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pgr_lineGraph - Proposed - pgRouting Manual (3.8)

`pgr_lineGraph` - Proposed

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

Proposed

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
- Function promoted to proposed.
- 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 \(-1\) .

Boost Graph Inside

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)


    

graph G {

v6 [label=6,shape=circle;style=filled;fixedsize=true;width=.4;color=deepskyblue,pos="0,0!"];
v7 [label=7,shape=circle;style=filled;fixedsize=true;width=.4;color=deepskyblue,pos="0,2!"];
v10 [label=10,shape=circle;style=filled;fixedsize=true;width=.4;color=deepskyblue,pos="2,0!"];
v11 [label=11,shape=circle;style=filled;fixedsize=true;width=.4;color=deepskyblue,pos="2,2!"];

v7--v6 [color=blue];
v7--v11 [color=blue];
v10--v6 [color=blue];
v10--v11 [color=blue];

s2 [label="2",shape=circle;style=filled;width=.4;color=yellow,pos="1,0!"];
s4 [label="4",shape=circle;style=filled;width=.4;color=yellow,pos="0,1!"];
s5 [label="5",shape=circle;style=filled;width=.4;color=yellow,pos="2,1!"];
s8 [label="8",shape=circle;style=filled;width=.4;color=yellow,pos="1,2!"];

s2--s4 [color=red];
s2--s5 [color=red];
s4--s8 [color=red];
s5--s8 [color=red];
}

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

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

digraph G {

subgraph clusterA {
style=invis;
edge [arrowsize=0.5,color=blue];
node [shape=circle;style=filled;fontsize=8;fixedsize=true;width=.4;color=deepskyblue];
v1 [label=1,pos="0,2!"];
v2 [label=2,pos="2,2!"];
v3 [label=3,pos="2,0!"];
v4 [label=4,pos="0,0!"];

v1->{v2,v4} [color=blue];
v3->{v2,v4} [dir=both,color=blue];
v3->v1 [arrowsize=0.5,color=blue ];
}
}

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