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.

Boost Graph Inside

Availability

• Version 3.2.0

  • New proposed function:

    • pgr_pushRelabel(Combinations)

• 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, source,  target)
pgr_pushRelabel(Edges SQL, sources, target)
pgr_pushRelabel(Edges SQL, source,  targets)
pgr_pushRelabel(Edges SQL, sources, targets)
pgr_pushRelabel(Edges SQL, Combinations SQL) -- Proposed on v3.2
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET

One to One

pgr_pushRelabel(Edges SQL, source,  target)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET

Example

From vertex \(6\) to vertex \(11\)

SELECT * FROM pgr_pushRelabel(
    'SELECT id,
            source,
            target,
            capacity,
            reverse_capacity
    FROM edge_table'
    , 6, 11
);
 seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
   1 |   10 |         5 |      10 |  100 |                30
   2 |    8 |         6 |       5 |  100 |                30
   3 |   11 |         6 |      11 |  130 |                 0
   4 |   12 |        10 |      11 |  100 |                 0
(4 rows)

One to Many

Calculates the flow on the graph edges that maximizes the flow from the source to all of the targets.

pgr_pushRelabel(Edges SQL, source,  targets)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET

Example

From vertex \(6\) to vertices \(\{11, 1, 13\}\)

SELECT * FROM pgr_pushRelabel(
    'SELECT id,
            source,
            target,
            capacity,
            reverse_capacity
    FROM edge_table'
    , 6, ARRAY[11, 1, 13]
);
 seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
   1 |    1 |         2 |       1 |  130 |                 0
   2 |    2 |         3 |       2 |   80 |                20
   3 |    3 |         4 |       3 |   80 |                50
   4 |    4 |         5 |       2 |   50 |                 0
   5 |    7 |         5 |       8 |   50 |                80
   6 |   10 |         5 |      10 |   80 |                50
   7 |    8 |         6 |       5 |  130 |                 0
   8 |    9 |         6 |       9 |   80 |                50
   9 |   11 |         6 |      11 |  130 |                 0
  10 |    6 |         7 |       8 |   50 |                 0
  11 |    6 |         8 |       7 |   50 |                50
  12 |    7 |         8 |       5 |   50 |                 0
  13 |   16 |         9 |       4 |   80 |                 0
  14 |   12 |        10 |      11 |   80 |                20
(14 rows)

Many to One

pgr_pushRelabel(Edges SQL, sources,  target)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET

Example

From vertices \(\{6, 8, 12\}\) to vertex \(11\)

SELECT * FROM pgr_pushRelabel(
    'SELECT id,
            source,
            target,
            capacity,
            reverse_capacity
    FROM edge_table'
    , ARRAY[6, 8, 12], 11
);
 seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
   1 |   10 |         5 |      10 |  100 |                30
   2 |    8 |         6 |       5 |  100 |                30
   3 |   11 |         6 |      11 |  130 |                 0
   4 |   12 |        10 |      11 |  100 |                 0
(4 rows)

Many to Many

pgr_pushRelabel(Edges SQL, sources,  targets)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET

Example

From vertices \(\{6, 8, 12\}\) to vertices \(\{1, 3, 11\}\)

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

SELECT * FROM pgr_pushRelabel(
    'SELECT id,
            source,
            target,
            capacity,
            reverse_capacity
    FROM edge_table',
    'SELECT * FROM ( VALUES (6, 1), (8, 3), (12, 11), (8, 1) ) AS t(source, target)'
);
 seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
   1 |    1 |         2 |       1 |   50 |                80
   2 |    3 |         4 |       3 |   80 |                50
   3 |    4 |         5 |       2 |   50 |                 0
   4 |   10 |         5 |      10 |  100 |                30
   5 |    8 |         6 |       5 |  130 |                 0
   6 |    9 |         6 |       9 |   30 |               100
   7 |   11 |         6 |      11 |  130 |                 0
   8 |    7 |         8 |       5 |   20 |                30
   9 |   16 |         9 |       4 |   80 |                 0
  10 |   12 |        10 |      11 |  100 |                 0
  11 |   15 |        12 |       9 |   50 |                 0
(11 rows)

Parameters

Column	Type	Default	Description
Edges SQL	TEXT		Edges query as described in Inner Queries.
Combinations SQL	TEXT		Combinations query as described in Inner Queries.
source	BIGINT		Identifier of the starting vertex of the flow.
sources	ARRAY[BIGINT]		Array of identifiers of the starting vertices of the flow.
target	BIGINT		Identifier of the ending vertex of the flow.
targets	ARRAY[BIGINT]		Array of identifiers of the ending vertices of the flow.

Inner queries

Edges SQL

an SQL query of a directed graph of capacities, which should return a set of rows with the following columns:

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) When negative: edge (source, target) does not exist, therefore it’s not part of the graph.
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

Combinations SQL

an SQL query which should return a set of rows with the following columns:

Column	Type	Default	Description
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.

Where:

ANY-INTEGER

SMALLINT, INTEGER, BIGINT

The function aggregates the sources and the targets, removes the duplicates, and then it calculates the result from the resultant source vertices to the target vertices.

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).

See Also

• Flow - Family of functions, pgr_boykovKolmogorov, pgr_edmondsKarp

• https://www.boost.org/libs/graph/doc/push_relabel_max_flow.html

• https://en.wikipedia.org/wiki/Push%E2%80%93relabel_maximum_flow_algorithm

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