pgr_pushRelabel

pgr_pushRelabel - Calculates the flow on the graph edges that maximizes the flow from the sources to the targets using Push Relabel Algorithm.

images/boost-inside.jpeg

Boost Graph Inside

Availability

  • Version 3.2.0

  • Version 3.0.0

    • Official function

  • Version 2.5.0

    • Renamed from pgr_maxFlowPushRelabel

    • Proposed function

  • Version 2.3.0

    • New Experimental function

Description

The main characteristics are:

  • The graph is directed .

  • Process is done only on edges with positive capacities.

  • When the maximum flow is 0 then there is no flow and EMPTY SET is returned.

    • There is no flow when a source is the same as a target .

  • Any duplicated value in the source(s) or target(s) are ignored.

  • Calculates the flow/residual capacity for each edge. In the output

    • Edges with zero flow are omitted.

  • Creates a super source and edges to all the source(s), and a super target and the edges from all the targets(s).

  • The maximum flow through the graph is guaranteed to be the value returned by pgr_maxFlow when executed with the same parameters and can be calculated:

    • By aggregation of the outgoing flow from the sources

    • By aggregation of the incoming flow to the targets

  • Running time: \(O( V ^ 3)\)

Signatures

Summary

pgr_pushRelabel( Edges SQL , start vid , end vid )
pgr_pushRelabel( Edges SQL , start vid , end vids )
pgr_pushRelabel( Edges SQL , start vids , end vid )
pgr_pushRelabel( Edges SQL , start vids , end vids )
pgr_pushRelabel( Edges SQL , Combinations SQL )
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET

One to One

pgr_pushRelabel( Edges SQL , start vid , end vid )
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET
Example :

From vertex \(11\) to vertex \(12\) 

SELECT * FROM pgr_pushRelabel(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  11, 12);
 seq  edge  start_vid  end_vid  flow  residual_capacity
-----+------+-----------+---------+------+-------------------
   1    10          7        8   100                 30
   2    12          8       12   100                  0
   3     8         11        7   100                 30
   4    11         11       12   130                  0
(4 rows)

One to Many

pgr_pushRelabel( Edges SQL , start vid , end vids )
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET
Example :

From vertex \(11\) to vertices \(\{5, 10, 12\}\) 

SELECT * FROM pgr_pushRelabel(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  11, ARRAY[5, 10, 12]);
 seq  edge  start_vid  end_vid  flow  residual_capacity
-----+------+-----------+---------+------+-------------------
   1     6          1        3    50                  0
   2     6          3        1    50                 50
   3     7          3        7    50                  0
   4     1          6        5    30                100
   5     7          7        3    50                 80
   6     4          7        6    30                 20
   7    10          7        8   100                 30
   8    12          8       12   100                  0
   9     8         11        7   130                  0
  10    11         11       12   130                  0
  11     9         11       16    80                 50
  12     3         15       10    80                 50
  13    16         16       15    80                  0
(13 rows)

Many to One

pgr_pushRelabel( Edges SQL , start vids , end vid )
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET
Example :

From vertices \(\{11, 3, 17\}\) to vertex \(12\) 

SELECT * FROM pgr_pushRelabel(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  ARRAY[11, 3, 17], 12);
 seq  edge  start_vid  end_vid  flow  residual_capacity
-----+------+-----------+---------+------+-------------------
   1    10          7        8   100                 30
   2    12          8       12   100                  0
   3     8         11        7   100                 30
   4    11         11       12   130                  0
(4 rows)

Many to Many

pgr_pushRelabel( Edges SQL , start vids , end vids )
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET
Example :

From vertices \(\{11, 3, 17\}\) to vertices \(\{5, 10, 12\}\) 

SELECT * FROM pgr_pushRelabel(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  ARRAY[11, 3, 17], ARRAY[5, 10, 12]);
 seq  edge  start_vid  end_vid  flow  residual_capacity
-----+------+-----------+---------+------+-------------------
   1     7          3        7    20                 30
   2     1          6        5    50                 80
   3     4          7        6    50                  0
   4    10          7        8   100                 30
   5    12          8       12   100                  0
   6     8         11        7   130                  0
   7    11         11       12   130                  0
   8     9         11       16    80                 50
   9     3         15       10    80                 50
  10    16         16       15    80                  0
(10 rows)

Combinations

pgr_pushRelabel( Edges SQL , Combinations SQL )
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET
Example :

Using a combinations table, equivalent to calculating result from vertices \(\{5, 6\}\) to vertices \(\{10, 15, 14\}\) .

The combinations table: 

SELECT source, target FROM combinations
WHERE target NOT IN (5, 6);
 source  target
--------+--------
      5      10
      6      15
      6      14
(3 rows)

The query: 

SELECT * FROM pgr_pushRelabel(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  'SELECT * FROM combinations WHERE target NOT IN (5, 6)');
 seq  edge  start_vid  end_vid  flow  residual_capacity
-----+------+-----------+---------+------+-------------------
   1     4          6        7    80                 20
   2     8          7       11    80                 20
   3    11         11       12    50                 80
   4     9         11       16    30                100
   5    13         12       17    50                 50
   6    16         16       15    80                  0
   7    15         17       16    50                  0
(7 rows)

Parameters

Column

Type

Description

Edges SQL

TEXT

Edges SQL as described below

Combinations SQL

TEXT

Combinations SQL as described below

start vid

BIGINT

Identifier of the starting vertex of the path.

start vids

ARRAY[BIGINT]

Array of identifiers of starting vertices.

end vid

BIGINT

Identifier of the ending vertex of the path.

end vids

ARRAY[BIGINT]

Array of identifiers of ending vertices.

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.

capacity

ANY-INTEGER

Weight of the edge ( source , target )

reverse_capacity

ANY-INTEGER

-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

Combinations SQL

Parameter

Type

Description

source

ANY-INTEGER

Identifier of the departure vertex.

target

ANY-INTEGER

Identifier of the arrival vertex.

Where:

ANY-INTEGER :

SMALLINT , INTEGER , BIGINT

Result Columns

Column

Type

Description

seq

INT

Sequential value starting from 1 .

edge

BIGINT

Identifier of the edge in the original query (edges_sql).

start_vid

BIGINT

Identifier of the first end point vertex of the edge.

end_vid

BIGINT

Identifier of the second end point vertex of the edge.

flow

BIGINT

Flow through the edge in the direction ( start_vid , end_vid ).

residual_capacity

BIGINT

Residual capacity of the edge in the direction ( start_vid , end_vid ).

Additional Examples

Example :

Manually assigned vertex combinations. 

SELECT * FROM pgr_pushRelabel(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  'SELECT * FROM (VALUES (5, 10), (6, 15), (6, 14)) AS t(source, target)');
 seq  edge  start_vid  end_vid  flow  residual_capacity
-----+------+-----------+---------+------+-------------------
   1     4          6        7    80                 20
   2     8          7       11    80                 20
   3    11         11       12    50                 80
   4     9         11       16    30                100
   5    13         12       17    50                 50
   6    16         16       15    80                  0
   7    15         17       16    50                  0
(7 rows)

See Also

Indices and tables