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 3.0.0
    • Official function
  • Version 2.5.0
    • Renamed from pgr_maxFlowPushRelabel
    • Proposed function
  • Version 2.3.0
    • New Experimental function

Support

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)
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)

Parameters

Column Type Default Description
Edges SQL TEXT   The edges SQL query as described in Inner Query.
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 query

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

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).