pgr_boykovKolmogorov

pgr_boykovKolmogorov — Calculates the flow on the graph edges that maximizes the flow from the sources to the targets using Boykov Kolmogorov algorithm.

_images/boost-inside.jpeg

Boost Graph Inside

Availability

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: Polynomial

Signatures

Summary

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

One to One

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

Many to One

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

Many to Many

pgr_boykovKolmogorov(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_boykovKolmogorov(
  '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 |   50 |                 0
   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 |  100 |                30
   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_boykovKolmogorov(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_boykovKolmogorov(
  '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 |    9 |        11 |      16 |   80 |                50
   4 |   16 |        16 |      15 |   80 |                 0
(4 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_boykovKolmogorov(
  '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 |    9 |        11 |      16 |   80 |                50
   4 |   16 |        16 |      15 |   80 |                 0
(4 rows)