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|
;;;
;;; GRAPHS - graph theory package for Maxima
;;;
;;; Copyright (C) 2007 Andrej Vodopivec <andrej.vodopivec@gmail.com>
;;;
;;; This program is free software; you can redistribute it and/or modify
;;; it under the terms of the GNU General Public License as published by
;;; the Free Software Foundation; either version 2 of the License, or
;;; (at your option) any later version.
;;;
;;; This program is distributed in the hope that it will be useful,
;;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
;;; GNU General Public License for more details.
;;;
;;; You should have received a copy of the GNU General Public License
;;; along with this program; if not, write to the Free Software
;;; Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
;;;
(in-package :maxima)
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;
;;; matching algorithms
;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defmfun $max_matching (gr)
(require-graph 'maximum_matching 1 gr)
(let ((partition (cdr ($bipartition gr))))
(if (null partition)
(maximum-matching-general gr)
(maximum-matching-bipartite gr (cdr (car partition)) (cdr (cadr partition))))))
;;;;;;;;;;
;;
;; the bipartite case
;;
(defun maximum-matching-bipartite (gr a b &optional cover)
(let ((matching (make-hash-table))
(done))
;; greedy matching
(loop for e in (edges gr) do
(let ((u (first e))
(v (second e)))
(when (and (null (gethash u matching))
(null (gethash v matching)))
(setf (gethash u matching) v)
(setf (gethash v matching) u))))
;; augment the matching
(loop while (not done) do
(setq done t)
(let ((au) (bu))
;; find unmatched vertices
(loop for u in a do
(when (null (gethash u matching))
(push u au)))
(loop for u in b do
(when (null (gethash u matching))
(push u bu)))
;; find augmenting path
(when (not (or (null au)
(null bu)))
(let ((prev (make-hash-table))
(new1 au) (new2) (x))
(loop while (not (null new1)) do
;; add edges in M^c
(setq new2 ())
(loop for u in new1 do
(loop for v in (neighbors u gr) do
(when (and (null (gethash v prev))
(or (null (gethash v matching))
(not (= (gethash v matching) u))))
(setf (gethash v prev) u)
(push v new2))))
(setq new1 ())
;; add edges in M
(loop for v in new2 do
(let ((u (gethash v matching)))
(unless (null u)
(push u new1)
(setf (gethash u prev) v)))))
;; chech for augmenting path
(loop for v in bu while (null x) do
(unless (null (gethash v prev))
(setq x v)))
;; extend the matching
(unless (null x)
(setq done nil)
(loop while (not (null x)) do
(let ((u x)
(v (gethash x prev)))
(setf (gethash u matching) v)
(setf (gethash v matching) u)
(setq x (gethash v prev))))) )) ))
(if (null cover)
;; we want the matching
(let ((mmatching ()))
(maphash #'(lambda (u v) (if (< u v)
(setq mmatching (cons `((mlist simp) ,u ,v) mmatching))))
matching)
(cons '(mlist simp) mmatching))
;; we want the vertex cover
(get-cover-from-matching gr a matching)) ))
(defun get-cover-from-matching (gr a matching)
(let ((au) (cov))
(loop for x in (cdr a) do
(when (null (gethash x matching))
(push x au)))
(if (null au)
`((mlist simp) ,@(sort (cdr a) #'<))
;; construct the Hungarian tree
(let ((prev (make-hash-table))
(new1 au) (new2))
(loop while (not (null new1)) do
;; add edges in M^c
(setq new2 ())
(loop for u in new1 do
(loop for v in (neighbors u gr) do
(when (and (null (gethash v prev))
(or (null (gethash v matching))
(not (= (gethash v matching) u))))
(setf (gethash v prev) u)
(push v new2))))
(setq new1 ())
;; add edges in M
(loop for v in new2 do
(let ((u (gethash v matching)))
(unless (null u)
(push u new1)
(setf (gethash u prev) v)))))
(maphash #'(lambda (u v)
(when (member v a)
(if (null (gethash v prev))
(push v cov)
(push u cov))))
matching)
`((mlist simp) ,@(sort cov #'<))))))
;;;;;;;;;;;
;;
;; set partiton using hash-tales
;;
(defun main-vertex (v sp)
(let ((m v))
(loop while (not (= m (gethash m sp))) do
(setq m (gethash m sp)))
(setf (gethash v sp) m)
m))
(defun join-sets (v u sp)
(setf (gethash (main-vertex u sp) sp)
(main-vertex v sp)))
(defun set-main-vertex (m v sp)
(setf (gethash (main-vertex v sp) sp) m)
(setf (gethash m sp) m))
;;;;;;;;;;;
;;
;; the general case
;;
(defvar *matching-state*)
(defvar *matching-matching*)
(defvar *matching-bridge*)
(defvar *matching-prev*)
(defvar *matching-path*)
(defun maximum-matching-general (gr)
(let ((*matching-matching* (make-hash-table))
