1/*  Part of SWI-Prolog
    2
    3    Author:        R.A.O'Keefe, Vitor Santos Costa, Jan Wielemaker
    4    E-mail:        J.Wielemaker@vu.nl
    5    WWW:           http://www.swi-prolog.org
    6    Copyright (c)  1984-2021, VU University Amsterdam
    7                              CWI, Amsterdam
    8                              SWI-Prolog Solutions .b.v
    9    All rights reserved.
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   36
   37:- module(ugraphs,
   38          [ add_edges/3,                % +Graph, +Edges, -NewGraph
   39            add_vertices/3,             % +Graph, +Vertices, -NewGraph
   40            complement/2,               % +Graph, -NewGraph
   41            compose/3,                  % +LeftGraph, +RightGraph, -NewGraph
   42            del_edges/3,                % +Graph, +Edges, -NewGraph
   43            del_vertices/3,             % +Graph, +Vertices, -NewGraph
   44            edges/2,                    % +Graph, -Edges
   45            neighbors/3,                % +Vertex, +Graph, -Vertices
   46            neighbours/3,               % +Vertex, +Graph, -Vertices
   47            reachable/3,                % +Vertex, +Graph, -Vertices
   48            top_sort/2,                 % +Graph, -Sort
   49            top_sort/3,                 % +Graph, -Sort0, -Sort
   50            transitive_closure/2,       % +Graph, -Closure
   51            transpose_ugraph/2,         % +Graph, -NewGraph
   52            vertices/2,                 % +Graph, -Vertices
   53            vertices_edges_to_ugraph/3, % +Vertices, +Edges, -Graph
   54            ugraph_union/3,             % +Graph1, +Graph2, -Graph
   55            connect_ugraph/3            % +Graph1, -Start, -Graph
   56          ]).

   79:- autoload(library(lists),[append/3]).   80:- autoload(library(ordsets),
   81	    [ord_subtract/3,ord_union/3,ord_add_element/3,ord_union/4]).

 vertices(+Graph, -Vertices)

Unify Vertices with all vertices appearing in Graph. Example:

?- vertices([1-[3,5],2-[4],3-[],4-[5],5-[]], L).
L = [1, 2, 3, 4, 5]

   90vertices([], []) :- !.
   91vertices([Vertex-_|Graph], [Vertex|Vertices]) :-
   92    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-[]]

  116vertices_edges_to_ugraph(Vertices, Edges, Graph) :-
  117    sort(Edges, EdgeSet),
  118    p_to_s_vertices(EdgeSet, IVertexBag),
  119    append(Vertices, IVertexBag, VertexBag),
  120    sort(VertexBag, VertexSet),
  121    p_to_s_group(VertexSet, EdgeSet, Graph).

 add_vertices(+Graph, +Vertices, -NewGraph)

Unify NewGraph with a new graph obtained by adding the list of Vertices to Graph. Example:

?- add_vertices([1-[3,5],2-[]], [0,1,2,9], NG).
NG = [0-[], 1-[3,5], 2-[], 9-[]]

  134add_vertices(Graph, Vertices, NewGraph) :-
  135    msort(Vertices, V1),
  136    add_vertices_to_s_graph(V1, Graph, NewGraph).
  137
  138add_vertices_to_s_graph(L, [], NL) :-
  139    !,
  140    add_empty_vertices(L, NL).
  141add_vertices_to_s_graph([], L, L) :- !.
  142add_vertices_to_s_graph([V1|VL], [V-Edges|G], NGL) :-
  143    compare(Res, V1, V),
  144    add_vertices_to_s_graph(Res, V1, VL, V, Edges, G, NGL).
  145
  146add_vertices_to_s_graph(=, _, VL, V, Edges, G, [V-Edges|NGL]) :-
  147    add_vertices_to_s_graph(VL, G, NGL).
  148add_vertices_to_s_graph(<, V1, VL, V, Edges, G, [V1-[]|NGL]) :-
  149    add_vertices_to_s_graph(VL, [V-Edges|G], NGL).
  150add_vertices_to_s_graph(>, V1, VL, V, Edges, G, [V-Edges|NGL]) :-
  151    add_vertices_to_s_graph([V1|VL], G, NGL).
  152
  153add_empty_vertices([], []).
  154add_empty_vertices([V|G], [V-[]|NG]) :-
  155    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-[]]

Compatibility

- 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.

