pgr_lineGraph - Proposed

pgr_lineGraph — Transforms the given graph into its corresponding edge-based graph.

Boost Graph Inside

Warning

Proposed functions for next mayor release.

• They are not officially in the current release.

• They will likely officially be part of the next mayor release:

  • The functions make use of ANY-INTEGER and ANY-NUMERICAL

  • Name might not change. (But still can)

  • Signature might not change. (But still can)

  • Functionality might not change. (But still can)

  • pgTap tests have being done. But might need more.

  • Documentation might need refinement.

Availability

• Version 3.7.0

  • Promoted to proposed signature.

  • Works for directed and undirected graphs.

• Version 2.5.0

  • New Experimental function

Description

Given a graph \(G\), its line graph \(L(G)\) is a graph such that:

• Each vertex of \(L(G)\) represents an edge of \(G\).

• Two vertices of \(L(G)\) are adjacent if and only if their corresponding edges share a common endpoint in \(G\)

The main characteristics are:

• Works for directed and undirected graphs.

• The cost and reverse_cost columns of the result represent existence of the edge.

• When the graph is directed the result is directed.

  • To get the complete Line Graph use unique identifiers on the double way edges (See Additional Examples).

• When the graph is undirected the result is undirected.

  • The reverse_cost is always \(-1\).

Signatures

pgr_lineGraph(Edges SQL, [directed])

Returns set of (seq, source, target, cost, reverse_cost)

OR EMPTY SET

Example:

For an undirected graph with edges :math:’{2,4,5,8}’

SELECT * FROM pgr_lineGraph(
  'SELECT id, source, target, cost, reverse_cost
  FROM edges WHERE id IN (2,4,5,8)',
  false);
 seq | source | target | cost | reverse_cost
-----+--------+--------+------+--------------
   1 |      2 |      4 |    1 |           -1
   2 |      2 |      5 |    1 |           -1
   3 |      4 |      8 |    1 |           -1
   4 |      5 |      8 |    1 |           -1
(4 rows)

Parameters

Parameter	Type	Description
Edges SQL	TEXT	Edges SQL as described below.

Optional parameters

Column	Type	Default	Description
directed	BOOLEAN	true	When true the graph is considered Directed When false the graph is considered as Undirected.

Inner Queries

Edges SQL

Column	Type	Default	Description
id	ANY-INTEGER		Identifier of the edge.
source	ANY-INTEGER		Identifier of the first end point vertex of the edge.
target	ANY-INTEGER		Identifier of the second end point vertex of the edge.
cost	ANY-NUMERICAL		Weight of the edge (source, target)
reverse_cost	ANY-NUMERICAL	-1	Weight of the edge (target, source) When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

Where:

ANY-INTEGER:

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

Result columns

Returns set of (seq, source, target, cost, reverse_cost)

Column	Type	Description
seq	INTEGER	Sequential value starting from 1. Gives a local identifier for the edge
source	BIGINT	Identifier of the source vertex of the current edge. When negative: the source is the reverse edge in the original graph.
target	BIGINT	Identifier of the target vertex of the current edge. When negative: the target is the reverse edge in the original graph.
cost	FLOAT	Weight of the edge (source, target). When negative: edge (source, target) does not exist, therefore it’s not part of the graph.
reverse_cost	FLOAT	Weight of the edge (target, source). When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

Additional Examples

Given the following directed graph

\(G(V,E) = G(\{1,2,3,4\},\{ 1 \rightarrow 2, 1 \rightarrow 4, 2 \rightarrow 3, 3 \rightarrow 1, 3 \rightarrow 2, 3 \rightarrow 4, 4 \rightarrow 3\})\)

Representation as directed with shared edge identifiers

For the simplicity, the design of the edges table on the database, has the edge’s identifiers are represented with 3 digits:

hundreds:

the source vertex

tens:

always 0, acts as a separator

units:

the target vertex

In this image,

• Single or double head arrows represent one edge (row) on the edges table.

• The numbers in the yellow shadow are the edge identifiers.

Two pair of edges share the same identifier when the reverse_cost column is used.

• Edges \({2 \rightarrow 3, 3 \rightarrow 2}\) are represented with one edge row with \(id=203\).

• Edges \({3 \rightarrow 4, 4 \rightarrow 3}\) are represented with one edge row with \(id=304\).

