;;; The SRFI-32 sort package -- three-way quick sort -*- Scheme -*-
;;; Copyright (c) 2002 by Olin Shivers.
;;; This code is open-source; see the end of the file for porting and
;;; more copyright information.
;;; Olin Shivers 2002/7.
;;; (quick-sort3! c v [start end]) -> unspecific
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Sort vector V[start,end) using three-way comparison function C:
;;; (c x y) < 0 => x<y
;;; (c x y) = 0 => x=y
;;; (c x y) > 0 => x>y
;;; That is, C acts as a sort of "subtraction" procedure; using - for the
;;; comparison function will cause numbers to be sorted into increasing order.
;;;
;;; This algorithm is more efficient than standard, two-way quicksort if there
;;; are many duplicate items in the data set and the comparison function is
;;; relatively expensive (e.g., comparing large strings). It is due to Jon
;;; Bentley & Doug McIlroy; I learned it from Bentley.
;;;
;;; The algorithm is a standard quicksort, but the partition loop is fancier,
;;; arranging the vector into a left part that is <, a middle region that is
;;; =, and a right part that is > the pivot. Here's how it is done:
;;; The partition loop divides the range being partitioned into five
;;; subranges:
;;; =======<<<<<<<<<?????????>>>>>>>=======
;;; where = marks a value that is equal the pivot, < marks a value that
;;; is less than the pivot, ? marks a value that hasn't been scanned, and
;;; > marks a value that is greater than the pivot. Let's consider the
;;; left-to-right scan. If it checks a ? value that is <, it keeps scanning.
;;; If the ? value is >, we stop the scan -- we are ready to start the
;;; right-to-left scan and then do a swap. But if the rightward scan checks
;;; a ? value that is =, we swap it *down* to the end of the initial chunk
;;; of ====='s -- we exchange it with the leftmost < value -- and then
;;; continue our rightward scan. The leftwards scan works in a similar
;;; fashion, scanning past > elements, stopping on a < element, and swapping
;;; up = elements. When we are done, we have a picture like this
;;; ========<<<<<<<<<<<<>>>>>>>>>>=========
;;; Then swap the = elements up into the middle of the vector to get
;;; this:
;;; <<<<<<<<<<<<=================>>>>>>>>>>
;;; Then recurse on the <'s and >'s. Work out all the tricky little
;;; boundary cases, and you're done.
;;;
;;; Other tricks that make this implementation industrial strength:
;;; - This quicksort makes some effort to pick the pivot well -- it uses the
;;; median of three elements as the partition pivot, so pathological n^2
;;; run time is much rarer (but not eliminated completely). If you really
;;; wanted to get fancy, you could use a random number generator to choose
;;; pivots. The key to this trick is that you only need to pick one random
;;; number for each *level* of recursion -- i.e. you only need (lg n) random
;;; numbers.
;;;
;;; - After the partition, we *recurse* on the smaller of the two pending
;;; regions, then *tail-recurse* (iterate) on the larger one. This guarantees
;;; we use no more than lg(n) stack frames, worst case.
;;;
;;; - There are two ways to finish off the sort.
;;; A. Recurse down to regions of size 10, then sort each such region using
;;; insertion sort.
;;; B. Recurse down to regions of size 10, then sort *the entire vector*
;;; using insertion sort.
;;; We do A. Each choice has a cost. Choice A has more overhead to invoke
;;; all the separate insertion sorts -- choice B only calls insertion sort
;;; once. But choice B will call the comparison function *more times* --
;;; it will unnecessarily compare elt 9 of one segment to elt 0 of the
;;; following segment. The overhead of choice A is linear in the length
;;; of the vector, but *otherwise independent of the algorithm's parameters*.
;;; I.e., it's a *fixed*, *small* constant factor. The cost of the extra
;;; comparisons made by choice B, however, is dependent on an externality:
;;; the comparison function passed in by the client. This can be made
;;; arbitrarily bad -- that is, the constant factor *isn't* fixed by the
;;; sort algorithm; instead, it's determined by the comparison function.
;;; If your comparison function is very, very slow, you want to eliminate
;;; every single one that you can. Choice A limits the potential badness,
;;; so that is what we do.
(define (vector-quick-sort3! c v . maybe-start+end)
(call-with-values
(lambda () (vector-start+end v maybe-start+end))
(lambda (start end)
(%quick-sort3! c v start end))))
(define (vector-quick-sort3 c v . maybe-start+end)
(call-with-values
(lambda () (vector-start+end v maybe-start+end))
(lambda (start end)
(let ((ans (make-vector (- end start))))
(vector-portion-copy! ans v start end)
(%quick-sort3! c ans 0 (- end start))
ans))))
;;; %QUICK-SORT3! is not exported.
;;; Preconditions:
;;; V vector
;;; START END fixnums
;;; 0 <= START, END <= (vector-length V)
;;; If these preconditions are ensured by the cover functions, you
;;; can safely change this code to use unsafe fixnum arithmetic and vector
;;; indexing ops, for *huge* speedup.
;;;
;;; We bail out to insertion sort for small ranges; feel free to tune the
;;; crossover -- it's just a random guess. If you don't have the insertion
;;; sort routine, just kill that branch of the IF and change the recursion
;;; test to (< 1 (- r l)) -- the code is set up to work that way.
(define (%quick-sort3! c v start end)
(define (swap l r n) ; Little utility -- swap the N
(if (> n 0)
(let ((x (vector-ref v l)) ; outer pairs of the range [l,r).
