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_maxFlowPushRelabel - Proposed function
- Renamed from
- 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)
|
|
| reverse_capacity | ANY-INTEGER |
-1 | Weight of the edge (target, source),
|
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). |