pgr_maxFlowMinCost - Experimental¶
pgr_maxFlowMinCost — Calculates the flow on the graph edges that maximizes
the flow and minimizes the cost from the sources to the targets.
Boost Graph Inside¶
Warning
Possible server crash
These functions might create a server crash
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
Availability
Version 3.2.0
New experimental function:
pgr_maxFlowMinCost(Combinations)
Version 3.0.0
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
TODO check which statement is true:
The cost value of all input edges must be nonnegative.
Process is done when the cost value of all input edges is nonnegative.
Process is done on edges with nonnegative cost.
Running time: \(O(U * (E + V * logV))\)
where \(U\) is the value of the max flow.
\(U\) is upper bound on number of iterations. In many real world cases number of iterations is much smaller than \(U\).
Signatures¶
Summary
pgr_maxFlowMinCost(Edges SQL, source, target)
pgr_maxFlowMinCost(Edges SQL, sources, target)
pgr_maxFlowMinCost(Edges SQL, source, targets)
pgr_maxFlowMinCost(Edges SQL, sources, targets)
pgr_maxFlowMinCost(Edges SQL, Combinations SQL) -- Experimental on v3.2
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
One to One¶
pgr_maxFlowMinCost(Edges SQL, source, target)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
- Example
From vertex \(2\) to vertex \(3\)
SELECT * FROM pgr_MaxFlowMinCost(
'SELECT id,
source, target,
capacity, reverse_capacity,
cost, reverse_cost FROM edge_table',
2, 3
);
seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
1 | 4 | 2 | 5 | 80 | 20 | 80 | 80
2 | 3 | 4 | 3 | 80 | 50 | 80 | 160
3 | 8 | 5 | 6 | 80 | 20 | 80 | 240
4 | 9 | 6 | 9 | 80 | 50 | 80 | 320
5 | 16 | 9 | 4 | 80 | 0 | 80 | 400
(5 rows)
One to Many¶
pgr_maxFlowMinCost(Edges SQL, source, targets)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
- Example
From vertex \(13\) to vertices \(\{7, 1, 4\}\)
SELECT * FROM pgr_MaxFlowMinCost(
'SELECT id,
source, target,
capacity, reverse_capacity,
cost, reverse_cost FROM edge_table',
13, ARRAY[7, 1, 4]
);
seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
1 | 1 | 2 | 1 | 50 | 80 | 50 | 50
2 | 4 | 5 | 2 | 50 | 0 | 50 | 100
3 | 16 | 9 | 4 | 50 | 30 | 50 | 150
4 | 10 | 10 | 5 | 50 | 0 | 50 | 200
5 | 12 | 10 | 11 | 50 | 50 | 50 | 250
6 | 13 | 11 | 12 | 50 | 50 | 50 | 300
7 | 15 | 12 | 9 | 50 | 0 | 50 | 350
8 | 14 | 13 | 10 | 100 | 30 | 100 | 450
(8 rows)
Many to One¶
pgr_maxFlowMinCost(Edges SQL, sources, target)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
- Example
From vertices \(\{1, 7, 14\}\) to vertex \(12\)
SELECT * FROM pgr_MaxFlowMinCost(
'SELECT id,
source, target,
capacity, reverse_capacity,
cost, reverse_cost FROM edge_table',
ARRAY[1, 7, 14], 12
);
seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
1 | 1 | 1 | 2 | 80 | 0 | 80 | 80
2 | 4 | 2 | 5 | 80 | 20 | 80 | 160
3 | 8 | 5 | 6 | 100 | 0 | 100 | 260
4 | 10 | 5 | 10 | 30 | 100 | 30 | 290
5 | 9 | 6 | 9 | 50 | 80 | 50 | 340
6 | 11 | 6 | 11 | 50 | 80 | 50 | 390
7 | 6 | 7 | 8 | 50 | 0 | 50 | 440
8 | 7 | 8 | 5 | 50 | 0 | 50 | 490
9 | 15 | 9 | 12 | 50 | 30 | 50 | 540
10 | 12 | 10 | 11 | 30 | 70 | 30 | 570
11 | 13 | 11 | 12 | 80 | 20 | 80 | 650
(11 rows)
Many to Many¶
pgr_maxFlowMinCost(Edges SQL, sources, targets)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
- Example
From vertices \(\{7, 13\}\) to vertices \(\{3, 9\}\)
SELECT * FROM pgr_MaxFlowMinCost(
'SELECT id,
source, target,
capacity, reverse_capacity,
cost, reverse_cost FROM edge_table',
ARRAY[7, 13], ARRAY[3, 9]
);
seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
1 | 8 | 5 | 6 | 100 | 0 | 100 | 100
2 | 9 | 6 | 9 | 100 | 30 | 100 | 200
3 | 6 | 7 | 8 | 50 | 0 | 50 | 250
4 | 7 | 8 | 5 | 50 | 0 | 50 | 300
5 | 10 | 10 | 5 | 50 | 0 | 50 | 350
6 | 12 | 10 | 11 | 50 | 50 | 50 | 400
7 | 13 | 11 | 12 | 50 | 50 | 50 | 450
8 | 15 | 12 | 9 | 50 | 0 | 50 | 500
9 | 14 | 13 | 10 | 100 | 30 | 100 | 600
(9 rows)
Combinations¶
pgr_maxFlowMinCost(Edges SQL, Combinations SQL)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
- Example
Using a combinations table, equivalent to calculating result from vertices \(\{7, 13\}\) to vertices \(\{3, 9\}\).
SELECT * FROM pgr_MaxFlowMinCost(
'SELECT id,
source, target,
capacity, reverse_capacity,
cost, reverse_cost FROM edge_table',
'SELECT * FROM ( VALUES (7, 3), (13, 9) ) AS t(source, target)'
);
seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
1 | 8 | 5 | 6 | 100 | 0 | 100 | 100
2 | 9 | 6 | 9 | 100 | 30 | 100 | 200
3 | 6 | 7 | 8 | 50 | 0 | 50 | 250
4 | 7 | 8 | 5 | 50 | 0 | 50 | 300
5 | 10 | 10 | 5 | 50 | 0 | 50 | 350
6 | 12 | 10 | 11 | 50 | 50 | 50 | 400
7 | 13 | 11 | 12 | 50 | 50 | 50 | 450
8 | 15 | 12 | 9 | 50 | 0 | 50 | 500
9 | 14 | 13 | 10 | 100 | 30 | 100 | 600
(9 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 |
|
Capacity of the edge (source, target)
|
|
reverse_capacity |
|
-1 |
Capacity of the edge (target, source),
|
cost |
|
Weight of the edge (source, target) if it exists. |
|
reverse_cost |
|
0 |
Weight of the edge (target, source) if it exists. |
Where:
- ANY-INTEGER
SMALLINT, INTEGER, BIGINT
- ANY-NUMERICAL
smallint, int, bigint, real, float
- 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:
- ANY-INTEGER
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). |
source |
|
Identifier of the first end point vertex of the edge. |
target |
|
Identifier of the second end point vertex of the edge. |
flow |
|
Flow through the edge in the direction (source, target). |
residual_capacity |
|
Residual capacity of the edge in the direction (source, target). |
cost |
|
The cost of sending this flow through the edge in the direction (source, target). |
agg_cost |
|
The aggregate cost. |
See Also¶
Indices and tables