Branch
Hash :
99ff1c7c
Author :
Thomas de Grivel
Date :
2024-05-08T12:58:39
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;; cl-facts
;; Copyright 2011, 2023 Thomas de Grivel <thodg@kmx.io>
;;
;; Permission is hereby granted to use this software granted
;; the above copyright notice and this permission paragraph
;; are included in all copies and substantial portions of this
;; software.
;;
;; THIS SOFTWARE IS PROVIDED "AS-IS" WITHOUT ANY GUARANTEE OF
;; PURPOSE AND PERFORMANCE. IN NO EVENT WHATSOEVER SHALL THE
;; AUTHOR BE CONSIDERED LIABLE FOR THE USE AND PERFORMANCE OF
;; THIS SOFTWARE.
(in-package :facts)
;;
;; Unlabeled Skip Lists
;; --------------------
#+nil
(pushnew :test *features*)
(declaim (optimize (debug 3) (speed 0)))
;; Types
(deftype positive-fixnum (&optional (low 0))
`(integer ,low ,most-positive-fixnum))
(deftype fixnum* (&optional (start '*) (end '*))
(let ((start (if (eq '* start) most-negative-fixnum start))
(end (if (eq '* end) most-positive-fixnum end)))
`(integer ,start ,end)))
(deftype fixnum-mult (mult &optional (start '*))
(declare (type (fixnum* 1) mult))
(let ((start (if (eq '* start)
(truncate most-negative-fixnum mult)
start)))
`(integer ,start
,(truncate most-positive-fixnum mult))))
(deftype fixnum-add (&optional (add '*) (start '*))
(let ((end (if (eq '* add)
(truncate most-positive-fixnum 2)
(- most-positive-fixnum (the positive-fixnum add)))))
`(fixnum* ,start ,end)))
(deftype fixnum-float (&optional (start '*) (end '*))
(let ((start (if (eq '* start) most-negative-fixnum start))
(end (if (eq '* end) most-positive-fixnum end)))
(declare (optimize (speed 1)))
`(long-float ,(coerce start 'long-float)
,(coerce end 'long-float))))
;; Unlabeled skip list node
(defstruct (usl-node (:constructor make-usl-node%))
(value nil)
(links nil :type simple-vector))
(defun make-usl-node (value height)
(declare (type (fixnum* 1) height))
(make-usl-node% :value value
:links (make-array height
:element-type '(or usl-node null)
:initial-element nil)))
(defun usl-node-next (node &optional (level 0))
(declare (type usl-node node)
(type (fixnum* 0) level))
(svref (usl-node-links node) level))
(define-setf-expander usl-node-next (node level &environment env)
(get-setf-expansion `(svref (usl-node-links (the usl-node ,node))
(the (fixnum* 0) ,level))
env))
(defun usl-node-height (node)
(declare (type usl-node node))
(the (fixnum* 1) (length (usl-node-links node))))
#+test
(defun test-usl-node ()
(let ((a (make-usl-node nil 2))
(b (make-usl-node #(1 2 3) 2))
(c (make-usl-node #(1 2 9) 1)))
(setf (usl-node-next a 0) b)
(setf (usl-node-next a 1) b)
(setf (usl-node-next b 0) c)
(setf (usl-node-next b 1) c)
a))
#+test
(test-usl-node)
;; Testing
#+test
(defun test-height-repartition (fun-fun max-height spacing)
(declare (type (fixnum* 1) max-height)
(type (function (fixnum fixnum) function) fun-fun))
(let ((height (make-array max-height
:element-type 'fixnum
:initial-element 0))
(rounds #1=1000000)
(fun (funcall fun-fun max-height spacing)))
(declare (type positive-fixnum rounds))
(format t "~&~S : ~D rounds, max-height ~D, spacing ~D~%"
fun-fun rounds max-height spacing)
(force-output)
(locally (declare (optimize speed))
(time
(dotimes (i rounds)
(incf (the (integer 0 #1#)
(aref height (1- (the positive-fixnum
(funcall fun)))))))))
(dotimes (i max-height)
(format t "~&height ~3D | p ~9,6F | 1/~D~%"
(1+ i)
(/ (aref height i) rounds)
(round rounds (1+ (aref height i)))))))
#+test
(defun cl-skip-list-random-level-fun (max-level spacing)
"Returns a random level for a new skip-list node, following Pugh's pattern of
L1: 50%, L2: 25%, L3: 12.5%, ..."
