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415 lines
21 KiB
Racket
415 lines
21 KiB
Racket
#lang typed/racket/base
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(require (for-syntax racket/base racket/syntax))
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(require/typed sugar/cache [make-caching-proc (Procedure . -> . Procedure)])
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(require racket/list sugar/debug rackunit racket/function racket/vector "logger-typed.rkt")
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(define-logger ocm)
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#|
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Totally monotone matrix searching algorithms.
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The offline algorithm in ConcaveMinima is from Agarwal, Klawe, Moran,
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Shor, and Wilbur, Geometric applications of a matrix searching algorithm,
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Algorithmica 2, pp. 195-208 (1987).
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The online algorithm in OnlineConcaveMinima is from Galil and Park,
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A linear time algorithm for concave one-dimensional dynamic programming,
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manuscript, 1989, which simplifies earlier work on the same problem
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by Wilbur (J. Algorithms 1988) and Eppstein (J. Algorithms 1990).
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D. Eppstein, March 2002, significantly revised August 2005
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|#
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;(provide smawky? make-ocm reduce reduce2 (prefix-out ocm- (combine-out min-value min-index)))
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(: select-elements ((Listof Any) (Listof Index-Type) . -> . (Listof Any)))
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(define (select-elements xs is)
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(map (λ([i : Index-Type]) ((inst list-ref Any) xs i)) is))
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(: odd-elements ((Listof Any) . -> . (Listof Any)))
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(define (odd-elements xs)
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(select-elements xs (range 1 (length xs) 2)))
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(: vector-odd-elements ((Vectorof Any) . -> . (Vectorof Any)))
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(define (vector-odd-elements xs)
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(for/vector ([i (in-range (vector-length xs))] #:when (odd? i))
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(vector-ref xs i)))
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(: even-elements ((Listof Any) . -> . (Listof Any)))
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(define (even-elements xs)
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(select-elements xs (range 0 (length xs) 2)))
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;; Wrapper for the matrix procedure
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;; that automatically maintains a hash cache of previously-calculated values
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;; because the minima operations tend to hit the same values.
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;; Assuming here that (matrix i j) is invariant
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;; and that the matrix function is more expensive than the cache lookup.
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(define-syntax-rule (vector-append-item xs value)
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((inst vector-append Any) xs (vector value)))
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(: vector-set (All (a) ((Vectorof a) Integer a -> (Vectorof a))))
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(define (vector-set vec idx val)
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(vector-set! vec idx val)
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vec)
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(define-syntax-rule (vector-cdr vec)
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(vector-drop vec 1))
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(define-syntax-rule (vector-empty? vec)
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(= 0 (vector-length vec)))
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(define (integers? x) (and (list? x) (andmap integer? x)))
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;; Reduce phase: make number of rows at most equal to number of cols
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(: reduce ((Vectorof Index-Type) (Vectorof Index-Type) Matrix-Proc-Type Entry->Value-Type . -> . (Vectorof Index-Type)))
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(define (reduce row-indices col-indices matrix-proc entry->value)
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;(vector? vector? procedure? procedure? . -> . vector?)
