pgr_maxFlow

pgr_maxFlow — Calculates the maximum flow in a directed graph from the source(s) to the targets(s) using the Push Relabel algorithm.

_images/boost-inside.jpeg

Boost Graph Inside

Availability

Description

The main characteristics are:

  • The graph is directed.
  • Calculates the maximum flow from the source(s) to the target(s).
    • When the maximum flow is 0 then there is no flow and 0 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.
  • Uses the pgr_pushRelabel algorithm.
  • Running time: \(O( V ^ 3)\)

Signatures

Summary

pgr_maxFlow(Edges SQL, start vid, end vid)
pgr_maxFlow(Edges SQL, start vid, end vids)
pgr_maxFlow(Edges SQL, start vids, end vid)
pgr_maxFlow(Edges SQL, start vids, end vids)
pgr_maxFlow(Edges SQL, Combinations SQL)
RETURNS BIGINT

One to One

pgr_maxFlow(Edges SQL, start vid, end vid)
RETURNS BIGINT
Example:From vertex \(11\) to vertex \(12\) 
SELECT * FROM pgr_maxFlow(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  11, 12);
 pgr_maxflow
-------------
         230
(1 row)

One to Many

pgr_maxFlow(Edges SQL, start vid, end vids)
RETURNS BIGINT
Example:From vertex \(11\) to vertices \(\{5, 10, 12\}\) 
SELECT * FROM pgr_maxFlow(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  11, ARRAY[5, 10, 12]);
 pgr_maxflow
-------------
         340
(1 row)

Many to One

pgr_maxFlow(Edges SQL, start vids, end vid)
RETURNS BIGINT
Example:From vertices \(\{11, 3, 17\}\) to vertex \(12\) 
SELECT * FROM pgr_maxFlow(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  ARRAY[11, 3, 17], 12);
 pgr_maxflow
-------------
         230
(1 row)

Many to Many

pgr_maxFlow(Edges SQL, start vids, end vids)
RETURNS BIGINT
Example:From vertices \(\{11, 3, 17\}\) to vertices \(\{5, 10, 12\}\) 
SELECT * FROM pgr_maxFlow(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  ARRAY[11, 3, 17], ARRAY[5, 10, 12]);
 pgr_maxflow
-------------
         360
(1 row)

Combinations

pgr_maxFlow(Edges SQL, Combinations SQL)
RETURNS BIGINT
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_maxFlow(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  'SELECT * FROM combinations WHERE target NOT IN (5, 6)');
 pgr_maxflow
-------------
          80
(1 row)

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

Type Description
BIGINT Maximum flow possible from the source(s) to the target(s)

Additional Examples

Example:Manually assigned vertex combinations. 
SELECT * FROM pgr_maxFlow(
  'SELECT id, source, target, capacity, reverse_capacity
  FROM edges',
  'SELECT * FROM (VALUES (5, 10), (6, 15), (6, 14)) AS t(source, target)');
 pgr_maxflow
-------------
          80
(1 row)