/* Part of SWI-Prolog
Author: R.A.O'Keefe, Vitor Santos Costa, Jan Wielemaker
E-mail: J.Wielemaker@cs.vu.nl
WWW: http://www.swi-prolog.org
Copyright (C): 1984-2012, VU University Amsterdam
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 library; if not, write to the Free Software
Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
As a special exception, if you link this library with other files,
compiled with a Free Software compiler, to produce an executable, this
library does not by itself cause the resulting executable to be covered
by the GNU General Public License. This exception does not however
invalidate any other reasons why the executable file might be covered by
the GNU General Public License.
The original code was distributed in the public domain and YAP
under the Perl artistic license. The current version is dual
licences and may be redistributed under the Artistic license,
version 2.0.
*/
% $Id$
%
% File : GRAPHS.PL
% Author : R.A.O'Keefe
% Updated: 20 March 1984
% Purpose: Graph-processing utilities.
:- module(ugraphs,
[ add_edges/3, % +Graph, +Edges, -NewGraph
add_vertices/3, % +Graph, +Vertices, -NewGraph
complement/2, % +Graph, -NewGraph
compose/3, % +LeftGraph, +RightGraph, -NewGraph
del_edges/3, % +Graph, +Edges, -NewGraph
del_vertices/3, % +Graph, +Vertices, -NewGraph
edges/2, % +Graph, -Edges
neighbors/3, % +Vertex, +Graph, -Vertices
neighbours/3, % +Vertex, +Graph, -Vertices
reachable/3, % +Vertex, +Graph, -Vertices
top_sort/2, % +Graph, -Sort
top_sort/3, % +Graph, -Sort0, -Sort
transitive_closure/2, % +Graph, -Closure
transpose/2, % +Graph, -NewGraph
vertices/2, % +Graph, -Vertices
vertices_edges_to_ugraph/3, % +Vertices, +Edges, -Graph
ugraph_union/3 % +Graph1, +Graph2, -Graph
]).
/** <module> Graph manipulation library
The S-representation of a graph is a list of (vertex-neighbours) pairs,
where the pairs are in standard order (as produced by keysort) and the
neighbours of each vertex are also in standard order (as produced by
sort). This form is convenient for many calculations.
A new UGraph from raw data can be created using
vertices_edges_to_ugraph/3.
Adapted to support some of the functionality of the SICStus ugraphs
library by Vitor Santos Costa.
Ported from YAP 5.0.1 to SWI-Prolog by Jan Wielemaker.
@author R.A.O'Keefe
@author Vitor Santos Costa
@author Jan Wielemaker
@license GPL+SWI-exception or Artistic 2.0
*/
:- use_module(library(lists), [
append/3,
member/2
]).
:- use_module(library(ordsets), [
ord_add_element/3,
ord_subtract/3,
ord_union/3,
ord_union/4
]).
/*
:- public
p_to_s_graph/2,
s_to_p_graph/2, % edges
s_to_p_trans/2,
p_member/3,
s_member/3,
p_transpose/2,
s_transpose/2,
compose/3,
top_sort/2,
vertices/2,
warshall/2.
:- mode
vertices(+, -),
p_to_s_graph(+, -),
p_to_s_vertices(+, -),
p_to_s_group(+, +, -),
p_to_s_group(+, +, -, -),
s_to_p_graph(+, -),
s_to_p_graph(+, +, -, -),
s_to_p_trans(+, -),
s_to_p_trans(+, +, -, -),
p_member(?, ?, +),
s_member(?, ?, +),
p_transpose(+, -),
s_transpose(+, -),
s_transpose(+, -, ?, -),
transpose_s(+, +, +, -),
compose(+, +, -),
compose(+, +, +, -),
compose1(+, +, +, -),
compose1(+, +, +, +, +, +, +, -),
top_sort(+, -),
vertices_and_zeros(+, -, ?),
count_edges(+, +, +, -),
incr_list(+, +, +, -),
select_zeros(+, +, -),
top_sort(+, -, +, +, +),
decr_list(+, +, +, -, +, -),
warshall(+, -),
warshall(+, +, -),
warshall(+, +, +, -).
*/
%% vertices(+S_Graph, -Vertices) is det.