(declare (type fixnum spacing max-level)
(optimize speed))
(assert (= 2 spacing))
(lambda ()
(do ((level 1 (the fixnum (1+ level))))
((or (= level max-level)
(= (random 4) 3)) ;;
level)
(declare (type fixnum level)))))
#+test
(test-height-repartition #'cl-skip-list-random-level-fun 8 2)
;;
;; Random height
;; -------------
;;
;; ∀ U ∈ ℕ : 1 < U
;; ∀ n ∈ ℕ*
;; ∀ r ∈ ℕ : r ≤ n
;; ∀ random : ℕ* ⟶ ℕ
;; random(n) ∈ [0..n-1]
;; ∀ i ∈ [0..n-1], P(random(n) = i) = n⁻¹ (i)
;; Qᵣ := random(Uⁿ) < Uⁿ⁻ʳ
;;
;; (i) ⇒ P(random(n) < v) = ∑ᵢ₌₀ᵛ⁻¹ P(random(n) = i)
;; ⇒ P(random(n) < v) = v . n⁻¹ (ii)
;;
;; ⇒ P(random(Uⁿ) < Uⁿ⁻ʳ) = Uⁿ⁻ʳ . (Uⁿ)⁻¹
;; ⇔ P(Qᵣ) = U⁻ʳ (iii)
;;
;; P(Qₙ) = P(random(Uⁿ) < U⁰)
;; = P(random(Uⁿ) < 1)
;; = P(random(Uⁿ) = 0)
;; = U⁻ⁿ
;;
;; R := maxᵣ(Qᵣ)
;; = maxᵣ(random(Uⁿ) < Uⁿ⁻ʳ)
;; = maxᵣ(random(Uⁿ) + 1 ≤ Uⁿ⁻ʳ)
;; = maxᵣ(logᵤ(random(Uⁿ) + 1) ≤ n - r)
;; = maxᵣ(⌈logᵤ(random(Uⁿ) + 1)⌉ ≤ n - r)
;; = maxᵣ(r ≤ n - ⌈logᵤ(random(Uⁿ) + 1)⌉)
;; = n - ⌈logᵤ(random(Uⁿ) + 1)⌉ (iv)
;;
;; 0 ≤ random(Uⁿ) < Uⁿ
;; ⇔ 1 ≤ random(Uⁿ)+1 ≤ Uⁿ
;; ⇔ logᵤ(1) ≤ logᵤ(random(Uⁿ)+1) ≤ logᵤ(Uⁿ)
;; ⇔ 0 ≤ ⌈logᵤ(random(Uⁿ)+1)⌉ ≤ n
;; ⇔ -n ≤ -⌈logᵤ(random(Uⁿ)+1)⌉ ≤ 0
;; ⇔ 0 ≤ n - ⌈logᵤ(random(Uⁿ)+1)⌉ ≤ n
;; ⇔ 0 ≤ R ≤ n (v)
;;
#+test
(defun usl-random-height-fun* (max-height spacing)
(let* ((u (coerce spacing '(single-float 0.0s0)))
(n (the (fixnum-add 1 1) max-height))
(max (the (fixnum-add 1 1) (ceiling (expt u n)))))
(lambda ()
(declare (optimize (speed 3)))
(the positive-fixnum
(1+ (mod (- n (the fixnum
(ceiling
(log (1+ (random max))
u))))
n))))))
(defconstant +e+ 2.718281828459045235360287471352662497757l0)
#+test
(test-height-repartition #'usl-random-height-fun* 8 2)
#+test
(test-height-repartition #'usl-random-height-fun* 8 +e+)
(defun usl-height-table (max-height spacing)
(declare (type (fixnum* 2) max-height)
(type (fixnum-float 1 16) spacing))
(let ((table (make-array max-height
:element-type '(fixnum* 1)
:initial-element 1)))
(do ((h 0 (1+ h))
(w spacing (* spacing (the fixnum-float w)))
(end 1.0l0 (+ end w))
(pos 0 end))
((= max-height h))
(setf (aref table h) (truncate (the fixnum-float end))))
table))
(defun usl-random-height-fun (max-height spacing)
(declare (type positive-fixnum max-height)
(optimize speed))
(let* ((table (the (simple-array (fixnum* 1) (*))
(usl-height-table max-height spacing)))
(max (the positive-fixnum (1- (aref table (1- (length table)))))))
(lambda ()
(let ((k (random max)))
(do ((i 0 (1+ (the fixnum-add i))))
((< k (aref table i)) (- max-height i)))))))
#+test(test-height-repartition #'usl-random-height-fun 8 2)
#+test(test-height-repartition #'usl-random-height-fun 8 3)
#+test(test-height-repartition #'usl-random-height-fun 8 +e+)
(defun usl-random-height (max-height spacing)
(declare (type (fixnum* 1) max-height)
(type (fixnum-float 1) spacing))
(let ((max-height* (load-time-value #1=2))
(spacing* (load-time-value #2=+e+))
(fun* (load-time-value (usl-random-height-fun #1# #2#))))
(declare (type (function () (values (fixnum* 1) &optional)) fun*))
(unless (and (= max-height* max-height)
(= spacing* spacing))
(setf max-height* max-height
spacing* spacing
fun* (usl-random-height-fun max-height spacing)))
(funcall fun*)))
#+test
(time (test-height-repartition #'usl-random-height 8 2))
#+test
(time (test-height-repartition #'usl-random-height 8 3))
;; Skip list
(defstruct (usl (:constructor make-usl%))
(compare-fun nil :type (function (t t) fixnum))
(head nil :type usl-node)
(length 0 :type (fixnum* 0))
(spacing nil :type (fixnum-float 1))
(height-fun nil :type (function () (fixnum* 1))))
(defun make-usl (&key (compare #'compare:compare) (height 3) (spacing +e+))
(make-usl% :compare-fun compare
:spacing spacing
:head (make-usl-node nil height)
:height-fun (usl-random-height-fun height spacing)))
(defun usl-height (usl)
(usl-node-height (usl-head usl)))
(defun usl-compare (usl a b)
(declare (type usl usl))
(funcall (the (function (t t) (values fixnum &optional))
(usl-compare-fun usl))
a b))
;; Find
(defun usl-find (usl value &optional pred)
"Return the in-tree value if found.
