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NAME
digraph - Directed Graphs
DESCRIPTION
The digraph module implements a version of labeled directed graphs. What makes the graphs
implemented here non-proper directed graphs is that multiple edges between vertices are
allowed. However, the customary definition of directed graphs will be used in the text
that follows.
A directed graph (or just "digraph") is a pair (V, E) of a finite set V of vertices and a
finite set E of directed edges (or just "edges"). The set of edges E is a subset of V x V
(the Cartesian product of V with itself). In this module, V is allowed to be empty; the so
obtained unique digraph is called the empty digraph. Both vertices and edges are
represented by unique Erlang terms.
Digraphs can be annotated with additional information. Such information may be attached to
the vertices and to the edges of the digraph. A digraph which has been annotated is called
a labeled digraph, and the information attached to a vertex or an edge is called a label.
Labels are Erlang terms.
An edge e = (v, w) is said to emanate from vertex v and to be incident on vertex w. The
out-degree of a vertex is the number of edges emanating from that vertex. The in-degree of
a vertex is the number of edges incident on that vertex. If there is an edge emanating
from v and incident on w, then w is said to be an out-neighbour of v, and v is said to be
an in-neighbour of w. A path P from v[1] to v[k] in a digraph (V, E) is a non-empty
sequence v[1], v[2], ..., v[k] of vertices in V such that there is an edge (v[i],v[i+1])
in E for 1 <= i < k. The length of the path P is k-1. P is simple if all vertices are
distinct, except that the first and the last vertices may be the same. P is a cycle if the
length of P is not zero and v[1] = v[k]. A loop is a cycle of length one. A simple cycle
is a path that is both a cycle and simple. An acyclic digraph is a digraph that has no
cycles.
DATA TYPES
d_type() = d_cyclicity() | d_protection()
d_cyclicity() = acyclic | cyclic
d_protection() = private | protected
digraph()
A digraph as returned by new/0,1.
edge()
label() = term()
vertex()
EXPORTS
add_edge(G, V1, V2) -> edge() | {error, add_edge_err_rsn()}
add_edge(G, V1, V2, Label) -> edge() | {error, add_edge_err_rsn()}
add_edge(G, E, V1, V2, Label) ->
edge() | {error, add_edge_err_rsn()}
Types:
G = digraph()
E = edge()
V1 = V2 = vertex()
Label = label()
add_edge_err_rsn() = {bad_edge, Path :: [vertex()]}
| {bad_vertex, V :: vertex()}
add_edge/5 creates (or modifies) the edge E of the digraph G, using Label as the
(new) label of the edge. The edge is emanating from V1 and incident on V2. Returns
E.
add_edge(G, V1, V2, Label) is equivalent to add_edge(G, E, V1, V2, Label), where E
is a created edge. The created edge is represented by the term ['$e' | N], where N
is an integer >= 0.
add_edge(G, V1, V2) is equivalent to add_edge(G, V1, V2, []).
If the edge would create a cycle in an acyclic digraph, then {error, {bad_edge,
Path}} is returned. If either of V1 or V2 is not a vertex of the digraph G, then
{error, {bad_vertex, V}} is returned, V = V1 or V = V2.
add_vertex(G) -> vertex()
add_vertex(G, V) -> vertex()
add_vertex(G, V, Label) -> vertex()
Types:
G = digraph()
V = vertex()
Label = label()
add_vertex/3 creates (or modifies) the vertex V of the digraph G, using Label as
the (new) label of the vertex. Returns V.
add_vertex(G, V) is equivalent to add_vertex(G, V, []).
add_vertex/1 creates a vertex using the empty list as label, and returns the
created vertex. The created vertex is represented by the term ['$v' | N], where N
is an integer >= 0.
del_edge(G, E) -> true
Types:
G = digraph()
E = edge()
Deletes the edge E from the digraph G.
del_edges(G, Edges) -> true
Types:
G = digraph()
Edges = [edge()]
Deletes the edges in the list Edges from the digraph G.
del_path(G, V1, V2) -> true
Types:
G = digraph()
V1 = V2 = vertex()
Deletes edges from the digraph G until there are no paths from the vertex V1 to the
vertex V2.
A sketch of the procedure employed: Find an arbitrary simple path v[1], v[2], ...,
v[k] from V1 to V2 in G. Remove all edges of G emanating from v[i] and incident to
v[i+1] for 1 <= i < k (including multiple edges). Repeat until there is no path
between V1 and V2.
del_vertex(G, V) -> true
Types:
G = digraph()
V = vertex()
Deletes the vertex V from the digraph G. Any edges emanating from V or incident on
V are also deleted.
del_vertices(G, Vertices) -> true
Types:
G = digraph()
Vertices = [vertex()]
Deletes the vertices in the list Vertices from the digraph G.
delete(G) -> true
Types:
G = digraph()
Deletes the digraph G. This call is important because digraphs are implemented with
ETS. There is no garbage collection of ETS tables. The digraph will, however, be
deleted if the process that created the digraph terminates.
edge(G, E) -> {E, V1, V2, Label} | false
Types:
G = digraph()
E = edge()
V1 = V2 = vertex()
Label = label()
Returns {E, V1, V2, Label} where Label is the label of the edge E emanating from V1
and incident on V2 of the digraph G. If there is no edge E of the digraph G, then
false is returned.
edges(G) -> Edges
Types:
G = digraph()
Edges = [edge()]
Returns a list of all edges of the digraph G, in some unspecified order.
edges(G, V) -> Edges
Types:
G = digraph()
V = vertex()
Edges = [edge()]
Returns a list of all edges emanating from or incident on V of the digraph G, in
some unspecified order.
get_cycle(G, V) -> Vertices | false
Types:
G = digraph()
V = vertex()
Vertices = [vertex(), ...]
