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 signature
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, 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 \(6\) 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¶
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 \(6\) to vertices \(\{1, 3, 11\}\)
SELECT * FROM pgr_pushRelabel(
'SELECT id, source, target, capacity, reverse_capacity
FROM edge_table',
6, 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 | 7 | 5 | 8 | 50 | 80
5 | 10 | 5 | 10 | 80 | 50
6 | 8 | 6 | 5 | 130 | 0
7 | 9 | 6 | 9 | 80 | 50
8 | 11 | 6 | 11 | 130 | 0
9 | 6 | 7 | 8 | 50 | 0
10 | 6 | 8 | 7 | 50 | 50
11 | 7 | 8 | 5 | 50 | 0
12 | 16 | 9 | 4 | 80 | 0
13 | 12 | 10 | 11 | 80 | 20
(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 \(\{6, 8, 12\}\) 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, start vids, end vids) 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 \(\{1, 2\}\) to vertices \(\{3, 4, 17\}\).
The combinations table:
SELECT source, target FROM combinations_table
WHERE target NOT IN (1, 2);
source | target
--------+--------
1 | 3
2 | 4
2 | 17
(3 rows)
The query:
SELECT * FROM pgr_pushRelabel(
'SELECT id, source, target, capacity, reverse_capacity
FROM edge_table',
'SELECT * FROM combinations_table
WHERE target NOT IN (1, 2)');
seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
1 | 4 | 2 | 5 | 80 | 20
2 | 8 | 5 | 6 | 80 | 20
3 | 9 | 6 | 9 | 80 | 50
4 | 16 | 9 | 4 | 80 | 0
(4 rows)
Parameters¶
Column |
Type |
Description |
---|---|---|
|
Edges SQL as described below |
|
|
Combinations SQL as described below |
|
start vid |
|
Identifier of the starting vertex of the path. |
start vids |
|
Array of identifiers of starting vertices. |
end vid |
|
Identifier of the ending vertex of the path. |
end vids |
|
Array of identifiers of ending vertices. |
Inner queries¶
Edges SQL¶
Column |
Type |
Default |
Description |
---|---|---|---|
|
ANY-INTEGER |
Identifier of the edge. |
|
|
ANY-INTEGER |
Identifier of the first end point vertex of the edge. |
|
|
ANY-INTEGER |
Identifier of the second end point vertex of the edge. |
|
|
ANY-INTEGER |
Weight of the edge ( |
|
|
ANY-INTEGER |
-1 |
Weight of the edge (
|
Where:
- ANY-INTEGER
SMALLINT
,INTEGER
,BIGINT
- ANY-NUMERICAL
SMALLINT
,INTEGER
,BIGINT
,REAL
,FLOAT
Combinations SQL¶
Parameter |
Type |
Description |
---|---|---|
|
ANY-INTEGER |
Identifier of the departure vertex. |
|
ANY-INTEGER |
Identifier of the arrival vertex. |
Where:
- ANY-INTEGER
SMALLINT
,INTEGER
,BIGINT
Return Columns¶
Column |
Type |
Description |
---|---|---|
seq |
|
Sequential value starting from 1. |
edge |
|
Identifier of the edge in the original query(edges_sql). |
start_vid |
|
Identifier of the first end point vertex of the edge. |
end_vid |
|
Identifier of the second end point vertex of the edge. |
flow |
|
Flow through the edge in the direction ( |
residual_capacity |
|
Residual capacity of the edge in the direction ( |
Additional Examples¶
- Example
Manually assigned vertex combinations.
SELECT * FROM pgr_pushRelabel(
'SELECT id, source, target, capacity, reverse_capacity
FROM edge_table',
'SELECT * FROM (VALUES (1, 3), (2, 4), (2, 17)) AS t(source, target)');
seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
1 | 4 | 2 | 5 | 80 | 20
2 | 8 | 5 | 6 | 80 | 20
3 | 9 | 6 | 9 | 80 | 50
4 | 16 | 9 | 4 | 80 | 0
(4 rows)
See Also¶
https://www.boost.org/libs/graph/doc/push_relabel_max_flow.html
https://en.wikipedia.org/wiki/Push%E2%80%93relabel_maximum_flow_algorithm
Indices and tables