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 function:
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, source, target)
pgr_pushRelabel(Edges SQL, sources, target)
pgr_pushRelabel(Edges SQL, source, targets)
pgr_pushRelabel(Edges SQL, sources, targets)
pgr_pushRelabel(Edges SQL, Combinations SQL)  Proposed on v3.2
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)
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 \(\{6, 8, 12\}\) to vertices \(\{1, 3, 11\}\).
SELECT * FROM pgr_pushRelabel(
'SELECT id,
source,
target,
capacity,
reverse_capacity
FROM edge_table',
'SELECT * FROM ( VALUES (6, 1), (8, 3), (12, 11), (8, 1) ) AS t(source, target)'
);
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 

Edges query as described in Inner Queries. 

Combinations SQL 

Combinations query as described in Inner Queries. 

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 queries¶
 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:
 ANYINTEGER
SMALLINT, INTEGER, BIGINT
 Combinations SQL
an SQL query which should return a set of rows with the following columns:
Column 
Type 
Default 
Description 

source 

Identifier of the first end point vertex of the edge. 

target 

Identifier of the second end point vertex of the edge. 
Where:
 ANYINTEGER
SMALLINT, INTEGER, BIGINT
The function aggregates the sources and the targets, removes the duplicates, and then it calculates the result from the resultant source vertices to the target vertices.
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 ( 
See Also¶
Flow  Family of functions, pgr_boykovKolmogorov, pgr_edmondsKarp
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