pgr_edmondsKarp
pgr_edmondsKarp
— Calculates the flow on the graph edges that maximizes the flow from the sources to the targets using Push Relabel Algorithm.
Availability
 Version 3.0.0
 Version 2.5.0
 Renamed from
pgr_maxFlowEdmondsKarp
 Proposed 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: \(O( V * E ^ 2)\)
Signatures
Summary
pgr_edmondsKarp(Edges SQL, source, target)
pgr_edmondsKarp(Edges SQL, sources, target)
pgr_edmondsKarp(Edges SQL, source, targets)
pgr_edmondsKarp(Edges SQL, sources, targets)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET
One to One
pgr_edmondsKarp(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_edmondsKarp(
'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_edmondsKarp(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 \(\{1, 3, 11\}\) 
SELECT * FROM pgr_edmondsKarp(
'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  10  5  10  80  50
5  8  6  5  130  0
6  9  6  9  80  50
7  11  6  11  130  0
8  16  9  4  80  0
9  12  10  11  80  20
(9 rows)
Many to One
pgr_edmondsKarp(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_edmondsKarp(
'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_edmondsKarp(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_edmondsKarp(
'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  80  50
7  11  6  11  130  0
8  7  8  5  20  30
9  16  9  4  80  0
10  12  10  11  100  0
(10 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 
ANYINTEGER 

Identifier of the edge. 
source 
ANYINTEGER 

Identifier of the first end point vertex of the edge. 
target 
ANYINTEGER 

Identifier of the second end point vertex of the edge. 
capacity 
ANYINTEGER 

Weight of the edge (source, target)
 When negative: edge (source, target) does not exist, therefore it’s not part of the graph.

reverse_capacity 
ANYINTEGER 
1 
Weight of the edge (target, source),
 When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

Where:
ANYINTEGER:  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 ). 
See Also
Indices and tables