  175del_vertices(Graph, Vertices, NewGraph) :-
  176    sort(Vertices, V1),             % JW: was msort
  177    (   V1 = []
  178    ->  Graph = NewGraph
  179    ;   del_vertices(Graph, V1, V1, NewGraph)
  180    ).
  181
  182del_vertices(G, [], V1, NG) :-
  183    !,
  184    del_remaining_edges_for_vertices(G, V1, NG).
  185del_vertices([], _, _, []).
  186del_vertices([V-Edges|G], [V0|Vs], V1, NG) :-
  187    compare(Res, V, V0),
  188    split_on_del_vertices(Res, V,Edges, [V0|Vs], NVs, V1, NG, NGr),
  189    del_vertices(G, NVs, V1, NGr).
  190
  191del_remaining_edges_for_vertices([], _, []).
  192del_remaining_edges_for_vertices([V0-Edges|G], V1, [V0-NEdges|NG]) :-
  193    ord_subtract(Edges, V1, NEdges),
  194    del_remaining_edges_for_vertices(G, V1, NG).
  195
  196split_on_del_vertices(<, V, Edges, Vs, Vs, V1, [V-NEdges|NG], NG) :-
  197    ord_subtract(Edges, V1, NEdges).
  198split_on_del_vertices(>, V, Edges, [_|Vs], Vs, V1, [V-NEdges|NG], NG) :-
  199    ord_subtract(Edges, V1, NEdges).
  200split_on_del_vertices(=, _, _, [_|Vs], Vs, _, NG, NG).

 add_edges(+Graph, +Edges, -NewGraph)

Unify NewGraph with a new graph obtained by adding the list of Edges to Graph. Example:

?- add_edges([1-[3,5],2-[4],3-[],4-[5],
              5-[],6-[],7-[],8-[]],
             [1-6,2-3,3-2,5-7,3-2,4-5],
             NL).
NL = [1-[3,5,6], 2-[3,4], 3-[2], 4-[5],
      5-[7], 6-[], 7-[], 8-[]]

  216add_edges(Graph, Edges, NewGraph) :-
  217    p_to_s_graph(Edges, G1),
  218    ugraph_union(Graph, G1, NewGraph).

 ugraph_union(+Graph1, +Graph2, -NewGraph)

NewGraph is the union of Graph1 and Graph2. Example:

?- ugraph_union([1-[2],2-[3]],[2-[4],3-[1,2,4]],L).
L = [1-[2], 2-[3,4], 3-[1,2,4]]

  229ugraph_union(Set1, [], Set1) :- !.
  230ugraph_union([], Set2, Set2) :- !.
  231ugraph_union([Head1-E1|Tail1], [Head2-E2|Tail2], Union) :-
  232    compare(Order, Head1, Head2),
  233    ugraph_union(Order, Head1-E1, Tail1, Head2-E2, Tail2, Union).
  234
  235ugraph_union(=, Head-E1, Tail1, _-E2, Tail2, [Head-Es|Union]) :-
  236    ord_union(E1, E2, Es),
  237    ugraph_union(Tail1, Tail2, Union).
  238ugraph_union(<, Head1, Tail1, Head2, Tail2, [Head1|Union]) :-
  239    ugraph_union(Tail1, [Head2|Tail2], Union).
  240ugraph_union(>, Head1, Tail1, Head2, Tail2, [Head2|Union]) :-
  241    ugraph_union([Head1|Tail1], Tail2, Union).

 del_edges(+Graph, +Edges, -NewGraph)

Unify NewGraph with a new graph obtained by removing the list of Edges from Graph. Notice that no vertices are deleted. Example:

?- del_edges([1-[3,5],2-[4],3-[],4-[5],5-[],6-[],7-[],8-[]],
             [1-6,2-3,3-2,5-7,3-2,4-5,1-3],
             NL).
NL = [1-[5],2-[4],3-[],4-[],5-[],6-[],7-[],8-[]]

  255del_edges(Graph, Edges, NewGraph) :-
  256    p_to_s_graph(Edges, G1),
  257    graph_subtract(Graph, G1, NewGraph).

 graph_subtract(+Set1, +Set2, ?Difference)

Is based on ord_subtract

  263graph_subtract(Set1, [], Set1) :- !.
  264graph_subtract([], _, []).
  265graph_subtract([Head1-E1|Tail1], [Head2-E2|Tail2], Difference) :-
  266    compare(Order, Head1, Head2),
  267    graph_subtract(Order, Head1-E1, Tail1, Head2-E2, Tail2, Difference).
  268
  269graph_subtract(=, H-E1,     Tail1, _-E2,     Tail2, [H-E|Difference]) :-
  270    ord_subtract(E1,E2,E),
  271    graph_subtract(Tail1, Tail2, Difference).
  272graph_subtract(<, Head1, Tail1, Head2, Tail2, [Head1|Difference]) :-
  273    graph_subtract(Tail1, [Head2|Tail2], Difference).
  274graph_subtract(>, Head1, Tail1, _,     Tail2, Difference) :-
  275    graph_subtract([Head1|Tail1], Tail2, Difference).

