recip)
|
|
(lambda (mat)
|
|
(let _-*-
|
|
((mat
|
|
mat))
|
|
(if (or (null? mat)
|
|
(null? (car mat)))
|
|
'()
|
|
(let _-**-
|
|
((in
|
|
mat)
|
|
(out
|
|
'()))
|
|
(if (null? in)
|
|
(map
|
|
(lambda (x)
|
|
(cons coef-zero x))
|
|
(_-*- out))
|
|
(let* ((prow
|
|
(car in))
|
|
(pivot
|
|
(car prow))
|
|
(prest
|
|
(cdr prow))
|
|
(in
|
|
(cdr in)))
|
|
(if (coef-zero? pivot)
|
|
(_-**- in
|
|
(cons prest out))
|
|
(let ((zap-row
|
|
(map
|
|
(let ((mult
|
|
(coef-recip pivot)))
|
|
(lambda (x)
|
|
(coef-* mult x)))
|
|
prest)))
|
|
(cons (cons coef-one zap-row)
|
|
(map
|
|
(lambda (x)
|
|
(cons coef-zero x))
|
|
(_-*-
|
|
(fold in
|
|
(lambda (row mat)
|
|
(cons
|
|
(let ((first-col
|
|
(car row))
|
|
(rest-row
|
|
(cdr row)))
|
|
(if (coef-zero? first-col)
|
|
rest-row
|
|
(map
|
|
(let ((mult
|
|
(coef-negate first-col)))
|
|
(lambda (f z)
|
|
(coef-+ f
|
|
(coef-* mult z))))
|
|
rest-row
|
|
zap-row)))
|
|
mat))
|
|
out))))))))))))))
|
|
|
|
|
|
; Given elements and operations on the base field, return a procedure which
|
|
; when given a matrix and a vector tests to see if the vector is in the
|
|
; row-space of the matrix. This returned function is curried.
|
|
; The items related to the base field which are needed are:
|
|
; coef-zero additive identity
|
|
; coef-one multiplicative identity
|
|
; coef-zero? test for additive identity
|
|
; coef-+ addition (two args)
|
|
; coef-negate additive inverse
|
|
; coef-* multiplication (two args)
|
|
; coef-recip multiplicative inverse
|
|
; Note, matricies are stored as lists of rows (i.e., lists of lists).
|
|
(define make-in-row-space?
|
|
(lambda (coef-zero coef-one coef-zero? coef-+ coef-negate coef-* coef-recip)
|
|
(let ((row-reduce
|
|
(make-row-reduce coef-zero
|
|
coef-one
|
|
coef-zero?
|
|
coef-+
|
|
coef-negate
|
|
coef-*
|
|
coef-recip)))
|
|
(lambda (mat)
|
|
(let ((mat
|
|
(row-reduce mat)))
|
|
(lambda (row)
|
|
(let _-*-
|
|
((row
|
|
row)
|
|
(mat
|
|
mat))
|
|
(if (null? row)
|
|
#t
|
|
(let ((r-first
|
|
(car row))
|
|
(r-rest
|
|
(cdr row)))
|
|
(cond ((coef-zero? r-first)
|
|
(_-*- r-rest
|
|
(map cdr
|
|
(if (or (null? mat)
|
|
(coef-zero? (caar mat)))
|
|
mat
|
|
(cdr mat)))))
|
|
((null? mat)
|
|
#f)
|
|
(else
|
|
(let* ((zap-row
|
|
(car mat))
|
|
(z-first
|
|
(car zap-row))
|
|
(z-rest
|
|
(cdr zap-row))
|
|
(mat
|
|
(cdr mat)))
|
|
(if (coef-zero? z-first)
|
|
#f
|
|
(_-*-
|
|
(map
|
|
(let ((mult
|
|
(coef-negate r-first)))
|
|
(lambda (r z)
|
|
(coef-+ r
|
|
(coef-* mult z))))
|
|
r-rest
|
|
z-rest)
|
|
(map cdr mat)))))))))))))))
|
|
|
|
|
|
; Given a prime number, return a procedure which takes integer matricies
|
|
; and returns their row-reduced form, modulo the prime.
|
|
(define make-modular-row-reduce
|
|
(lambda (modulus)
|
|
((make-modular modulus)
|
|
make-row-reduce)))
|
|
|
|
|
|
(define make-modular-in-row-space?