%
% Strips off the neighbours lists of an S-representation to
% produce a list of the vertices of the graph. (It is a
% characteristic of S-representations that *every* vertex appears,
% even if it has no neighbours.). Vertices is in the standard
% order of terms.
vertices([], []) :- !.
vertices([Vertex-_|Graph], [Vertex|Vertices]) :-
vertices(Graph, Vertices).
%% vertices_edges_to_ugraph(+Vertices, +Edges, -UGraph) is det.
%
% Create a UGraph from Vertices and edges. Given a graph with a
% set of Vertices and a set of Edges, Graph must unify with the
% corresponding S-representation. Note that the vertices without
% edges will appear in Vertices but not in Edges. Moreover, it is
% sufficient for a vertice to appear in Edges.
%
% ==
% ?- vertices_edges_to_ugraph([],[1-3,2-4,4-5,1-5], L).
% L = [1-[3,5], 2-[4], 3-[], 4-[5], 5-[]]
% ==
%
% In this case all vertices are defined implicitly. The next
% example shows three unconnected vertices:
%
% ==
% ?- vertices_edges_to_ugraph([6,7,8],[1-3,2-4,4-5,1-5], L).
% L = [1-[3,5], 2-[4], 3-[], 4-[5], 5-[], 6-[], 7-[], 8-[]]
% ==
vertices_edges_to_ugraph(Vertices, Edges, Graph) :-
sort(Edges, EdgeSet),
p_to_s_vertices(EdgeSet, IVertexBag),
append(Vertices, IVertexBag, VertexBag),
sort(VertexBag, VertexSet),
p_to_s_group(VertexSet, EdgeSet, Graph).
add_vertices(Graph, Vertices, NewGraph) :-
msort(Vertices, V1),
add_vertices_to_s_graph(V1, Graph, NewGraph).
add_vertices_to_s_graph(L, [], NL) :- !,
add_empty_vertices(L, NL).
add_vertices_to_s_graph([], L, L) :- !.
add_vertices_to_s_graph([V1|VL], [V-Edges|G], NGL) :-
compare(Res, V1, V),
add_vertices_to_s_graph(Res, V1, VL, V, Edges, G, NGL).
add_vertices_to_s_graph(=, _, VL, V, Edges, G, [V-Edges|NGL]) :-
add_vertices_to_s_graph(VL, G, NGL).
add_vertices_to_s_graph(<, V1, VL, V, Edges, G, [V1-[]|NGL]) :-
add_vertices_to_s_graph(VL, [V-Edges|G], NGL).
add_vertices_to_s_graph(>, V1, VL, V, Edges, G, [V-Edges|NGL]) :-
add_vertices_to_s_graph([V1|VL], G, NGL).
add_empty_vertices([], []).
add_empty_vertices([V|G], [V-[]|NG]) :-
add_empty_vertices(G, NG).
%% del_vertices(+Graph, +Vertices, -NewGraph) is det.
%
% Unify NewGraph with a new graph obtained by deleting the list of
% Vertices and all the edges that start from or go to a vertex in
% Vertices to the Graph. Example:
%
% ==
% ?- del_vertices([1-[3,5],2-[4],3-[],4-[5],5-[],6-[],7-[2,6],8-[]],
% [2,1],
% NL).
% NL = [3-[],4-[5],5-[],6-[],7-[6],8-[]]
% ==
%
% @compat Upto 5.6.48 the argument order was (+Vertices, +Graph,
% -NewGraph). Both YAP and SWI-Prolog have changed the argument
% order for compatibility with recent SICStus as well as
% consistency with del_edges/3.
del_vertices(Graph, Vertices, NewGraph) :-
sort(Vertices, V1), % JW: was msort
( V1 = []
-> Graph = NewGraph
; del_vertices(Graph, V1, V1, NewGraph)
).
del_vertices(G, [], V1, NG) :- !,
del_remaining_edges_for_vertices(G, V1, NG).
del_vertices([], _, _, []).
del_vertices([V-Edges|G], [V0|Vs], V1, NG) :-
compare(Res, V, V0),
split_on_del_vertices(Res, V,Edges, [V0|Vs], NVs, V1, NG, NGr),
del_vertices(G, NVs, V1, NGr).
del_remaining_edges_for_vertices([], _, []).
del_remaining_edges_for_vertices([V0-Edges|G], V1, [V0-NEdges|NG]) :-
ord_subtract(Edges, V1, NEdges),
del_remaining_edges_for_vertices(G, V1, NG).
split_on_del_vertices(<, V, Edges, Vs, Vs, V1, [V-NEdges|NG], NG) :-
ord_subtract(Edges, V1, NEdges).
split_on_del_vertices(>, V, Edges, [_|Vs], Vs, V1, [V-NEdges|NG], NG) :-
ord_subtract(Edges, V1, NEdges).
split_on_del_vertices(=, _, _, [_|Vs], Vs, _, NG, NG).
add_edges(Graph, Edges, NewGraph) :-
p_to_s_graph(Edges, G1),
ugraph_union(Graph, G1, NewGraph).