 edges(+Graph, -Edges)

Unify Edges with all edges appearing in Graph. Example:

?- edges([1-[3,5],2-[4],3-[],4-[5],5-[]], L).
L = [1-3, 1-5, 2-4, 4-5]

  284edges(Graph, Edges) :-
  285    s_to_p_graph(Graph, Edges).
  286
  287p_to_s_graph(P_Graph, S_Graph) :-
  288    sort(P_Graph, EdgeSet),
  289    p_to_s_vertices(EdgeSet, VertexBag),
  290    sort(VertexBag, VertexSet),
  291    p_to_s_group(VertexSet, EdgeSet, S_Graph).
  292
  293
  294p_to_s_vertices([], []).
  295p_to_s_vertices([A-Z|Edges], [A,Z|Vertices]) :-
  296    p_to_s_vertices(Edges, Vertices).
  297
  298
  299p_to_s_group([], _, []).
  300p_to_s_group([Vertex|Vertices], EdgeSet, [Vertex-Neibs|G]) :-
  301    p_to_s_group(EdgeSet, Vertex, Neibs, RestEdges),
  302    p_to_s_group(Vertices, RestEdges, G).
  303
  304
  305p_to_s_group([V1-X|Edges], V2, [X|Neibs], RestEdges) :- V1 == V2,
  306    !,
  307    p_to_s_group(Edges, V2, Neibs, RestEdges).
  308p_to_s_group(Edges, _, [], Edges).
  309
  310
  311
  312s_to_p_graph([], []) :- !.
  313s_to_p_graph([Vertex-Neibs|G], P_Graph) :-
  314    s_to_p_graph(Neibs, Vertex, P_Graph, Rest_P_Graph),
  315    s_to_p_graph(G, Rest_P_Graph).
  316
  317
  318s_to_p_graph([], _, P_Graph, P_Graph) :- !.
  319s_to_p_graph([Neib|Neibs], Vertex, [Vertex-Neib|P], Rest_P) :-
  320    s_to_p_graph(Neibs, Vertex, P, Rest_P).

 transitive_closure(+Graph, -Closure)

Generate the graph Closure as the transitive closure of Graph. Example:

?- transitive_closure([1-[2,3],2-[4,5],4-[6]],L).
L = [1-[2,3,4,5,6], 2-[4,5,6], 4-[6]]

  332transitive_closure(Graph, Closure) :-
  333    warshall(Graph, Graph, Closure).
  334
  335warshall([], Closure, Closure) :- !.
  336warshall([V-_|G], E, Closure) :-
  337    memberchk(V-Y, E),      %  Y := E(v)
  338    warshall(E, V, Y, NewE),
  339    warshall(G, NewE, Closure).
  340
  341
  342warshall([X-Neibs|G], V, Y, [X-NewNeibs|NewG]) :-
  343    memberchk(V, Neibs),
  344    !,
  345    ord_union(Neibs, Y, NewNeibs),
  346    warshall(G, V, Y, NewG).
  347warshall([X-Neibs|G], V, Y, [X-Neibs|NewG]) :-
  348    !,
  349    warshall(G, V, Y, NewG).
  350warshall([], _, _, []).

 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-[]]

Compatibility

- 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.

  370transpose_ugraph(Graph, NewGraph) :-
  371    edges(Graph, Edges),
  372    vertices(Graph, Vertices),
  373    flip_edges(Edges, TransposedEdges),
  374    vertices_edges_to_ugraph(Vertices, TransposedEdges, NewGraph).
  375
  376flip_edges([], []).
  377flip_edges([Key-Val|Pairs], [Val-Key|Flipped]) :-
  378    flip_edges(Pairs, Flipped).

 compose(+LeftGraph, +RightGraph, -NewGraph)

Compose NewGraph by connecting the drains of LeftGraph to the sources of RightGraph. Example:

?- compose([1-[2],2-[3]],[2-[4],3-[1,2,4]],L).
L = [1-[4], 2-[1,2,4], 3-[]]

