digraph
Directed graphs.
This module provides a version of labeled directed graphs. What makes the graphs provided here non-proper directed graphs is that multiple edges between vertices are allowed. However, the customary definition of directed graphs is used here.
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 × 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 more information. Such information can be attached to the vertices and to the edges of the digraph. An annotated digraph 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 an edge is emanating from v and incident on w, then w is said to be an out-neighbor of v, and v is said to be an in-neighbor 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 path P is k-1.
Path P is simple if all vertices are distinct, except that the first and the last vertices can be the same.
Path 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 without cycles.
d_type() = d_cyclicity() | d_protection()
d_cyclicity() = acyclic | cyclic
d_protection() = private | protected
graph()
A digraph as returned by new/0,1
.
label() = term()
edge()
| {error, add_edge_err_rsn()}edge()
| {error, add_edge_err_rsn()}edge()
| {error, add_edge_err_rsn()}add_edge/5
creates (or modifies) edge E
of 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 term ['$e' | N]
, where N
is an integer >= 0.
add_edge(G, V1, V2)
is equivalent to add_edge(G, V1, V2, [])
.
If the edge would create a cycle in an acyclic digraph
, {error, {bad_edge, Path}}
is returned. If G
already has an edge with value E
connecting a different pair of vertices, {error, {bad_edge, [V1, V2]}}
is returned. If either of V1
or V2
is not a vertex of digraph G
, {error, {bad_vertex,
V}}
is returned, V = V1
or V = V2
.
vertex()
vertex()
vertex()
add_vertex/3
creates (or modifies) vertex V
of digraph G
, using Label
as the (new) label
of the vertex. Returns V
.
add_vertex(G, V)
is equivalent to add_vertex(G, V, [])
.
add_vertex/1
creates a vertex using the empty list as label, and returns the created vertex. The created vertex is represented by term ['$v' | N]
, where N
is an integer >= 0.
Deletes edge E
from digraph G
.
Deletes the edges in list Edges
from digraph G
.
Deletes edges from digraph G
until there are no paths
from vertex V1
to 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
.
Deletes vertex V
from digraph G
. Any edges emanating
from V
or incident
on V
are also deleted.
Deletes the vertices in list Vertices
from digraph G
.
Deletes digraph G
. This call is important as digraphs are implemented with ETS. There is no garbage collection of ETS tables. However, the digraph is deleted if the process that created the digraph terminates.
Returns {E, V1, V2, Label}
, where Label
is the label
of edge E
emanating
from V1
and incident
on V2
of digraph G
. If no edge E
of digraph G
exists, false
is returned.
Returns a list of all edges of digraph G
, in some unspecified order.
Returns a list of all edges emanating
from or incident
onV
of digraph G
, in some unspecified order.
If a simple cycle
of length two or more exists through vertex V
, the cycle is returned as a list [V, ..., V]
of vertices. If a loop
through V
exists, the loop is returned as a list [V]
. If no cycles through V
exist, false
is returned.
get_path/3
is used for finding a simple cycle through V
.
Tries to find a simple path
from vertex V1
to vertex V2
of 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.
Digraph G
is traversed in a depth-first manner, and the first found path is returned.
Tries to find an as short as possible simple cycle
through vertex V
of digraph G
. Returns the cycle as a list [V, ..., V]
of vertices, or false
if no simple cycle through V
exists. Notice that a loop
through V
is returned as list [V, V]
.
get_short_path/3
is used for finding a simple cycle through V
.
Tries to find an as short as possible simple path
from vertex V1
to vertex V2
of 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.
Digraph G
is traversed in a breadth-first manner, and the first found path is returned.
Returns the in-degree
of vertex V
of digraph G
.
Returns a list of all edges incident
on V
of digraph G
, in some unspecified order.
Returns a list of all in-neighbors
of V
of digraph G
, in some unspecified order.
Returns a list of {Tag, Value}
pairs describing 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
.
graph()
Equivalent to new([])
.
graph()
Returns an empty digraph
with properties according to the options in Type
:
cyclic
Allows 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 specified or Type
is not a proper list, a badarg
exception is raised.
Returns the number of edges of digraph G
.
Returns the number of vertices of digraph G
.
Returns the out-degree
of vertex V
of digraph G
.
Returns a list of all edges emanating
from V
of digraph G
, in some unspecified order.
Returns a list of all out-neighbors
of V
of digraph G
, in some unspecified order.
Returns {V, Label}
, where Label
is the label
of the vertex V
of digraph G
, or false
if no vertex V
of digraph G
exists.
Returns a list of all vertices of digraph G
, in some unspecified order.
© 2010–2017 Ericsson AB
Licensed under the Apache License, Version 2.0.