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21 digraph

%--------------------------------------------------%
% vim: ft=mercury ts=4 sw=4 et
%--------------------------------------------------%
% Copyright (C) 1995-1999,2002-2007,2010-2012 The University of Melbourne.
% This file may only be copied under the terms of the GNU Library General
% Public License - see the file COPYING.LIB in the Mercury distribution.
%--------------------------------------------------%
%
% File: digraph.m
% Main author: bromage, petdr
% Stability: medium
%
% This module defines a data type representing directed graphs. A directed
% graph of type digraph(T) is logically equivalent to a set of vertices of
% type T, and a set of edges of type pair(T). The endpoints of each edge
% must be included in the set of vertices; cycles and loops are allowed.
%
%--------------------------------------------------%
%--------------------------------------------------%

:- module digraph.
:- interface.

:- import_module assoc_list.
:- import_module enum.
:- import_module list.
:- import_module map.
:- import_module pair.
:- import_module set.
:- import_module sparse_bitset.

%--------------------------------------------------%

    % The type of directed graphs with vertices in T.
    %
:- type digraph(T).

    % The abstract type that indexes vertices in a digraph. Each key is only
    % valid with the digraph it was created from -- predicates and functions
    % in this module may throw an exception if an invalid key is used.
    %
:- type digraph_key(T).

:- instance enum(digraph_key(T)).

:- type digraph_key_set(T) == sparse_bitset(digraph_key(T)).

    % init creates an empty digraph.
    %
:- func init = digraph(T).
:- pred init(digraph(T)::out) is det.

    % add_vertex adds a vertex to the domain of a digraph.
    % Returns the old key if one already exists for this vertex,
    % otherwise it allocates a new key.
    %
:- pred add_vertex(T::in, digraph_key(T)::out,
    digraph(T)::in, digraph(T)::out) is det.

    % search_key returns the key associated with a vertex.
    % Fails if the vertex is not in the graph.
    %
:- pred search_key(digraph(T)::in, T::in, digraph_key(T)::out) is semidet.

    % lookup_key returns the key associated with a vertex.
    % Aborts if the vertex is not in the graph.
    %
:- func lookup_key(digraph(T), T) = digraph_key(T).
:- pred lookup_key(digraph(T)::in, T::in, digraph_key(T)::out) is det.

    % lookup_vertex returns the vertex associated with a key.
    %
:- func lookup_vertex(digraph(T), digraph_key(T)) = T.
:- pred lookup_vertex(digraph(T)::in, digraph_key(T)::in, T::out) is det.

    % add_edge adds an edge to the digraph if it doesn't already
    % exist, and leaves the digraph unchanged otherwise.
    %
:- func add_edge(digraph_key(T), digraph_key(T), digraph(T)) = digraph(T).
:- pred add_edge(digraph_key(T)::in, digraph_key(T)::in,
    digraph(T)::in, digraph(T)::out) is det.

    % add_vertices_and_edge adds a pair of vertices and an edge
    % between them to the digraph.
    %
    % add_vertices_and_edge(X, Y, !G) :-
    %    add_vertex(X, XKey, !G),
    %    add_vertex(Y, YKey, !G),
    %    add_edge(XKey, YKey, !G).
    %
:- func add_vertices_and_edge(T, T, digraph(T)) = digraph(T).
:- pred add_vertices_and_edge(T::in, T::in,
    digraph(T)::in, digraph(T)::out) is det.

    % As above, but takes a pair of vertices in a single argument.
    %
:- func add_vertex_pair(pair(T), digraph(T)) = digraph(T).
:- pred add_vertex_pair(pair(T)::in, digraph(T)::in, digraph(T)::out) is det.

    % add_assoc_list adds a list of edges to a digraph.
    %
:- func add_assoc_list(assoc_list(digraph_key(T), digraph_key(T)),
    digraph(T)) = digraph(T).
:- pred add_assoc_list(assoc_list(digraph_key(T), digraph_key(T))::in,
    digraph(T)::in, digraph(T)::out) is det.

