%   $Id$
%
%   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.

As the original code was distributed in  the public domain and YAP under
the  Perl  artistic  license  the  code  can  be  used  with  SWI-Prolog
applications without consequences to the   overall system or proprietary
code linked to SWI-Prolog

@author R.A.O'Keefe
@author Vitor Santos Costa
@author Jan Wielemaker
*/

:- 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(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(S_Graph, Transpose) :-
	s_transpose(S_Graph, Base, Base, Transpose).

s_transpose([], [], Base, Base) :- !.
s_transpose([Vertex-Neibs|Graph], [Vertex-[]|RestBase], Base, Transpose) :-
	s_transpose(Graph, RestBase, Base, SoFar),
	transpose_s(SoFar, Neibs, Vertex, Transpose).

transpose_s([Neib-Trans|SoFar], [Neib|Neibs], Vertex,
		[Neib-[Vertex|Trans]|Transpose]) :- !,
	transpose_s(SoFar, Neibs, Vertex, Transpose).
transpose_s([Head|SoFar], Neibs, Vertex, [Head|Transpose]) :- !,
	transpose_s(SoFar, Neibs, Vertex, Transpose).
transpose_s([], [], _, []).


%%	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).

