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; Copyright (c) 1993-2008 by Richard Kelsey. See file COPYING.
; Code to determine the separation vertices of a graph
; NODES is a list of nodes
; (TO node) returns a list of the nodes which are connected to this one
; (SLOT-NODE node) and (SET-SLOT! node value) are used by the algorithm to
; associate data with nodes (in the absence of a tables).
(define (separation-vertices nodes to slot set-slot!)
(cond ((null? nodes)
(values '() '()))
((null? (cdr nodes))
(values nodes (list nodes)))
(else
(receive (separators components)
(real-separation-vertices (make-vertices nodes to slot set-slot!))
(for-each (lambda (n) (set-slot! n #f)) nodes)
(values separators components)))))
(define-record-type vertex :vertex
(really-make-vertex data edges dfs-index)
vertex?
(data vertex-data) ; user's data
(edges vertex-edges ; list of edges from this vertex
set-vertex-edges!)
(dfs-index vertex-dfs-index ; ordering from depth-first-search
set-vertex-dfs-index!)
(level vertex-level ; value used in algorithm...
set-vertex-level!)
(parent vertex-parent ; parent of this node in DFS tree
set-vertex-parent!))
(define (make-vertex data)
(really-make-vertex data '() 0))
(define-record-type edge :edge
(make-edge from to unused?)
(from edge-from) ; two (unordered) vertices
(to edge-to)
(unused? edge-unused ; used to mark edges that have been traversed
set-edge-unused!))
(define (make-edge from to)
(really-make-edge from to #t))
(define (other-vertex edge v)
(if (eq? v (edge-from edge))
(edge-to edge)
(edge-from edge)))
(define (maybe-add-edge from to)
(if (and (not (eq? from to))
(not (any? (lambda (e)
(or (eq? to (edge-from e))
(eq? to (edge-to e))))
(vertex-edges from))))
(let ((e (make-edge from to)))
(set-vertex-edges! from (cons e (vertex-edges from)))
(set-vertex-edges! to (cons e (vertex-edges to))))))
(define (make-vertices nodes to slot set-slot!)
(let ((vertices (map (lambda (n)
(let ((v (make-vertex n)))
(set-slot! n v)
v))
nodes)))
(for-each (lambda (n)
(for-each (lambda (n0)
(maybe-add-edge (slot n) (slot n0)))
(to n)))
nodes)
vertices))
; The numbers are the algorithm step numbers from page 62 of Graph Algorithms,
; Shimon Even, Computer Science Press, 1979.
; Them Us
; L(v) (vertex-level v)
; k(v) (vertex-dfs-index v)
; f(v) (vertex-parent v)
; S stack
; s start
(define (real-separation-vertices vertices)
(do-vertex (car vertices) 0 '() (car vertices) '() '()))
; 2
(define (do-vertex v i stack start v-res c-res)
(let ((i (+ i 1)))
(set-vertex-level! v i)
(set-vertex-dfs-index! v i)
(find-unused-edge v i (cons v stack) start v-res c-res)))
; 3
(define (find-unused-edge v i stack start v-res c-res)
(let ((e (first edge-unused? (vertex-edges v))))
(if e
(do-edge e v i stack start v-res c-res)
(no-unused-edge v i stack start v-res c-res))))
; 4
(define (do-edge e v i stack start v-res c-res)
(let ((u (other-vertex e v)))
(set-edge-unused?! e #f)
(cond ((= 0 (vertex-dfs-index u))
(set-vertex-parent! u v)
(do-vertex u i stack start v-res c-res))
(else
(if (> (vertex-level v)
(vertex-dfs-index u))
(set-vertex-level! v (vertex-dfs-index u)))
(find-unused-edge v i stack start v-res c-res)))))
; 5
(define (no-unused-edge v i stack start v-res c-res)
(let* ((parent (vertex-parent v))
(p-dfs-index (vertex-dfs-index parent)))
(cond ((= 1 p-dfs-index)
(gather-nonseparable-with-start v i stack start v-res c-res))
((< (vertex-level v) p-dfs-index)
(if (< (vertex-level v)
(vertex-level parent))
(set-vertex-level! parent (vertex-level v)))
(find-unused-edge parent i stack start v-res c-res))
(else
(gather-nonseparable v i stack start v-res c-res)))))
; 7
(define (gather-nonseparable v i stack start v-res c-res)
(let* ((parent (vertex-parent v))
(data (vertex-data parent)))
(receive (vertices stack)
(pop-down-to stack v)
(find-unused-edge parent
i
stack
start
(if (not (memq? data v-res))
(cons data v-res)
v-res)
(cons (cons data (map vertex-data vertices)) c-res)))))
; 9
(define (gather-nonseparable-with-start v i stack start v-res c-res)
(receive (vertices stack)
(pop-down-to stack v)
(let* ((data (vertex-data start))
(c-res (cons (cons data (map vertex-data vertices)) c-res)))
(if (not (any? edge-unused? (vertex-edges start)))
(values v-res c-res)
(find-unused-edge start
i
stack
start
(if (not (memq? data v-res))
(cons data v-res)
v-res)
c-res)))))
(define (pop-down-to stack v)
(do ((stack stack (cdr stack))
(res '() (cons (car stack) res)))
((eq? v (car stack))
(values (cons v res) (cdr stack)))))
(define (test-separation-vertices graph)
(let ((nodes (map (lambda (n)
(vector (car n) #f #f))
graph)))
(for-each (lambda (data node)
(vector-set! node 1 (map (lambda (s)
(first (lambda (v)
(eq? s (vector-ref v 0)))
nodes))
(cdr data))))
graph
nodes)
(receive (separation-vertices components)
(separation-vertices nodes
(lambda (v) (vector-ref v 1))
(lambda (v) (vector-ref v 2))
(lambda (v val) (vector-set! v 2 val)))
(values (map (lambda (v) (vector-ref v 0)) separation-vertices)
(map (lambda (l)
(map (lambda (v) (vector-ref v 0))
l))
components)))))
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