# pgr_pushRelabel - Proposed¶

## Synopsis¶

pgr_pushRelabel — Calculates the flow on the graph edges that maximizes the flow from the sources to the targets using Push Relabel Algorithm.

Availability:

• Renamed 2.5.0, Previous name pgr_maxFlowPushRelabel
• New in 2.3.0

Warning

Experimental functions

• They are not officially of the current release.
• They likely will not be officially be part of the next release:
• The functions might not make use of ANY-INTEGER and ANY-NUMERICAL
• Name might change.
• Signature might change.
• Functionality might change.
• pgTap tests might be missing.
• Might need c/c++ coding.
• May lack documentation.
• Documentation if any might need to be rewritten.
• Documentation examples might need to be automatically generated.
• Might need a lot of feedback from the comunity.
• Might depend on a proposed function of pgRouting
• Might depend on a deprecated function of pgRouting

Characteristics

• 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: $$O( V ^ 3)$$

## Signature Summary¶

pgr_pushRelabel(edges_sql, source,  target) - Proposed
pgr_pushRelabel(edges_sql, sources, target) - Proposed
pgr_pushRelabel(edges_sql, source,  targets) - Proposed
pgr_pushRelabel(edges_sql, sources, targets) - Proposed
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET


### One to One¶

Calculates the flow on the graph edges that maximizes the flow from the source to the target.

pgr_pushRelabel(edges_sql, source,  target)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET

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

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¶

Calculates the flow on the graph edges that maximizes the flow from all of the sources to the target.

pgr_pushRelabel(edges_sql, sources,  target)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET

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¶

Calculates the flow on the graph edges that maximizes the flow from all of the sources to all of the targets.

pgr_pushRelabel(edges_sql, sources,  targets)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET

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)



## Description of the Signatures¶

### Description of the edges_sql query for Max-flow like functions¶

edges_sql: an SQL query, 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)

• When negative: edge (source, target) does not exist, therefore it’s not part of the graph.
reverse_capacity ANY-INTEGER -1

Weight of the edge (target, source),

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

Where:

ANY-INTEGER: SMALLINT, INTEGER, BIGINT

### Description of the Parameters of the Flow Signatures¶

Column Type Default Description
edges_sql TEXT   The edges SQL query as described above.
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.

### Description of the Return Values¶

Column Type Description
seq INT Sequential value starting from 1.
edge_id BIGINT Identifier of the edge in the original query(edges_sql).
source BIGINT Identifier of the first end point vertex of the edge.
target BIGINT Identifier of the second end point vertex of the edge.
flow BIGINT Flow through the edge in the direction (source, target).
residual_capacity BIGINT Residual capacity of the edge in the direction (source, target).