    % delete_edge deletes an edge from the digraph if it exists,
    % and leaves the digraph unchanged otherwise.
    %
:- func delete_edge(digraph_key(T), digraph_key(T), digraph(T)) = digraph(T).
:- pred delete_edge(digraph_key(T)::in, digraph_key(T)::in,
    digraph(T)::in, digraph(T)::out) is det.

    % delete_assoc_list deletes a list of edges from a digraph.
    %
:- func delete_assoc_list(assoc_list(digraph_key(T), digraph_key(T)),
    digraph(T)) = digraph(T).
:- pred delete_assoc_list(
    assoc_list(digraph_key(T), digraph_key(T))::in,
    digraph(T)::in, digraph(T)::out) is det.

    % is_edge checks to see if an edge is in the digraph.
    %
:- pred is_edge(digraph(T), digraph_key(T), digraph_key(T)).
:- mode is_edge(in, in, out) is nondet.
:- mode is_edge(in, in, in) is semidet.

    % is_edge_rev is equivalent to is_edge, except that
    % the nondet mode works in the reverse direction.
    %
:- pred is_edge_rev(digraph(T), digraph_key(T), digraph_key(T)).
:- mode is_edge_rev(in, out, in) is nondet.
:- mode is_edge_rev(in, in, in) is semidet.

    % Given key x, lookup_from returns the set of keys y such that
    % there is an edge (x,y) in the digraph.
    %
:- func lookup_from(digraph(T), digraph_key(T)) = set(digraph_key(T)).
:- pred lookup_from(digraph(T)::in, digraph_key(T)::in,
    set(digraph_key(T))::out) is det.

    % As above, but returns a digraph_key_set.
    %
:- func lookup_key_set_from(digraph(T), digraph_key(T)) = digraph_key_set(T).
:- pred lookup_key_set_from(digraph(T)::in, digraph_key(T)::in,
    digraph_key_set(T)::out) is det.

    % Given a key y, lookup_to returns the set of keys x such that
    % there is an edge (x,y) in the digraph.
    %
:- func lookup_to(digraph(T), digraph_key(T)) = set(digraph_key(T)).
:- pred lookup_to(digraph(T)::in, digraph_key(T)::in,
    set(digraph_key(T))::out) is det.

    % As above, but returns a digraph_key_set.
    %
:- func lookup_key_set_to(digraph(T), digraph_key(T)) = digraph_key_set(T).
:- pred lookup_key_set_to(digraph(T)::in, digraph_key(T)::in,
    digraph_key_set(T)::out) is det.

%--------------------------------------------------%

    % to_assoc_list turns a digraph into a list of pairs of vertices,
    % one for each edge.
    %
:- func to_assoc_list(digraph(T)) = assoc_list(T, T).
:- pred to_assoc_list(digraph(T)::in, assoc_list(T, T)::out) is det.

    % to_key_assoc_list turns a digraph into a list of pairs of keys,
    % one for each edge.
    %
:- func to_key_assoc_list(digraph(T)) =
    assoc_list(digraph_key(T), digraph_key(T)).
:- pred to_key_assoc_list(digraph(T)::in,
    assoc_list(digraph_key(T), digraph_key(T))::out) is det.

    % from_assoc_list turns a list of pairs of vertices into a digraph.
    %
:- func from_assoc_list(assoc_list(T, T)) = digraph(T).
:- pred from_assoc_list(assoc_list(T, T)::in, digraph(T)::out) is det.

%--------------------------------------------------%

    % dfs(G, Key, Dfs) is true if Dfs is a depth-first sorting of G
    % starting at Key. The set of keys in the list Dfs is equal to the
    % set of keys reachable from Key.
    %
:- func dfs(digraph(T), digraph_key(T)) = list(digraph_key(T)).
:- pred dfs(digraph(T)::in, digraph_key(T)::in,
    list(digraph_key(T))::out) is det.