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(log-ocm-debug "starting reduce phase with")
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(log-ocm-debug "row-indices = ~a" row-indices)
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(log-ocm-debug "col-indices = ~a" col-indices)
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(: process-stack ((Vectorof Index-Type) Index-Type . -> . (Vectorof Index-Type)))
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(define (process-stack stack row-idx)
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(log-ocm-debug "row stack = ~a" stack)
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(let ([last-stack-idx (sub1 (vector-length stack))])
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(cond
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[(and (>= (vector-length stack) 1)
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(log-ocm-debug "comparing row values at column ~a" (vector-ref col-indices last-stack-idx))
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(log-ocm-debug "end of row stack (~a) value at column ~a = ~a" (vector-ref stack last-stack-idx) (vector-ref col-indices last-stack-idx) (entry->value (matrix-proc (vector-ref stack last-stack-idx) (vector-ref col-indices last-stack-idx))))
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(log-ocm-debug "challenger row (~a) value at column ~a = ~a" row-idx (vector-ref col-indices last-stack-idx) (entry->value (matrix-proc row-idx (vector-ref col-indices last-stack-idx))))
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(> (entry->value (matrix-proc (vector-ref stack last-stack-idx) (vector-ref col-indices last-stack-idx)))
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(entry->value (matrix-proc row-idx (vector-ref col-indices last-stack-idx)))))
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(log-ocm-debug "challenger row (~a) wins with a new minimum ~a, so end of row stack (~a) is removed" row-idx (entry->value (matrix-proc row-idx (vector-ref col-indices last-stack-idx))) (vector-ref stack last-stack-idx))
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(process-stack (vector-drop-right stack 1) row-idx)]
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[else
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(log-ocm-debug (if (< (vector-length stack) 1)
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(format "row stack too short for challenge, pushing row ~a" row-idx)
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(format "challenger row (~a) loses to end of row stack (~a), so ~a joins stack" row-idx (vector-ref stack last-stack-idx) row-idx)))
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stack])))
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(define reduced-row-indexes
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(for/fold : (Vectorof Index-Type) ([stack (cast (vector) (Vectorof Index-Type))]) ([row-idx (in-vector row-indices)])
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(let ([stack (process-stack stack row-idx)])
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(if (= (vector-length stack) (vector-length col-indices))
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stack
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((inst vector-append Index-Type) stack (vector row-idx))))))
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(log-ocm-debug "finished reduce. row indexes = ~v" reduced-row-indexes)
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reduced-row-indexes)
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(: reduce2 ((Vectorof Index-Type) (Vectorof Index-Type) Matrix-Proc-Type Entry->Value-Type . -> . (Vectorof Index-Type)))
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(define (reduce2 row-indices col-indices matrix-proc entry->value)
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(let find-survivors ([rows row-indices][survivors : (Listof Index-Type) empty])
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(cond
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[(vector-empty? rows) ((inst list->vector Index-Type) (reverse survivors))]
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[else
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(define challenger-row (vector-ref rows 0))
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(cond
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;; no survivors yet, so push first row and keep going
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[(empty? survivors) (find-survivors (vector-cdr rows) (cons challenger-row survivors))]
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[else
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(define index-of-last-survivor (sub1 (length survivors)))
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(define col-head (vector-ref col-indices index-of-last-survivor))
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(define-syntax-rule (test-function r) (entry->value (matrix-proc r col-head)))
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(cond
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;; this is the challenge: is the head cell of challenger a new minimum?
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;; use < not <=, so the recorded winner is the earliest row with the new minimum, not the latest row
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;; if yes, challenger wins. pop element from stack, and let challenger try again (= leave rows alone)
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[(< (test-function challenger-row) (test-function (car survivors))) (find-survivors rows (cdr survivors))]
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;; if not, challenger lost.
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;; If we're in the last column, ignore the loser by recurring on the same values
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[(= col-head (vector-last col-indices)) (find-survivors (vector-cdr rows) survivors)]
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;; otherwise challenger lost and we're not in last column,
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;; so add challenger to survivor stack
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[else (find-survivors (vector-cdr rows) (cons challenger-row survivors))])])])))
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;; define a special type so it can be reused in `interpolate`
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;; it is (cons value row-idx)
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(define-type Make-Minimum-Input (Pair Any Index-Type))
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(: make-minimum (Make-Minimum-Input . -> . (HashTable Any Any)))
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(define (make-minimum value-rowidx-pair)
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(define ht ((inst make-hash Any Any)))
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(! ht 'value (car value-rowidx-pair))
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(! ht 'row-idx (cdr value-rowidx-pair))
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ht)
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;; Interpolate phase: in the minima hash, add results for even rows
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(define-syntax-rule (vector-last v)
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(vector-ref v (sub1 (vector-length v))))
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(: interpolate ((HashTable Any Any) (Vectorof Index-Type) (Vectorof Index-Type) Matrix-Proc-Type Entry->Value-Type . -> . (HashTable Any Any)))
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(define (interpolate minima row-indices col-indices matrix-proc entry->value)
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;(hash? vector? vector? procedure? procedure? . -> . hash?)