(r-1 (- r 1)))
(vector-set! v l (vector-ref v r-1))
(vector-set! v r-1 x)
(swap (+ l 1) r-1 (- n 1)))))
(define (sort3 v1 v2 v3)
(call-with-values
(lambda () (if (< (c v1 v2) 0) (values v1 v2) (values v2 v1)))
(lambda (little big)
(if (< (c big v3) 0)
(values little big v3)
(if (< (c little v3) 0)
(values little v3 big)
(values v3 little big))))))
(define (elt< v1 v2)
(negative? (c v1 v2)))
(let recur ((l start) (r end)) ; Sort the range [l,r).
(if (< 10 (- r l)) ; 10: the gospel according to Sedgewick.
;; Choose the median of V[l], V[r-1], and V[middle] for the pivot.
;; We do this by sorting these three elts; call the results LO, PIVOT
;; & HI. Put LO, PIVOT & HI where they should go in the vector. We
;; will kick off the partition step with one elt (PIVOT) in the left=
;; range, one elt (LO) in the < range, one elt (HI) in in the > range
;; & no elts in the right= range.
(let* ((r-1 (- r 1)) ; Three handy
(mid (quotient (+ l r) 2)) ; common
(l+1 (+ l 1)) ; subexpressions
(pivot (call-with-values
(lambda ()
(sort3 (vector-ref v l)
(vector-ref v mid)
(vector-ref v r-1)))
(lambda (lo piv hi)
(let ((tmp (vector-ref v l+1))) ; Put LO, PIV & HI
(vector-set! v l piv) ; back into V
(vector-set! v r-1 hi) ; where they belong,
(vector-set! v l+1 lo)
(vector-set! v mid tmp)
piv))))) ; and return PIV as pivot.
;; Everything in these loops is driven by the invariants expressed
;; in the little pictures, the corresponding l,i,j,k,m,r indices,
;; & the associated ranges.
;; =======<<<<<<<<<?????????>>>>>>>======= (picture)
;; l i j k m r (indices)
;; [l,i) [i,j) [j,k] (k,m] (m,r) (ranges )
(letrec ((lscan (lambda (i j k m) ; left-to-right scan
(let lp ((i i) (j j))
(if (> j k)
(done i j m)
(let* ((x (vector-ref v j))
(sign (c x pivot)))
(cond ((< sign 0) (lp i (+ j 1)))
((= sign 0)
(if (< i j)
(begin (vector-set! v j (vector-ref v i))
(vector-set! v i x)))
(lp (+ i 1) (+ j 1)))
((> sign 0) (rscan i j k m))))))))
;; =======<<<<<<<<<>????????>>>>>>>=======
;; l i j k m r
;; [l,i) [i,j) j (j,k] (k,m] (m,r)
(rscan (lambda (i j k m) ; right-to-left scan
(let lp ((k k) (m m))
(if (<= k j)
(done i j m)
(let* ((x (vector-ref v k))
(sign (c x pivot)))
(cond ((> sign 0) (lp (- k 1) m))
((= sign 0)
(if (< k m)
(begin (vector-set! v k (vector-ref v m))
(vector-set! v m x)))
(lp (- k 1) (- m 1)))
((< sign 0) ; Swap j & k & lscan.
(vector-set! v k (vector-ref v j))
(vector-set! v j x)
(lscan i (+ j 1) (- k 1) m))))))))
;; =======<<<<<<<<<<<<<>>>>>>>>>>>=======
;; l i j m r
;; [l,i) [i,j) [j,m] (m,r)
(done (lambda (i j m)
(let ((num< (- j i))
(num> (+ 1 (- m j)))
(num=l (- i l))
(num=r (- (- r m) 1)))
(swap l j (min num< num=l)) ; Swap ='s into
(swap j r (min num> num=r)) ; the middle.
;; Recur on the <'s and >'s. Recurring on the
;; smaller range and iterating on the bigger
;; range ensures O(lg n) stack frames, worst case.
(cond ((<= num< num>)
(recur l (+ l num<))
(recur (- r num>) r))
(else
(recur (- r num>) r)
(recur l (+ l num<))))))))
;; To repeat: We kick off the partition step with one elt (PIVOT)
;; in the left= range, one elt (LO) in the < range, one elt (HI)
;; in the > range & no elts in the right= range.
(lscan l+1 (+ l 2) (- r 2) r-1)))
;; Small segment => punt to insert sort.
;; Use the dangerous subprimitive.
(%vector-insert-sort! elt< v l r))))
;;; Copyright
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; This code is
;;; Copyright (c) 1998 by Olin Shivers.
;;; The terms are: You may do as you please with this code, as long as
;;; you do not delete this notice or hold me responsible for any outcome
;;; related to its use.
;;;
;;; Blah blah blah.
;;; Code tuning & porting
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; - The quicksort recursion bottoms out in a call to an insertion sort
;;; routine, %INSERT-SORT!. But you could even punt this and go with pure
;;; recursion in a pinch.
;;;
;;; This code is *tightly* bummed as far as I can go in portable Scheme.
;;;
;;; The internal primitive %QUICK-SORT! that does the real work can be
;;; converted to use unsafe vector-indexing and fixnum-specific arithmetic ops
;;; *if* you alter the two small cover functions to enforce the invariants.
;;; This should provide *big* speedups. In fact, all the code bumming I've
;;; done pretty much disappears in the noise unless you have a good compiler
;;; and also can dump the vector-index checks and generic arithmetic -- so
;;; I've really just set things up for you to exploit.
;;;
;;; The optional-arg parsing, defaulting, and error checking is done with a
;;; portable R4RS macro. But if your Scheme has a faster mechanism (e.g.,
;;; Chez), you should definitely port over to it. Note that argument defaulting
;;; and error-checking are interleaved -- you don't have to error-check
;;; defaulted START/END args to see if they are fixnums that are legal vector
;;; indices for the corresponding vector, etc.