File: ugraphs.pl

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/*  Part of SWI-Prolog

    Author:        R.A.O'Keefe, Vitor Santos Costa, Jan Wielemaker
    E-mail:        J.Wielemaker@vu.nl
    WWW:           http://www.swi-prolog.org
    Copyright (c)  1984-2012, VU University Amsterdam
    All rights reserved.

    Redistribution and use in source and binary forms, with or without
    modification, are permitted provided that the following conditions
    are met:

    1. Redistributions of source code must retain the above copyright
       notice, this list of conditions and the following disclaimer.

    2. Redistributions in binary form must reproduce the above copyright
       notice, this list of conditions and the following disclaimer in
       the documentation and/or other materials provided with the
       distribution.

    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
    "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
    LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
    FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
    COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
    INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
    BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
    LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
    CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
    LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
    ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
    POSSIBILITY OF SUCH DAMAGE.
*/

:- 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_ugraph/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_ugraph(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-[]]
%     ==
%
%   @compat  This  predicate  used  to   be  known  as  transpose/2.
%   Following  SICStus  4,  we  reserve    transpose/2   for  matrix
%   transposition    and    renamed    ugraph    transposition    to
%   transpose_ugraph/2.

transpose_ugraph(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).