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(for ([col-idx (in-range 0 (vector-length col-indices) 2)]) ;; even-col-indices
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(define col (vector-ref col-indices col-idx))
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(define idx-of-last-row
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(cast (if (= col-idx (sub1 (vector-length col-indices)))
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(vector-last row-indices)
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(hash-ref (cast (hash-ref minima (vector-ref col-indices (add1 col-idx))) HashTableTop) 'row-idx)) Index-Type))
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(define smallest-value-entry
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((inst vector-argmin Make-Minimum-Input) (λ(x) (entry->value (car x)))
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(for/vector : (Vectorof Make-Minimum-Input)
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([row-idx (in-list ((inst dropf-right Index-Type) (vector->list row-indices) (λ(x) (not (= x idx-of-last-row)))))])
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(cons (matrix-proc row-idx col) row-idx))))
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(! minima col (make-minimum smallest-value-entry)))
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minima)
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(: interpolate2 ((HashTable Any Any) (Vectorof Index-Type) (Vectorof Index-Type) Matrix-Proc-Type Entry->Value-Type . -> . (HashTable Any Any)))
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(define (interpolate2 minima row-indices col-indices matrix-proc entry->value)
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(define idx-of-last-col (sub1 (vector-length col-indices)))
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(define (smallest-value-entry [col : Index-Type] [idx-of-last-row : Index-Type])
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((inst argmin Make-Minimum-Input) (λ(x) (entry->value (car x)))
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(for/list ([row-idx (stop-after (in-vector row-indices) (λ(x) (= idx-of-last-row x)))])
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(cons (matrix-proc row-idx col) row-idx))))
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(for ([([col : Index-Type] col-idx) (in-indexed col-indices)] #:when (even? col-idx))
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(define idx-of-last-row (cast (if (= col-idx idx-of-last-col)
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(vector-last row-indices)
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(hash-ref (cast (hash-ref minima (vector-ref col-indices (add1 col-idx))) HashTableTop) 'row-idx)) Index-Type))
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(! minima col (make-minimum (smallest-value-entry col idx-of-last-row))))
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minima)
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#|
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Search for the minimum value in each column of a matrix.
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The return value is a dictionary mapping ColIndices to pairs
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(value,rowindex). We break ties in favor of earlier rows.
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The matrix is defined implicitly as a function, passed
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as the third argument to this routine, where Matrix(i,j)
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gives the matrix value at row index i and column index j.
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The matrix must be concave, that is, satisfy the property
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Matrix(i,j) > Matrix(i',j) => Matrix(i,j') > Matrix(i',j')
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for every i<i' and j<j'; that is, in every submatrix of
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the input matrix, the positions of the column minima
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must be monotonically nondecreasing.
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The rows and columns of the matrix are labeled by the indices
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given in order by the first two arguments. In most applications,
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these arguments can simply be integer ranges.
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|#
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;; The return value `minima` is a hash:
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;; the keys are col-indices (integers)
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;; the values are pairs of (value row-index).
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(: concave-minima (((Vectorof Index-Type)) ((Vectorof Index-Type) Matrix-Proc-Type Entry->Value-Type) . ->* . HashTableTop))
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(define (concave-minima row-indices [col-indices (vector)] [matrix-proc (cast (make-caching-proc identity) Matrix-Proc-Type)] [entry->value (cast identity Entry->Value-Type)])
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;((vector?) ((or/c #f vector?) procedure? procedure?) . ->* . hash?)
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(define reduce-proc reduce2)
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(define interpolate-proc interpolate2)
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(if (= 0 (vector-length col-indices))
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(make-hash)
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(let ([row-indices (reduce-proc row-indices col-indices matrix-proc entry->value)])
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(define odd-column-minima (concave-minima row-indices (cast (vector-odd-elements (cast col-indices (Vectorof Any))) (Vectorof Index-Type)) matrix-proc entry->value))
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(interpolate-proc (cast odd-column-minima (HashTable Any Any)) row-indices col-indices matrix-proc entry->value))))
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#|
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Online concave minimization algorithm of Galil and Park.