  388compose(G1, G2, Composition) :-
  389    vertices(G1, V1),
  390    vertices(G2, V2),
  391    ord_union(V1, V2, V),
  392    compose(V, G1, G2, Composition).
  393
  394compose([], _, _, []) :- !.
  395compose([Vertex|Vertices], [Vertex-Neibs|G1], G2,
  396        [Vertex-Comp|Composition]) :-
  397    !,
  398    compose1(Neibs, G2, [], Comp),
  399    compose(Vertices, G1, G2, Composition).
  400compose([Vertex|Vertices], G1, G2, [Vertex-[]|Composition]) :-
  401    compose(Vertices, G1, G2, Composition).
  402
  403
  404compose1([V1|Vs1], [V2-N2|G2], SoFar, Comp) :-
  405    compare(Rel, V1, V2),
  406    !,
  407    compose1(Rel, V1, Vs1, V2, N2, G2, SoFar, Comp).
  408compose1(_, _, Comp, Comp).
  409
  410
  411compose1(<, _, Vs1, V2, N2, G2, SoFar, Comp) :-
  412    !,
  413    compose1(Vs1, [V2-N2|G2], SoFar, Comp).
  414compose1(>, V1, Vs1, _, _, G2, SoFar, Comp) :-
  415    !,
  416    compose1([V1|Vs1], G2, SoFar, Comp).
  417compose1(=, V1, Vs1, V1, N2, G2, SoFar, Comp) :-
  418    ord_union(N2, SoFar, Next),
  419    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.

  437top_sort(Graph, Sorted) :-
  438    vertices_and_zeros(Graph, Vertices, Counts0),
  439    count_edges(Graph, Vertices, Counts0, Counts1),
  440    select_zeros(Counts1, Vertices, Zeros),
  441    top_sort(Zeros, Sorted, Graph, Vertices, Counts1).
  442
  443top_sort(Graph, Sorted0, Sorted) :-
  444    vertices_and_zeros(Graph, Vertices, Counts0),
  445    count_edges(Graph, Vertices, Counts0, Counts1),
  446    select_zeros(Counts1, Vertices, Zeros),
  447    top_sort(Zeros, Sorted, Sorted0, Graph, Vertices, Counts1).
  448
  449
  450vertices_and_zeros([], [], []) :- !.
  451vertices_and_zeros([Vertex-_|Graph], [Vertex|Vertices], [0|Zeros]) :-
  452    vertices_and_zeros(Graph, Vertices, Zeros).
  453
  454
  455count_edges([], _, Counts, Counts) :- !.
  456count_edges([_-Neibs|Graph], Vertices, Counts0, Counts2) :-
  457    incr_list(Neibs, Vertices, Counts0, Counts1),
  458    count_edges(Graph, Vertices, Counts1, Counts2).
  459
  460
  461incr_list([], _, Counts, Counts) :- !.
  462incr_list([V1|Neibs], [V2|Vertices], [M|Counts0], [N|Counts1]) :-
  463    V1 == V2,
  464    !,
  465    N is M+1,
  466    incr_list(Neibs, Vertices, Counts0, Counts1).
  467incr_list(Neibs, [_|Vertices], [N|Counts0], [N|Counts1]) :-
  468    incr_list(Neibs, Vertices, Counts0, Counts1).
  469
  470
  471select_zeros([], [], []) :- !.
  472select_zeros([0|Counts], [Vertex|Vertices], [Vertex|Zeros]) :-
  473    !,
  474    select_zeros(Counts, Vertices, Zeros).
  475select_zeros([_|Counts], [_|Vertices], Zeros) :-
  476    select_zeros(Counts, Vertices, Zeros).
  477
  478
  479
  480top_sort([], [], Graph, _, Counts) :-
  481    !,
  482    vertices_and_zeros(Graph, _, Counts).
  483top_sort([Zero|Zeros], [Zero|Sorted], Graph, Vertices, Counts1) :-
  484    graph_memberchk(Zero-Neibs, Graph),
  485    decr_list(Neibs, Vertices, Counts1, Counts2, Zeros, NewZeros),
  486    top_sort(NewZeros, Sorted, Graph, Vertices, Counts2).
  487
  488top_sort([], Sorted0, Sorted0, Graph, _, Counts) :-
  489    !,
  490    vertices_and_zeros(Graph, _, Counts).
  491top_sort([Zero|Zeros], [Zero|Sorted], Sorted0, Graph, Vertices, Counts1) :-
  492    graph_memberchk(Zero-Neibs, Graph),
  493    decr_list(Neibs, Vertices, Counts1, Counts2, Zeros, NewZeros),
  494    top_sort(NewZeros, Sorted, Sorted0, Graph, Vertices, Counts2).
  495
  496graph_memberchk(Element1-Edges, [Element2-Edges2|_]) :-
  497    Element1 == Element2,
  498    !,
  499    Edges = Edges2.
  500graph_memberchk(Element, [_|Rest]) :-
  501    graph_memberchk(Element, Rest).
  502
  503
  504decr_list([], _, Counts, Counts, Zeros, Zeros) :- !.
  505decr_list([V1|Neibs], [V2|Vertices], [1|Counts1], [0|Counts2], Zi, Zo) :-
  506    V1 == V2,
  507    !,
  508    decr_list(Neibs, Vertices, Counts1, Counts2, [V2|Zi], Zo).
  509decr_list([V1|Neibs], [V2|Vertices], [N|Counts1], [M|Counts2], Zi, Zo) :-
  510    V1 == V2,
  511    !,
  512    M is N-1,
  513    decr_list(Neibs, Vertices, Counts1, Counts2, Zi, Zo).
  514decr_list(Neibs, [_|Vertices], [N|Counts1], [N|Counts2], Zi, Zo) :-
  515    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. Example:

?- neighbours(4,[1-[3,5],2-[4],3-[],
                 4-[1,2,7,5],5-[],6-[],7-[],8-[]], NL).
NL = [1,2,7,5]

  530neighbors(Vertex, Graph, Neig) :-
  531    neighbours(Vertex, Graph, Neig).
  532
  533neighbours(V,[V0-Neig|_],Neig) :-
  534    V == V0,
  535    !.
  536neighbours(V,[_|G],Neig) :-
  537    neighbours(V,G,Neig).

 connect_ugraph(+UGraphIn, -Start, -UGraphOut) is det

Adds Start as an additional vertex that is connected to all vertices in UGraphIn. This can be used to create an topological sort for a not connected graph. Start is before any vertex in UGraphIn in the standard order of terms. No vertex in UGraphIn can be a variable.

Can be used to order a not-connected graph as follows:

top_sort_unconnected(Graph, Vertices) :-
    (   top_sort(Graph, Vertices)
    ->  true
    ;   connect_ugraph(Graph, Start, Connected),
        top_sort(Connected, Ordered0),
        Ordered0 = [Start|Vertices]
    ).

  559connect_ugraph([], 0, []) :- !.
  560connect_ugraph(Graph, Start, [Start-Vertices|Graph]) :-
  561    vertices(Graph, Vertices),
  562    Vertices = [First|_],
  563    before(First, Start).

 before(+Term, -Before) is det

Unify Before to a term that comes before Term in the standard order of terms.

Errors

- instantiation_error if Term is unbound.

  572before(X, _) :-
  573    var(X),
  574    !,
  575    instantiation_error(X).
  576before(Number, Start) :-
  577    number(Number),
  578    !,
  579    Start is Number - 1.
  580before(_, 0).

 complement(+UGraphIn, -UGraphOut)

UGraphOut is a ugraph with an edge between all vertices that are not connected in UGraphIn and all edges from UGraphIn removed. Example:

?- complement([1-[3,5],2-[4],3-[],
               4-[1,2,7,5],5-[],6-[],7-[],8-[]], NL).
NL = [1-[2,4,6,7,8],2-[1,3,5,6,7,8],3-[1,2,4,5,6,7,8],
      4-[3,5,6,8],5-[1,2,3,4,6,7,8],6-[1,2,3,4,5,7,8],
      7-[1,2,3,4,5,6,8],8-[1,2,3,4,5,6,7]]

To be done

- Simple two-step algorithm. You could be smarter, I suppose.

  599complement(G, NG) :-
  600    vertices(G,Vs),
  601    complement(G,Vs,NG).
  602
  603complement([], _, []).
  604complement([V-Ns|G], Vs, [V-INs|NG]) :-
  605    ord_add_element(Ns,V,Ns1),
  606    ord_subtract(Vs,Ns1,INs),
  607    complement(G, Vs, NG).

 reachable(+Vertex, +UGraph, -Vertices)

True when Vertices is an ordered set of vertices reachable in UGraph, including Vertex. Example:

?- reachable(1,[1-[3,5],2-[4],3-[],4-[5],5-[]],V).
V = [1, 3, 5]

  617reachable(N, G, Rs) :-
  618    reachable([N], G, [N], Rs).
  619
  620reachable([], _, Rs, Rs).
  621reachable([N|Ns], G, Rs0, RsF) :-
  622    neighbours(N, G, Nei),
  623    ord_union(Rs0, Nei, Rs1, D),
  624    append(Ns, D, Nsi),
  625    reachable(Nsi, G, Rs1, RsF)