    % dfsrev(G, Key, DfsRev) is true if DfsRev is a reverse
    % depth-first sorting of G starting at Key. The set of keys in the
    % list DfsRev is equal to the set of keys reachable from Key.
    %
:- func dfsrev(digraph(T), digraph_key(T)) = list(digraph_key(T)).
:- pred dfsrev(digraph(T)::in, digraph_key(T)::in,
    list(digraph_key(T))::out) is det.

    % dfs(G, Dfs) is true if Dfs is a depth-first sorting of G.
    % If one considers each edge to point from a parent node to a child node,
    % then Dfs will be a list of all the keys in G such that all keys for
    % the children of a vertex are placed in the list before the parent key.
    %
    % If the digraph is cyclic, the position in which cycles are broken
    % (that is, in which a child is placed *after* its parent) is undefined.
    %
:- func dfs(digraph(T)) = list(digraph_key(T)).
:- pred dfs(digraph(T)::in, list(digraph_key(T))::out) is det.

    % dfsrev(G, DfsRev) is true if DfsRev is a reverse depth-first
    % sorting of G. That is, DfsRev is the reverse of Dfs from dfs/2.
    %
:- func dfsrev(digraph(T)) = list(digraph_key(T)).
:- pred dfsrev(digraph(T)::in, list(digraph_key(T))::out) is det.

    % dfs(G, Key, !Visit, Dfs) is true if Dfs is a depth-first
    % sorting of G starting at Key, assuming we have already visited !.Visit
    % vertices. That is, Dfs is a list of vertices such that all the
    % unvisited children of a vertex are placed in the list before the
    % parent. !.Visit allows us to initialise a set of previously visited
    % vertices. !:Visit is Dfs + !.Visit.
    %
:- pred dfs(digraph(T)::in, digraph_key(T)::in, digraph_key_set(T)::in,
    digraph_key_set(T)::out, list(digraph_key(T))::out) is det.

    % dfsrev(G, Key, !Visit, DfsRev) is true if DfsRev is a
    % reverse depth-first sorting of G starting at Key providing we have
    % already visited !.Visit nodes, ie the reverse of Dfs from dfs/5.
    % !:Visit is !.Visit + DfsRev.
    %
:- pred dfsrev(digraph(T)::in, digraph_key(T)::in,
    digraph_key_set(T)::in, digraph_key_set(T)::out,
    list(digraph_key(T))::out) is det.

%--------------------------------------------------%

    % vertices returns the set of vertices in a digraph.
    %
:- func vertices(digraph(T)) = set(T).
:- pred vertices(digraph(T)::in, set(T)::out) is det.

    % inverse(G, G') is true iff the domains of G and G' are equal,
    % and for all x, y in this domain, (x,y) is an edge in G iff (y,x) is
    % an edge in G'.
    %
:- func inverse(digraph(T)) = digraph(T).
:- pred inverse(digraph(T)::in, digraph(T)::out) is det.

    % compose(G1, G2, G) is true if G is the composition
    % of the digraphs G1 and G2. That is, there is an edge (x,y) in G iff
    % there exists vertex m such that (x,m) is in G1 and (m,y) is in G2.
    %
:- func compose(digraph(T), digraph(T)) = digraph(T).
:- pred compose(digraph(T)::in, digraph(T)::in, digraph(T)::out)
    is det.

    % is_dag(G) is true iff G is a directed acyclic graph.
    %
:- pred is_dag(digraph(T)::in) is semidet.

    % components(G, Comp) is true if Comp is the set of the
    % connected components of G.
    %
:- func components(digraph(T)) = set(set(digraph_key(T))).
:- pred components(digraph(T)::in, set(set(digraph_key(T)))::out)
    is det.

    % cliques(G, Cliques) is true if Cliques is the set of the
    % cliques (strongly connected components) of G.
    %
:- func cliques(digraph(T)) = set(set(digraph_key(T))).
:- pred cliques(digraph(T)::in, set(set(digraph_key(T)))::out) is det.