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OnlineConcaveMinima(Matrix,initial) creates a sequence of pairs
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(self.value(j),self.index(j)), where
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self.value(0) = initial,
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self.value(j) = min { Matrix(i,j) | i < j } for j > 0,
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and where self.index(j) is the value of j that provides the minimum.
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Matrix(i,j) must be concave, in the same sense as for ConcaveMinima.
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We never call Matrix(i,j) until value(i) has already been computed,
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so that the Matrix function may examine previously computed values.
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Calling value(i) for an i that has not yet been computed forces
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the sequence to be continued until the desired index is reached.
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Calling iter(self) produces a sequence of (value,index) pairs.
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Matrix(i,j) should always return a value, rather than raising an
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exception, even for j larger than the range we expect to compute.
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If j is out of range, a suitable value to return that will not
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violate concavity is Matrix(i,j) = -i. It will not work correctly
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to return a flag value such as None for large j, because the ties
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formed by the equalities among such flags may violate concavity.
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|#
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;; Online Concave Minima object
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;(struct $ocm (values indices finished matrix-proc base tentative) #:transparent #:mutable)
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;; State used by self.value(), self.index(), and iter(self) =
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;; $ocm-values, $ocm-indices, $ocm-finished
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#|
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State used by the internal algorithm:
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$ocm-matrix, $ocm-base, $ocm-tentative
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We allow self._values to be nonempty for indices > finished,
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keeping invariant that
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(1) self._values[i] = Matrix(self._indices[i], i),
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(2) if the eventual correct value of self.index(i) < base,
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then self._values[i] is nonempty and correct.
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In addition, we keep a column index self._tentative, such that
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(3) if i <= tentative, and the eventual correct value of
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self.index(i) <= finished, then self._values[i] is correct.
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|#
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(define no-value 'none)
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(define-syntax-rule (@ hashtable key)
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(hash-ref hashtable key))
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(define-syntax-rule (! hashtable key value)
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(hash-set! hashtable key value))
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(: ocm-ref (OCM-Type Index-Type . -> . Any))
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(define (ocm-ref ocm key)
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(vector-ref ocm key))
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(: ocm-set! (OCM-Type Index-Type Any . -> . Void))
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(define (ocm-set! ocm key value)
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(vector-set! ocm key value))
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(define-type OCM-Type (Vector Any Any Any Any Any Any Any))
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(define-type Index-Type Nonnegative-Integer)
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(define-type Matrix-Proc-Type (Index-Type Index-Type . -> . Any))
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(define-type Entry->Value-Type (Any . -> . Flonum))
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(define o:min-values 0)
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(define o:min-row-indices 1)
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(define o:finished 2)
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(define o:matrix-proc 3)
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(define o:entry->value 4)
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(define o:base 5)
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(define o:tentative 6)
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(: make-ocm ((Procedure) (Any Procedure) . ->* . OCM-Type))
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(define (make-ocm matrix-proc [initial-value 0][entry->value identity])
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(log-ocm-debug "making new ocm")
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(define ocm (cast (make-vector 7) OCM-Type))
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(ocm-set! ocm o:min-values (vector initial-value))
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(ocm-set! ocm o:min-row-indices (vector no-value))
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(ocm-set! ocm o:finished 0)
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(ocm-set! ocm o:matrix-proc (make-caching-proc matrix-proc))
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(ocm-set! ocm o:entry->value entry->value) ; for converting matrix values to an integer
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(ocm-set! ocm o:base 0)
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(ocm-set! ocm o:tentative 0)
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(cast ocm OCM-Type))
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;; Return min { Matrix(i,j) | i < j }.