%% ugraph_union(+Set1, +Set2, ?Union)
%
% Is true when Union is the union of Set1 and Set2. This code is a
% copy of set union
ugraph_union(Set1, [], Set1) :- !.
ugraph_union([], Set2, Set2) :- !.
ugraph_union([Head1-E1|Tail1], [Head2-E2|Tail2], Union) :-
compare(Order, Head1, Head2),
ugraph_union(Order, Head1-E1, Tail1, Head2-E2, Tail2, Union).
ugraph_union(=, Head-E1, Tail1, _-E2, Tail2, [Head-Es|Union]) :-
ord_union(E1, E2, Es),
ugraph_union(Tail1, Tail2, Union).
ugraph_union(<, Head1, Tail1, Head2, Tail2, [Head1|Union]) :-
ugraph_union(Tail1, [Head2|Tail2], Union).
ugraph_union(>, Head1, Tail1, Head2, Tail2, [Head2|Union]) :-
ugraph_union([Head1|Tail1], Tail2, Union).
del_edges(Graph, Edges, NewGraph) :-
p_to_s_graph(Edges, G1),
graph_subtract(Graph, G1, NewGraph).
%% graph_subtract(+Set1, +Set2, ?Difference)
%
% Is based on ord_subtract
graph_subtract(Set1, [], Set1) :- !.
graph_subtract([], _, []).
graph_subtract([Head1-E1|Tail1], [Head2-E2|Tail2], Difference) :-
compare(Order, Head1, Head2),
graph_subtract(Order, Head1-E1, Tail1, Head2-E2, Tail2, Difference).
graph_subtract(=, H-E1, Tail1, _-E2, Tail2, [H-E|Difference]) :-
ord_subtract(E1,E2,E),
graph_subtract(Tail1, Tail2, Difference).
graph_subtract(<, Head1, Tail1, Head2, Tail2, [Head1|Difference]) :-
graph_subtract(Tail1, [Head2|Tail2], Difference).
graph_subtract(>, Head1, Tail1, _, Tail2, Difference) :-
graph_subtract([Head1|Tail1], Tail2, Difference).
%% edges(+UGraph, -Edges) is det.
%
% Edges is the set of edges in UGraph. Each edge is represented as
% a pair From-To, where From and To are vertices in the graph.
edges(Graph, Edges) :-
s_to_p_graph(Graph, Edges).
p_to_s_graph(P_Graph, S_Graph) :-
sort(P_Graph, EdgeSet),
p_to_s_vertices(EdgeSet, VertexBag),
sort(VertexBag, VertexSet),
p_to_s_group(VertexSet, EdgeSet, S_Graph).
p_to_s_vertices([], []).
p_to_s_vertices([A-Z|Edges], [A,Z|Vertices]) :-
p_to_s_vertices(Edges, Vertices).
p_to_s_group([], _, []).
p_to_s_group([Vertex|Vertices], EdgeSet, [Vertex-Neibs|G]) :-
p_to_s_group(EdgeSet, Vertex, Neibs, RestEdges),
p_to_s_group(Vertices, RestEdges, G).
p_to_s_group([V1-X|Edges], V2, [X|Neibs], RestEdges) :- V1 == V2, !,
p_to_s_group(Edges, V2, Neibs, RestEdges).
p_to_s_group(Edges, _, [], Edges).
s_to_p_graph([], []) :- !.
s_to_p_graph([Vertex-Neibs|G], P_Graph) :-
s_to_p_graph(Neibs, Vertex, P_Graph, Rest_P_Graph),
s_to_p_graph(G, Rest_P_Graph).
s_to_p_graph([], _, P_Graph, P_Graph) :- !.
s_to_p_graph([Neib|Neibs], Vertex, [Vertex-Neib|P], Rest_P) :-
s_to_p_graph(Neibs, Vertex, P, Rest_P).