(done))
;; greedy matching
(loop for e in (edges gr) do
(let ((u (first e))
(v (second e)))
(when (and (null (gethash u *matching-matching*))
(null (gethash v *matching-matching*)))
(setf (gethash u *matching-matching*) v)
(setf (gethash v *matching-matching*) u))))
;; augment the matching
(loop while (not done) do
(setq done t)
(let ((unmatched-vertices)
(dfsnum)
(dfsnum-curr)
(vertex-partition)
(*matching-state*)
(*matching-bridge* (make-hash-table))
(active-edges)
(*matching-prev* (make-hash-table))
(x))
;; find unmatched vertices
(loop for v in (vertices gr) do
(when (null (gethash v *matching-matching*))
(push v unmatched-vertices)))
(loop for first-vertex in unmatched-vertices while (null x) do
(setq dfsnum (make-hash-table))
(setq dfsnum-curr 0)
(setq *matching-state* (make-hash-table))
(setq *matching-bridge* (make-hash-table))
(setq vertex-partition (make-hash-table))
(setq *matching-state* (make-hash-table))
(setf (gethash first-vertex *matching-state*) 'A)
(setq active-edges ())
(loop for v in (vertices gr) do
(setf (gethash v vertex-partition) v))
(loop for u in (neighbors first-vertex gr) do
(push (list first-vertex u) active-edges))
;; find augmenting path
(loop while (and (not (null active-edges)) (null x)) do
(let* ((e (pop active-edges))
(v (first e))
(w (second e))
(v-main (main-vertex v vertex-partition))
(w-main (main-vertex w vertex-partition)))
(when (null (gethash v-main dfsnum))
(setf (gethash v-main dfsnum) dfsnum-curr)
(incf dfsnum-curr))
(cond
;; we found an agugmented path
((and (member w-main unmatched-vertices)
(not (= w-main first-vertex)))
(setf (gethash w-main *matching-prev*) v)
(setq x w-main))
;; we didn't visit w yet
((null (gethash w-main *matching-state*))
(setf (gethash w-main *matching-state*) 'B)
(let ((z (gethash w-main *matching-matching*)))
(setf (gethash z *matching-state*) 'A)
(loop for x in (neighbors z gr) do
(unless (= x w-main)
(push (list z x) active-edges))))
(setf (gethash w-main *matching-prev*) v))
;; found a blossom
((and (not (= w-main v-main)) ;; in the same blossom already
(eql (gethash w-main *matching-state*) 'A))
(let ((fst) (lst) (b))
(if (> (gethash w-main dfsnum)
(gethash v-main dfsnum))
(setq fst w-main
lst v-main
b (list w v))
(setq fst v-main
lst w-main
b (list v w)))
(let ((tmp-vrt fst))
(loop while (not (= tmp-vrt lst)) do
(setf (gethash tmp-vrt *matching-bridge*) b)
(join-sets lst tmp-vrt vertex-partition)
(if (eql (gethash tmp-vrt *matching-state*) 'A)
(progn
(setq tmp-vrt (main-vertex (gethash tmp-vrt *matching-matching*)
vertex-partition))
(loop for u in (neighbors tmp-vrt gr) do
(setq u (main-vertex u vertex-partition))
(push (list tmp-vrt u) active-edges)))
(progn
(setq tmp-vrt (main-vertex (gethash tmp-vrt *matching-prev*)
vertex-partition))))))) )
)))
;; augment the path
(unless (null x)
(setq done nil)
(let ((*matching-path* (list x)))
(find-augmenting-path first-vertex (gethash x *matching-prev*))
(loop while *matching-path* do
(let ((a (car *matching-path*))
(b (cadr *matching-path*)))
(setf (gethash a *matching-matching*) b)
(setf (gethash b *matching-matching*) a)
(setf *matching-path* (cddr *matching-path*))))
))) ))
(let ((matching ()))
(maphash #'(lambda (u v) (if (< u v) (setq matching (cons `((mlist simp) ,u ,v) matching))))
*matching-matching*)
(cons '(mlist simp) matching))))
;; returns the path from b to a
(defun find-augmenting-path (a b)
(cond
((= a b)
(push a *matching-path*))
((eql (gethash b *matching-state*) 'A)
(push b *matching-path*)
(push (gethash b *matching-matching*) *matching-path*)
(find-augmenting-path a (gethash (gethash b *matching-matching*) *matching-prev*)))
(t
(push b *matching-path*)
(find-augmenting-path-1 (gethash b *matching-matching*) (first (gethash b *matching-bridge*)))
(find-augmenting-path a (second (gethash b *matching-bridge*)))) ))
(defun find-augmenting-path-1 (a b)
(cond
((= a b)
(push a *matching-path*))
((eql (gethash b *matching-state*) 'A)
(find-augmenting-path-1 a (gethash (gethash b *matching-matching*) *matching-prev*))
(push (gethash b *matching-matching*) *matching-path*)
(push b *matching-path*))
(t
(find-augmenting-path-1 a (second (gethash b *matching-bridge*)))
(find-augmenting-path (gethash b *matching-matching*) (first (gethash b *matching-bridge*)))
(push b *matching-path*)) ))
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