If there is a simple cycle of length two or more through the vertex V, then the
cycle is returned as a list [V, ..., V] of vertices, otherwise if there is a loop
through V, then the loop is returned as a list [V]. If there are no cycles through
V, then false is returned.
get_path/3 is used for finding a simple cycle through V.
get_path(G, V1, V2) -> Vertices | false
Types:
G = digraph()
V1 = V2 = vertex()
Vertices = [vertex(), ...]
Tries to find a simple path from the vertex V1 to the vertex V2 of the digraph G.
Returns the path as a list [V1, ..., V2] of vertices, or false if no simple path
from V1 to V2 of length one or more exists.
The digraph G is traversed in a depth-first manner, and the first path found is
returned.
get_short_cycle(G, V) -> Vertices | false
Types:
G = digraph()
V = vertex()
Vertices = [vertex(), ...]
Tries to find an as short as possible simple cycle through the vertex V of the
digraph G. Returns the cycle as a list [V, ..., V] of vertices, or false if no
simple cycle through V exists. Note that a loop through V is returned as the list
[V, V].
get_short_path/3 is used for finding a simple cycle through V.
get_short_path(G, V1, V2) -> Vertices | false
Types:
G = digraph()
V1 = V2 = vertex()
Vertices = [vertex(), ...]
Tries to find an as short as possible simple path from the vertex V1 to the vertex
V2 of the digraph G. Returns the path as a list [V1, ..., V2] of vertices, or false
if no simple path from V1 to V2 of length one or more exists.
The digraph G is traversed in a breadth-first manner, and the first path found is
returned.
in_degree(G, V) -> integer() >= 0
Types:
G = digraph()
V = vertex()
Returns the in-degree of the vertex V of the digraph G.
in_edges(G, V) -> Edges
Types:
G = digraph()
V = vertex()
Edges = [edge()]
Returns a list of all edges incident on V of the digraph G, in some unspecified
order.
in_neighbours(G, V) -> Vertex
Types:
G = digraph()
V = vertex()
Vertex = [vertex()]
Returns a list of all in-neighbours of V of the digraph G, in some unspecified
order.
info(G) -> InfoList
Types:
G = digraph()
InfoList =
[{cyclicity, Cyclicity :: d_cyclicity()} |
{memory, NoWords :: integer() >= 0} |
{protection, Protection :: d_protection()}]
d_cyclicity() = acyclic | cyclic
d_protection() = private | protected
Returns a list of {Tag, Value} pairs describing the digraph G. The following pairs
are returned:
* {cyclicity, Cyclicity}, where Cyclicity is cyclic or acyclic, according to the
options given to new.
* {memory, NoWords}, where NoWords is the number of words allocated to the ETS
tables.
* {protection, Protection}, where Protection is protected or private, according
to the options given to new.
new() -> digraph()
Equivalent to new([]).
new(Type) -> digraph()
Types:
Type = [d_type()]
d_type() = d_cyclicity() | d_protection()
d_cyclicity() = acyclic | cyclic
d_protection() = private | protected
Returns an empty digraph with properties according to the options in Type:
cyclic:
Allow cycles in the digraph (default).
acyclic:
The digraph is to be kept acyclic.
protected:
Other processes can read the digraph (default).
private:
The digraph can be read and modified by the creating process only.
If an unrecognized type option T is given or Type is not a proper list, there will
be a badarg exception.
no_edges(G) -> integer() >= 0
Types:
G = digraph()
Returns the number of edges of the digraph G.
no_vertices(G) -> integer() >= 0
Types:
G = digraph()
Returns the number of vertices of the digraph G.
out_degree(G, V) -> integer() >= 0
Types:
G = digraph()
V = vertex()
Returns the out-degree of the vertex V of the digraph G.
out_edges(G, V) -> Edges
Types:
G = digraph()
V = vertex()
Edges = [edge()]
Returns a list of all edges emanating from V of the digraph G, in some unspecified
order.
out_neighbours(G, V) -> Vertices
Types:
G = digraph()
V = vertex()
Vertices = [vertex()]
Returns a list of all out-neighbours of V of the digraph G, in some unspecified
order.
vertex(G, V) -> {V, Label} | false
Types:
G = digraph()
V = vertex()
Label = label()
Returns {V, Label} where Label is the label of the vertex V of the digraph G, or
false if there is no vertex V of the digraph G.
vertices(G) -> Vertices
Types:
G = digraph()
Vertices = [vertex()]
Returns a list of all vertices of the digraph G, in some unspecified order.
SEE ALSO
digraph_utils(3erl), ets(3erl)