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 3.0.0
Official function
Version 2.5.0
Renamed from
pgr_maxFlowPushRelabelProposed 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 |
|
The edges SQL query as described in Inner Query. |
|
source |
|
Identifier of the starting vertex of the flow. |
|
sources |
|
Array of identifiers of the starting vertices of the flow. |
|
target |
|
Identifier of the ending vertex of the flow. |
|
targets |
|
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 |
|
Identifier of the edge. |
|
source |
|
Identifier of the first end point vertex of the edge. |
|
target |
|
Identifier of the second end point vertex of the edge. |
|
capacity |
|
Weight of the edge (source, target)
|
|
reverse_capacity |
|
-1 |
Weight of the edge (target, source),
|
Where:
- ANY-INTEGER
SMALLINT, INTEGER, BIGINT
Result 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 ( |