transitive_closure(Graph, Closure) :-
warshall(Graph, Graph, Closure).
warshall([], Closure, Closure) :- !.
warshall([V-_|G], E, Closure) :-
memberchk(V-Y, E), % Y := E(v)
warshall(E, V, Y, NewE),
warshall(G, NewE, Closure).
warshall([X-Neibs|G], V, Y, [X-NewNeibs|NewG]) :-
memberchk(V, Neibs),
!,
ord_union(Neibs, Y, NewNeibs),
warshall(G, V, Y, NewG).
warshall([X-Neibs|G], V, Y, [X-Neibs|NewG]) :- !,
warshall(G, V, Y, NewG).
warshall([], _, _, []).
%% transpose(Graph, NewGraph) is det.
%
% Unify NewGraph with a new graph obtained from Graph by replacing
% all edges of the form V1-V2 by edges of the form V2-V1. The cost
% is O(|V|*log(|V|)). Notice that an undirected graph is its own
% transpose. Example:
%
% ==
% ?- transpose([1-[3,5],2-[4],3-[],4-[5],
% 5-[],6-[],7-[],8-[]], NL).
% NL = [1-[],2-[],3-[1],4-[2],5-[1,4],6-[],7-[],8-[]]
% ==
transpose(Graph, NewGraph) :-
edges(Graph, Edges),
vertices(Graph, Vertices),
flip_edges(Edges, TransposedEdges),
vertices_edges_to_ugraph(Vertices, TransposedEdges, NewGraph).
flip_edges([], []).
flip_edges([Key-Val|Pairs], [Val-Key|Flipped]) :-
flip_edges(Pairs, Flipped).
%% compose(G1, G2, Composition)
%
% Calculates the composition of two S-form graphs, which need not
% have the same set of vertices.
compose(G1, G2, Composition) :-
vertices(G1, V1),
vertices(G2, V2),
ord_union(V1, V2, V),
compose(V, G1, G2, Composition).
compose([], _, _, []) :- !.
compose([Vertex|Vertices], [Vertex-Neibs|G1], G2,
[Vertex-Comp|Composition]) :- !,
compose1(Neibs, G2, [], Comp),
compose(Vertices, G1, G2, Composition).
compose([Vertex|Vertices], G1, G2, [Vertex-[]|Composition]) :-
compose(Vertices, G1, G2, Composition).
compose1([V1|Vs1], [V2-N2|G2], SoFar, Comp) :-
compare(Rel, V1, V2), !,
compose1(Rel, V1, Vs1, V2, N2, G2, SoFar, Comp).
compose1(_, _, Comp, Comp).
compose1(<, _, Vs1, V2, N2, G2, SoFar, Comp) :- !,
compose1(Vs1, [V2-N2|G2], SoFar, Comp).
compose1(>, V1, Vs1, _, _, G2, SoFar, Comp) :- !,
compose1([V1|Vs1], G2, SoFar, Comp).
compose1(=, V1, Vs1, V1, N2, G2, SoFar, Comp) :-
ord_union(N2, SoFar, Next),
compose1(Vs1, G2, Next, Comp).
%% top_sort(+Graph, -Sorted) is semidet.
%% top_sort(+Graph, -Sorted, ?Tail) is semidet.
%
% Sorted is a topological sorted list of nodes in Graph. A
% toplogical sort is possible if the graph is connected and
% acyclic. In the example we show how topological sorting works
% for a linear graph:
%
% ==
% ?- top_sort([1-[2], 2-[3], 3-[]], L).
% L = [1, 2, 3]
% ==
%
% The predicate top_sort/3 is a difference list version of
% top_sort/2.
top_sort(Graph, Sorted) :-
vertices_and_zeros(Graph, Vertices, Counts0),
count_edges(Graph, Vertices, Counts0, Counts1),
select_zeros(Counts1, Vertices, Zeros),
top_sort(Zeros, Sorted, Graph, Vertices, Counts1).
top_sort(Graph, Sorted0, Sorted) :-
vertices_and_zeros(Graph, Vertices, Counts0),
count_edges(Graph, Vertices, Counts0, Counts1),
select_zeros(Counts1, Vertices, Zeros),
top_sort(Zeros, Sorted, Sorted0, Graph, Vertices, Counts1).
vertices_and_zeros([], [], []) :- !.
