/* 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) 1984-2020, VU University Amsterdam CWI, 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 connect_ugraph/3 % +Graph1, -Start, -Graph ]). /** 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 */ :- autoload(library(lists),[append/3]). :- autoload(library(ordsets), [ord_subtract/3,ord_union/3,ord_add_element/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). %! 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] % ). % ``` connect_ugraph([], 0, []) :- !. connect_ugraph(Graph, Start, [Start-Vertices|Graph]) :- vertices(Graph, Vertices), Vertices = [First|_], before(First, Start). %! before(+Term, -Before) is det. % % Unify Before to a term that comes before Term in the standard % order of terms. % % @error instantiation_error if Term is unbound. before(X, _) :- var(X), !, instantiation_error(X). before(Number, Start) :- number(Number), !, Start is Number - 1. before(_, 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. % % @tbd 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(+Vertex, +UGraph, -Vertices) % % True when Vertices is an ordered set of vertices reachable in % UGraph, including Vertex. 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).