The graph can be created as follows:

CREATE TABLE edges_shared (
    id BIGINT,
    source BIGINT,
    target BIGINT,
    cost FLOAT,
    reverse_cost FLOAT,
    geom geometry
);
CREATE TABLE
INSERT INTO edges_shared (id, source, target, cost, reverse_cost, geom) VALUES
  (102, 1, 2, 1, -1, ST_MakeLine(ST_POINT(0,  2), ST_POINT(2,  2))),
  (104, 1, 4, 1, -1, ST_MakeLine(ST_POINT(0,  2), ST_POINT(0,  0))),
  (301, 3, 1, 1, -1, ST_MakeLine(ST_POINT(2,  0), ST_POINT(0,  2))),
  (203, 2, 3, 1,  1, ST_MakeLine(ST_POINT(2,  2), ST_POINT(2,  0))),
  (304, 3, 4, 1,  1, ST_MakeLine(ST_POINT(0,  0), ST_POINT(2,  0)));

Line Graph of a directed graph represented with shared edges

SELECT seq, source, target, cost, reverse_cost
FROM pgr_lineGraph(
  'SELECT id, source, target, cost, reverse_cost FROM edges_shared',
  true);
 seq | source | target | cost | reverse_cost
-----+--------+--------+------+--------------
   1 |    102 |    203 |    1 |           -1
   2 |    104 |    304 |    1 |           -1
   3 |    203 |    203 |    1 |            1
   4 |    203 |    301 |    1 |           -1
   5 |    203 |    304 |    1 |            1
   6 |    301 |    102 |    1 |           -1
   7 |    301 |    104 |    1 |           -1
   8 |    304 |    301 |    1 |           -1
   9 |    304 |    304 |    1 |            1
(9 rows)

• The result is a directed graph.

• For \(seq=4\) from \(203 \leftrightarrow 304\) represent two edges

• For all the other values of seq represent one edge.

• The cost and reverse_cost values represent the existence of the edge.

  • When positive: the edge exists.

  • When negative: the edge does not exist.

Representation as directed with unique edge identifiers

For the simplicity, the design of the edges table on the database, has the edge’s identifiers are represented with 3 digits:

hundreds:

the source vertex

tens:

always 0, acts as a separator

units:

the target vertex

In this image,

• Single head arrows represent one edge (row) on the edges table.

• There are no double head arrows

• The numbers in the yellow shadow are the edge identifiers.

Two pair of edges share the same ending nodes and the reverse_cost column is not used.

• Edges \({2 \rightarrow 3, 3 \rightarrow 2}\) are represented with two edges \(id=203\) and \(id=302\) respectively.

• Edges \({3 \rightarrow 4, 4 \rightarrow 3}\) are represented with two edges \(id=304\) and \(id=403\) respectively.

The graph can be created as follows:

CREATE TABLE edges_unique (
    id BIGINT,
    source BIGINT,
    target BIGINT,
    cost FLOAT,
    geom geometry
);
CREATE TABLE
INSERT INTO edges_unique (id, source, target, cost, geom) VALUES
  (102, 1, 2, 1, ST_MakeLine(ST_POINT(0,  2), ST_POINT(2,  2))),
  (104, 1, 4, 1, ST_MakeLine(ST_POINT(0,  2), ST_POINT(0,  0))),
  (301, 3, 1, 1, ST_MakeLine(ST_POINT(2,  0), ST_POINT(0,  2))),
  (203, 2, 3, 1, ST_MakeLine(ST_POINT(2,  2), ST_POINT(2,  0))),
  (304, 3, 4, 1, ST_MakeLine(ST_POINT(2,  0), ST_POINT(0,  0))),
  (302, 3, 2, 1, ST_MakeLine(ST_POINT(2,  0), ST_POINT(2,  2))),
  (403, 4, 3, 1, ST_MakeLine(ST_POINT(0,  0), ST_POINT(2,  0)));

Line Graph of a directed graph represented with unique edges

SELECT seq, source, target, cost, reverse_cost
FROM pgr_lineGraph(
  'SELECT id, source, target, cost FROM edges_unique',
  true);
 seq | source | target | cost | reverse_cost
-----+--------+--------+------+--------------
   1 |    102 |    203 |    1 |           -1
   2 |    104 |    403 |    1 |           -1
   3 |    203 |    301 |    1 |           -1
   4 |    203 |    304 |    1 |           -1
   5 |    301 |    102 |    1 |           -1
   6 |    301 |    104 |    1 |           -1
   7 |    302 |    203 |    1 |            1
   8 |    304 |    403 |    1 |            1
   9 |    403 |    301 |    1 |           -1
  10 |    403 |    302 |    1 |           -1
(10 rows)

• The result is a directed graph.

• For \(seq=7\) from \(203 \leftrightarrow 302\) represent two edges.

• For \(seq=8\) from \(304 \leftrightarrow 403\) represent two edges.

• For all the other values of seq represent one edge.

• The cost and reverse_cost values represent the existence of the edge.

  • When positive: the edge exists.

  • When negative: the edge does not exist.

See Also

• wikipedia: Line Graph

• mathworld: Line Graph

• Sample Data

Indices and tables

• Index

• Search Page