vertices_and_zeros([Vertex-_|Graph], [Vertex|Vertices], [0|Zeros]) :-
vertices_and_zeros(Graph, Vertices, Zeros).
count_edges([], _, Counts, Counts) :- !.
count_edges([_-Neibs|Graph], Vertices, Counts0, Counts2) :-
incr_list(Neibs, Vertices, Counts0, Counts1),
count_edges(Graph, Vertices, Counts1, Counts2).
incr_list([], _, Counts, Counts) :- !.
incr_list([V1|Neibs], [V2|Vertices], [M|Counts0], [N|Counts1]) :-
V1 == V2, !,
N is M+1,
incr_list(Neibs, Vertices, Counts0, Counts1).
incr_list(Neibs, [_|Vertices], [N|Counts0], [N|Counts1]) :-
incr_list(Neibs, Vertices, Counts0, Counts1).
select_zeros([], [], []) :- !.
select_zeros([0|Counts], [Vertex|Vertices], [Vertex|Zeros]) :- !,
select_zeros(Counts, Vertices, Zeros).
select_zeros([_|Counts], [_|Vertices], Zeros) :-
select_zeros(Counts, Vertices, Zeros).
top_sort([], [], Graph, _, Counts) :- !,
vertices_and_zeros(Graph, _, Counts).
top_sort([Zero|Zeros], [Zero|Sorted], Graph, Vertices, Counts1) :-
graph_memberchk(Zero-Neibs, Graph),
decr_list(Neibs, Vertices, Counts1, Counts2, Zeros, NewZeros),
top_sort(NewZeros, Sorted, Graph, Vertices, Counts2).
top_sort([], Sorted0, Sorted0, Graph, _, Counts) :- !,
vertices_and_zeros(Graph, _, Counts).
top_sort([Zero|Zeros], [Zero|Sorted], Sorted0, Graph, Vertices, Counts1) :-
graph_memberchk(Zero-Neibs, Graph),
decr_list(Neibs, Vertices, Counts1, Counts2, Zeros, NewZeros),
top_sort(NewZeros, Sorted, Sorted0, Graph, Vertices, Counts2).
graph_memberchk(Element1-Edges, [Element2-Edges2|_]) :-
Element1 == Element2, !,
Edges = Edges2.
graph_memberchk(Element, [_|Rest]) :-
graph_memberchk(Element, Rest).
decr_list([], _, Counts, Counts, Zeros, Zeros) :- !.
decr_list([V1|Neibs], [V2|Vertices], [1|Counts1], [0|Counts2], Zi, Zo) :-
V1 == V2, !,
decr_list(Neibs, Vertices, Counts1, Counts2, [V2|Zi], Zo).
decr_list([V1|Neibs], [V2|Vertices], [N|Counts1], [M|Counts2], Zi, Zo) :-
V1 == V2, !,
M is N-1,
decr_list(Neibs, Vertices, Counts1, Counts2, Zi, Zo).
decr_list(Neibs, [_|Vertices], [N|Counts1], [N|Counts2], Zi, Zo) :-
decr_list(Neibs, Vertices, Counts1, Counts2, Zi, Zo).
%% neighbors(+Vertex, +Graph, -Neigbours) is det.
%% neighbours(+Vertex, +Graph, -Neigbours) is det.
%
% Neigbours is a sorted list of the neighbours of Vertex in Graph.
neighbors(Vertex, Graph, Neig) :-
neighbours(Vertex, Graph, Neig).
neighbours(V,[V0-Neig|_],Neig) :-
V == V0, !.
neighbours(V,[_|G],Neig) :-
neighbours(V,G,Neig).
%
% Simple two-step algorithm. You could be smarter, I suppose.
%
complement(G, NG) :-
vertices(G,Vs),
complement(G,Vs,NG).
complement([], _, []).
complement([V-Ns|G], Vs, [V-INs|NG]) :-
ord_add_element(Ns,V,Ns1),
ord_subtract(Vs,Ns1,INs),
complement(G, Vs, NG).
reachable(N, G, Rs) :-
reachable([N], G, [N], Rs).
reachable([], _, Rs, Rs).
reachable([N|Ns], G, Rs0, RsF) :-
neighbours(N, G, Nei),
ord_union(Rs0, Nei, Rs1, D),
append(Ns, D, Nsi),
reachable(Nsi, G, Rs1, RsF).