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(: min-value (OCM-Type Index-Type . -> . Any))
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(define (min-value ocm j)
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(if (< (cast (ocm-ref ocm o:finished) Real) j)
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(begin (advance! ocm) (min-value ocm j))
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(vector-ref (cast (ocm-ref ocm o:min-values) VectorTop) j)))
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;; Return argmin { Matrix(i,j) | i < j }.
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(: min-index (OCM-Type Index-Type . -> . Index-Type))
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(define (min-index ocm j)
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(if (< (cast (ocm-ref ocm o:finished) Real) j)
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(begin (advance! ocm) (min-index ocm j))
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((inst vector-ref Index-Type) (cast (ocm-ref ocm o:min-row-indices) (Vectorof Index-Type)) j)))
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;; Finish another value,index pair.
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(: advance! (OCM-Type . -> . Void))
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(define (advance! ocm)
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(define next (add1 (cast (ocm-ref ocm o:finished) Index-Type)))
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(log-ocm-debug "advance! ocm to next = ~a" (add1 (cast (ocm-ref ocm o:finished) Number)))
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(cond
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;; First case: we have already advanced past the previous tentative
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;; value. We make a new tentative value by applying ConcaveMinima
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;; to the largest square submatrix that fits under the base.
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[(> next (cast (ocm-ref ocm o:tentative) Real))
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(log-ocm-debug "advance: first case because next (~a) > tentative (~a)" next (ocm-ref ocm o:tentative))
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(define rows : (Vectorof Index-Type) (list->vector (range (cast (ocm-ref ocm o:base) Index-Type) next)))
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(ocm-set! ocm o:tentative (+ (cast (ocm-ref ocm o:finished) Number) (vector-length rows)))
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(define cols : (Vectorof Index-Type) (list->vector (range next (add1 (cast (ocm-ref ocm o:tentative) Index-Type)))))
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(define minima (concave-minima rows cols (cast (ocm-ref ocm o:matrix-proc) Matrix-Proc-Type) (cast (ocm-ref ocm o:entry->value) Entry->Value-Type)))
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(for ([col (in-vector cols)])
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(cond
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[(>= col (vector-length (cast (ocm-ref ocm o:min-values) VectorTop)))
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(ocm-set! ocm o:min-values (vector-append-item (cast (ocm-ref ocm o:min-values) (Vectorof Any)) (@ (cast (@ minima col) HashTableTop) 'value)))
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(ocm-set! ocm o:min-row-indices (vector-append-item (cast (ocm-ref ocm o:min-row-indices) (Vectorof Any)) (@ (cast (@ minima col) HashTableTop) 'row-idx)))]
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[(< ((cast (ocm-ref ocm o:entry->value) Entry->Value-Type) (@ (cast (@ minima col) HashTableTop) 'value)) ((cast (ocm-ref ocm o:entry->value) Entry->Value-Type) (vector-ref (cast (ocm-ref ocm o:min-values) VectorTop) col)))
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(ocm-set! ocm o:min-values ((inst vector-set Index-Type) (cast (ocm-ref ocm o:min-values) (Vectorof Index-Type)) col (cast (@ (cast (@ minima col) HashTableTop) 'value) Index-Type)))
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(ocm-set! ocm o:min-row-indices ((inst vector-set Index-Type) (cast (ocm-ref ocm o:min-row-indices) (Vectorof Index-Type)) col (cast (@ (cast (@ minima col) HashTableTop) 'row-idx) Index-Type)))]))
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(ocm-set! ocm o:finished next)]
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[else
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;; Second case: the new column minimum is on the diagonal.
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;; All subsequent ones will be at least as low,
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;; so we can clear out all our work from higher rows.
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;; As in the fourth case, the loss of tentative is
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;; amortized against the increase in base.