    % reduced(G, R) is true if R is the reduced digraph (digraph of cliques)
    % obtained from G.
    %
:- func reduced(digraph(T)) = digraph(set(T)).
:- pred reduced(digraph(T)::in, digraph(set(T))::out) is det.

    % As above, but also return a map from each key in the original digraph
    % to the key for its clique in the reduced digraph.
    %
:- pred reduced(digraph(T)::in, digraph(set(T))::out,
    map(digraph_key(T), digraph_key(set(T)))::out) is det.

    % tsort(G, TS) is true if TS is a topological sorting of G.
    %
    % If we view each edge in the digraph as representing a <from, to>
    % relationship, then TS will contain a vertex "from" *before*
    % all the other vertices "to" for which a <from, to> edge exists
    % in the graph. In other words, TS will be in from-to order.
    %
    % tsort fails if G is cyclic.
    %
:- pred tsort(digraph(T)::in, list(T)::out) is semidet.

    % Both these predicates do a topological sort of G.
    %
    % return_vertices_in_from_to_order(G, TS) is a synonym for tsort(G, TS).
    % return_vertices_in_to_from_order(G, TS) is identical to both
    % except for the fact that it returns the vertices in the opposite order.
    %
:- pred return_vertices_in_from_to_order(digraph(T)::in, list(T)::out)
    is semidet.
:- pred return_vertices_in_to_from_order(digraph(T)::in, list(T)::out)
    is semidet.

    % atsort(G, ATS) is true if ATS is a topological sorting
    % of the strongly connected components (SCCs) in G.
    %
    % If we view each edge in the digraph as representing a <from, to>
    % relationship, then ATS will contain SCC A before all SCCs B
    % for which there is a vertex <from, to> with "from" being in SCC A
    % and "to" being in SCC B. In other words, ATS will be in from-to order.
    %
:- func atsort(digraph(T)) = list(set(T)).
:- pred atsort(digraph(T)::in, list(set(T))::out) is det.

    % Both these predicates do a topological sort of the strongly connected
    % components (SCCs) of G.
    %
    % return_sccs_in_from_to_order(G) = ATS is a synonym for atsort(G) = ATS.
    % return_sccs_in_to_from_order(G) = ATS is identical to both
    % except for the fact that it returns the SCCs in the opposite order.
    %
:- func return_sccs_in_from_to_order(digraph(T)) = list(set(T)).
:- func return_sccs_in_to_from_order(digraph(T)) = list(set(T)).

    % sc(G, SC) is true if SC is the symmetric closure of G.
    % That is, (x,y) is in SC iff either (x,y) or (y,x) is in G.
    %
:- func sc(digraph(T)) = digraph(T).
:- pred sc(digraph(T)::in, digraph(T)::out) is det.

    % tc(G, TC) is true if TC is the transitive closure of G.
    %
:- func tc(digraph(T)) = digraph(T).
:- pred tc(digraph(T)::in, digraph(T)::out) is det.

    % rtc(G, RTC) is true if RTC is the reflexive transitive closure of G.
    %
:- func rtc(digraph(T)) = digraph(T).
:- pred rtc(digraph(T)::in, digraph(T)::out) is det.

    % traverse(G, ProcessVertex, ProcessEdge) will traverse a digraph
    % calling ProcessVertex for each vertex in the digraph and ProcessEdge for
    % each edge in the digraph. Each vertex is processed followed by all the
    % edges originating at that vertex, until all vertices have been processed.
    %
:- pred traverse(digraph(T), pred(T, A, A), pred(T, T, A, A), A, A).
:- mode traverse(in, pred(in, di, uo) is det,
    pred(in, in, di, uo) is det, di, uo) is det.
:- mode traverse(in, pred(in, in, out) is det,
    pred(in, in, in, out) is det, in, out) is det.

%--------------------------------------------------%
%--------------------------------------------------%


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