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.
Availability
- Version 3.2.0
- New proposed signature
pgr_pushRelabel
(Combinations)
- New proposed signature
- 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
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 \(11\) to vertex \(12\) |
---|
SELECT * FROM pgr_pushRelabel(
'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_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 \(11\) to vertices \(\{5, 10, 12\}\) |
---|
SELECT * FROM pgr_pushRelabel(
'SELECT id, source, target, capacity, reverse_capacity
FROM edges',
11, ARRAY[5, 10, 12]);
seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
1 | 6 | 1 | 3 | 50 | 0
2 | 6 | 3 | 1 | 50 | 50
3 | 7 | 3 | 7 | 50 | 0
4 | 1 | 6 | 5 | 30 | 100
5 | 7 | 7 | 3 | 50 | 80
6 | 4 | 7 | 6 | 30 | 20
7 | 10 | 7 | 8 | 100 | 30
8 | 12 | 8 | 12 | 100 | 0
9 | 8 | 11 | 7 | 130 | 0
10 | 11 | 11 | 12 | 130 | 0
11 | 9 | 11 | 16 | 80 | 50
12 | 3 | 15 | 10 | 80 | 50
13 | 16 | 16 | 15 | 80 | 0
(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 \(\{11, 3, 17\}\) to vertex \(12\) |
---|
SELECT * FROM pgr_pushRelabel(
'SELECT id, source, target, capacity, reverse_capacity
FROM edges',
ARRAY[11, 3, 17], 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)
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 \(\{11, 3, 17\}\) to vertices \(\{5, 10, 12\}\) |
---|
SELECT * FROM pgr_pushRelabel(
'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 | 20 | 30
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 | 130 | 0
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_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 \(\{5, 6\}\) 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_pushRelabel(
'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 | 11 | 11 | 12 | 50 | 80
4 | 9 | 11 | 16 | 30 | 100
5 | 13 | 12 | 17 | 50 | 50
6 | 16 | 16 | 15 | 80 | 0
7 | 15 | 17 | 16 | 50 | 0
(7 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 (
|
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_pushRelabel(
'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 | 11 | 11 | 12 | 50 | 80
4 | 9 | 11 | 16 | 30 | 100
5 | 13 | 12 | 17 | 50 | 50
6 | 16 | 16 | 15 | 80 | 0
7 | 15 | 17 | 16 | 50 | 0
(7 rows)