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(define diag ((cast (ocm-ref ocm o:matrix-proc) Matrix-Proc-Type) (sub1 next) next))
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(cond
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[(< ((cast (ocm-ref ocm o:entry->value) Entry->Value-Type) diag) ((cast (ocm-ref ocm o:entry->value) Entry->Value-Type) (vector-ref (cast (ocm-ref ocm o:min-values) (Vectorof Any)) next)))
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(log-ocm-debug "advance: second case because column minimum is on the diagonal")
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(ocm-set! ocm o:min-values (vector-set (cast (ocm-ref ocm o:min-values) (Vectorof Any)) next diag))
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(ocm-set! ocm o:min-row-indices (vector-set (cast (ocm-ref ocm o:min-row-indices) (Vectorof Any)) next (sub1 next)))
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(ocm-set! ocm o:base (sub1 next))
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(ocm-set! ocm o:tentative next)
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(ocm-set! ocm o:finished next)]
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;; Third case: row i-1 does not supply a column minimum in
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;; any column up to tentative. We simply advance finished
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;; while maintaining the invariant.
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[(>= ((cast (ocm-ref ocm o:entry->value) Entry->Value-Type) ((cast (ocm-ref ocm o:matrix-proc) Matrix-Proc-Type) (sub1 next) (cast (ocm-ref ocm o:tentative) Index-Type)))
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((cast (ocm-ref ocm o:entry->value) Entry->Value-Type) (vector-ref (cast (ocm-ref ocm o:min-values) (Vectorof Any)) (cast (ocm-ref ocm o:tentative) Index-Type))))
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(log-ocm-debug "advance: third case because row i-1 does not suppply a column minimum")
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(ocm-set! ocm o:finished next)]
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;; Fourth and final case: a new column minimum at self._tentative.
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;; This allows us to make progress by incorporating rows
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;; prior to finished into the base. The base invariant holds
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;; because these rows cannot supply any later column minima.
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;; The work done when we last advanced tentative (and undone by
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;; this step) can be amortized against the increase in base.
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[else
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(log-ocm-debug "advance: fourth case because new column minimum")
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(ocm-set! ocm o:base (sub1 next))
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(ocm-set! ocm o:tentative next)
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(ocm-set! ocm o:finished next)])]))
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(: print (OCM-Type . -> . Void))
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(define (print ocm)
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(displayln (ocm-ref ocm o:min-values))
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(displayln (ocm-ref ocm o:min-row-indices)))
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(: smawky? ((Listof (Listof Real)) . -> . Boolean))
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(define (smawky? m)
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(: position-of-minimum ((Listof Real) . -> . Index-Type))
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(define (position-of-minimum xs)
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;; put each element together with its list index
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(let ([xs : (Listof (Pairof Index-Type Real)) (map (inst cons Index-Type Real) (range (length xs)) xs)])
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;; find the first one with the min value, and grab the list index
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(car ((inst argmin (Pairof Index-Type Real)) cdr (filter (λ([x : (Pairof Index-Type Real)]) (not (negative? (cdr x)))) xs)))))
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;; tests if penalty matrix is monotone for non-negative values.
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(define increasing-minima? (apply <= (cast (map position-of-minimum m) (List* Real Real (Listof Real)))))
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|
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(define monotone? : Boolean
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|
(for/and ([ridx (in-range 1 (length m))])
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(for/and : Boolean ([cidx (in-range (sub1 (length (car m))))])
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(cast (let* ([prev-row : (Listof Real) ((inst list-ref (Listof Real)) m (sub1 ridx))]
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[row : (Listof Real) (list-ref m ridx)]
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[a : Real (list-ref prev-row cidx)]
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[b : Real (list-ref prev-row (add1 cidx))]
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|
[c : Real (list-ref row cidx)]
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|
[d : Real (list-ref row (add1 cidx))])
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|
(if (andmap (λ([x : Real]) (not (negative? x))) (list a b c d)) ;; smawk disregards negative values
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|
(cond
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[(< c d) (if (< a b) #t (error (format "Submatrix ~a not monotone in ~a" (list (list a b) (list c d)) m)))]
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[(= c d) (if (<= a b) #t (error (format "Submatrix ~a not monotone in ~a" (list (list a b) (list c d)) m)))]
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[else #t])
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|
#t)) Boolean))))
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|
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(and increasing-minima? monotone?))
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