|
|
(lambda (modulus)
|
|
((make-modular modulus)
|
|
make-in-row-space?)))
|
|
|
|
|
|
|
|
; Usual utilities.
|
|
|
|
|
|
|
|
; Given a bound, find a prime greater than the bound.
|
|
(define find-prime
|
|
(lambda (bound)
|
|
(let* ((primes
|
|
(list 2))
|
|
(last
|
|
(chez-box primes))
|
|
(is-next-prime?
|
|
(lambda (trial)
|
|
(let _-*-
|
|
((primes
|
|
primes))
|
|
(or (null? primes)
|
|
(let ((p
|
|
(car primes)))
|
|
(or (< trial (* p p))
|
|
(and (not (zero? (mod trial p)))
|
|
(_-*- (cdr primes))))))))))
|
|
(if (> 2 bound)
|
|
2
|
|
(let _-*-
|
|
((trial
|
|
3))
|
|
(if (is-next-prime? trial)
|
|
(let ((entry
|
|
(list trial)))
|
|
(set-cdr! (chez-unbox last) entry)
|
|
(chez-set-box! last entry)
|
|
(if (> trial bound)
|
|
trial
|
|
(_-*- (+ trial 2))))
|
|
(_-*- (+ trial 2))))))))
|
|
|
|
; Given the size of a square matrix consisting only of +1's and -1's,
|
|
; return an upper bound on the determinant.
|
|
(define det-upper-bound
|
|
(lambda (size)
|
|
(let ((main-part
|
|
(expt size
|
|
(div size 2))))
|
|
(if (even? size)
|
|
main-part
|
|
(* main-part
|
|
(do ((i 0 (+ i 1)))
|
|
((>= (* i i) size)
|
|
i)))))))
|
|
|
|
; Fold over all maximal matrices.
|
|
(define go
|
|
(lambda (number-of-cols inv-size folder state)
|
|
(let* ((in-row-space?
|
|
(make-modular-in-row-space?
|
|
(find-prime
|
|
(det-upper-bound inv-size))))
|
|
(make-tester
|
|
(lambda (mat)
|
|
(let ((tests
|
|
(let ((old-mat
|
|
(cdr mat))
|
|
(new-row
|
|
(car mat)))
|
|
(fold-over-subs-of-size old-mat
|
|
(- inv-size 2)
|
|
(lambda (sub tests)
|
|
(cons
|
|
(in-row-space?
|
|
(cons new-row sub))
|
|
tests))
|
|
'()))))
|
|
(lambda (row)
|
|
(let _-*-
|
|
((tests
|
|
tests))
|
|
(and (not (null? tests))
|
|
(or ((car tests) row)
|
|
(_-*- (cdr tests)))))))))
|
|
(all-rows ; all rows starting with +1 in decreasing order
|
|
(fold
|
|
(fold-over-rows (- number-of-cols 1)
|
|
cons
|
|
'())
|
|
(lambda (row rows)
|
|
(cons (cons 1 row)
|
|
rows))
|
|
'())))
|
|
(let _-*-
|
|
((number-of-rows
|
|
1)
|
|
(rev-mat
|
|
(list
|
|
(car all-rows)))
|
|
(possible-future
|
|
(cdr all-rows))
|
|
(state
|
|
state))
|
|
(let ((zulu-future
|
|
(remove-in-order
|
|
(if (< number-of-rows inv-size)
|
|
(in-row-space? rev-mat)
|
|
(make-tester rev-mat))
|
|
possible-future)))
|
|
(if (null? zulu-future)
|
|
(folder (reverse rev-mat)
|
|
state)
|
|
(let _-**-
|
|
((zulu-future
|
|
zulu-future)
|
|
(state
|
|
state))
|
|
(if (null? zulu-future)
|
|
state
|
|
(let ((rest-of-future
|
|
(cdr zulu-future)))
|
|
(_-**- rest-of-future
|
|
(let* ((first
|
|
(car zulu-future))
|
|
(new-rev-mat
|
|
(cons first rev-mat)))
|
|
(if (maximal? (reverse new-rev-mat))
|
|
(_-*- (+ number-of-rows 1)
|
|
new-rev-mat
|
|
rest-of-future
|
|
state)
|
|
state))))))))))))
|
|
|
|
(define go-folder
|
|
(lambda (mat bsize.blen.blist)
|
|
(let ((bsize
|
|
(car bsize.blen.blist))
|
|
(size
|
|
(length mat)))