PRED if given must be an array, fill it with predecessor usl nodes."
(declare (type usl usl)
(type (or null (simple-array t (*))) pred))
(with-slots (head) usl
(let ((node head))
(do ((level (1- (the (fixnum* 1) (usl-node-height head)))
(1- (the (fixnum* 0) level))))
((< level 0))
(do ((n node (usl-node-next n level)))
((or (null n)
(not (= -1 (usl-compare usl (usl-node-value n) value)))))
(setf node n))
(when (and pred (< level (length pred)))
(setf (aref pred level) node)))
(let* ((next (usl-node-next node 0))
(found (when next
(let ((next-value (usl-node-value next)))
(unless (= -1 (usl-compare usl value next-value))
value)))))
found))))
(defun usl-cursor (usl &optional value)
(let ((pred (make-array 1)))
(usl-find usl value pred)
(aref pred 0)))
#+test
(let ((usl (make-usl)))
(usl-cursor usl nil))
;; Insert
(defun usl-node-insert (pred new)
(declare (type (simple-array usl-node (*)) pred))
(dotimes (i (length pred))
(let ((node (aref pred i)))
(setf (usl-node-next new i) (usl-node-next node i)
(usl-node-next node i) new))))
(defun usl-insert (usl value)
(with-slots (head height-fun) usl
(declare (type (function nil (fixnum* 1)) height-fun))
(let* ((height (funcall height-fun))
(pred (make-array (the (fixnum* 1) height))))
(or (usl-find usl value pred)
(progn (usl-node-insert pred (make-usl-node value height))
(incf (usl-length usl))
value)))))
#+test
(defun test-usl-insert ()
(let ((usl (make-usl)))
(usl-insert usl #(1 1 1))
(usl-insert usl #(2 2 2))
(usl-insert usl #(0 0 0))
(usl-insert usl #(1 2 2))
(usl-insert usl #(0 0 1))
(usl-insert usl #(3 3 3))
usl))
#+test
(let ((*print-circle* t))
(format t "~%~A~&" (test-usl-insert)))
;; Get
(defun usl-get (usl value)
(usl-find usl value))
#+test
(let ((usl (test-usl-insert)))
(mapcar (lambda (x)
(format t "~&Get ~S -> ~S" x (usl-get usl x)))
(list 1 #(0 0 0) #(1 2 2) #(0 0 1) #(1 0 0) #(3 3 3) #(2 2 2))))
;; Each
(defun usl-each (usl fn &key start end)
(declare (type (function (t) t) fn))
(do ((node (usl-node-next (usl-cursor usl start)) (usl-node-next node)))
((or (null node)
(when end
(= -1 (usl-compare usl end (usl-node-value node))))))
(funcall fn (usl-node-value node))))
#+test
(defun test-usl-each ()
(usl-each (test-usl-insert)
(lambda (fact)
(format t "~&each ~S~%" fact))))
#+test
(test-usl-each)
;; Delete
(defun usl-delete (usl value)
(let* ((pred (make-array (the (fixnum* 1) (usl-height usl))))
(found (usl-find usl value pred)))
(when found
(let* ((node (usl-node-next (aref pred 0)))
(height (the (fixnum* 1) (usl-node-height node))))
(dotimes (i height)
(setf (usl-node-next (aref pred i) i)
(usl-node-next node i))))
(decf (usl-length usl))
found)))
#+test
(defun test-usl-delete ()
(let ((usl (test-usl-insert)))
(mapcar (lambda (x)
(format t "~&delete ~S -> ~S~%"
x (usl-delete usl x)))
(list #(1 2 2) #(0 0 0) #(1 2 2) #(3 3 3)))
(format t "~&~S~%" usl)))
#+test
(test-usl-delete)