|
|
(if (< size bsize)
|
|
bsize.blen.blist
|
|
(let ((blen
|
|
(cadr bsize.blen.blist))
|
|
(blist
|
|
(cddr bsize.blen.blist)))
|
|
(if (= size bsize)
|
|
(let ((blen
|
|
(+ blen 1)))
|
|
; (if
|
|
; (let _-*-
|
|
; ((blen
|
|
; blen))
|
|
; (or (< blen 10)
|
|
; (and (zero? (mod blen 10))
|
|
; (_-*- (div blen 10)))))
|
|
;
|
|
; (begin
|
|
; (display blen)
|
|
; (display " of size ")
|
|
; (display bsize)
|
|
; (newline)))
|
|
|
|
(cons bsize
|
|
(cons blen
|
|
(cond ((< blen 3000)
|
|
(cons mat blist))
|
|
((= blen 3000)
|
|
(cons "..." blist))
|
|
(else
|
|
blist)))))
|
|
; (begin
|
|
; (newline)
|
|
; (display "First of size ")
|
|
; (display size)
|
|
; (display ":")
|
|
; (newline)
|
|
; (for-each
|
|
; (lambda (row)
|
|
; (display " ")
|
|
; (for-each
|
|
; (lambda (e)
|
|
; (case e
|
|
; ((1)
|
|
; (display " 1"))
|
|
; ((-1)
|
|
; (display " -1"))))
|
|
; row)
|
|
; (newline))
|
|
; mat)
|
|
|
|
(list size 1 mat)))))))
|
|
|
|
(define really-go
|
|
(lambda (number-of-cols inv-size)
|
|
(cddr
|
|
(go number-of-cols
|
|
inv-size
|
|
go-folder
|
|
(list -1 -1)))))
|
|
|
|
(define remove-in-order
|
|
(lambda (remove? lst)
|
|
(reverse
|
|
(fold lst
|
|
(lambda (e lst)
|
|
(if (remove? e)
|
|
lst
|
|
(cons e lst)))
|
|
'()))))
|
|
|
|
; The first fold-over-rows is slower than the second one, but folds
|
|
; over rows in lexical order (large to small).
|
|
(define fold-over-rows
|
|
(lambda (number-of-cols folder state)
|
|
(if (zero? number-of-cols)
|
|
(folder '()
|
|
state)
|
|
(fold-over-rows (- number-of-cols 1)
|
|
(lambda (tail state)
|
|
(folder (cons -1 tail)
|
|
state))
|
|
(fold-over-rows (- number-of-cols 1)
|
|
(lambda (tail state)
|
|
(folder (cons 1 tail)
|
|
state))
|
|
state)))))
|
|
|
|
; Fold over subsets of a given size.
|
|
(define fold-over-subs-of-size
|
|
(lambda (universe size folder state)
|
|
(let ((usize
|
|
(length universe)))
|
|
(if (< usize size)
|
|
state
|
|
(let _-*-
|
|
((size
|
|
size)
|
|
(universe
|
|
universe)
|
|
(folder
|
|
folder)
|
|
(csize
|
|
(- usize size))
|
|
(state
|
|
state))
|
|
(cond ((zero? csize)
|
|
(folder universe state))
|
|
((zero? size)
|
|
(folder '() state))
|
|
(else
|
|
(let ((first-u
|
|
(car universe))
|
|
(rest-u
|
|
(cdr universe)))
|
|
(_-*- size
|
|
rest-u
|
|
folder
|
|
(- csize 1)
|
|
(_-*- (- size 1)
|
|
rest-u
|
|
(lambda (tail state)
|
|
(folder (cons first-u tail)
|
|
state))
|
|
csize
|
|
state))))))))))
|
|
|
|
(define (main)
|
|
(let* ((count (read))
|
|
(input1 (read))
|
|
(input2 (read))
|
|
(output (read))
|
|
(s3 (number->string count))
|
|
(s2 (number->string input2))
|
|
(s1 (number->string input1))
|
|
(name "matrix"))
|
|
(run-r7rs-benchmark
|
|
(string-append name ":" s1 ":" s2 ":" s3)
|
|
count
|
|
(lambda () (really-go (hide count input1) (hide count input2)))
|
|
(lambda (result) (equal? result output)))))
|
|
|
|
(include "src/common.sch")
|