# Flow - Family of functions¶

Experimental

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

## Flow Functions General Information¶

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

pgr_maxFlow is the maximum Flow and that maximum is guaranteed to be the same on the functions pgr_pushRelabel, pgr_edmondsKarp, pgr_boykovKolmogorov, but the actual flow through each edge may vary.

## 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

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

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

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

Capacity 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

Capacity of the edge (target, source),

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

cost

ANY-NUMERICAL

Weight of the edge (source, target) if it exists.

reverse_cost

ANY-NUMERICAL

0

Weight of the edge (target, source) if it exists.

Where:

ANY-INTEGER

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL

smallint, int, bigint, real, float

## 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).

Column

Type

Description

seq

INT

Sequential value starting from 1.

edge

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).

cost

FLOAT

The cost of sending this flow through the edge in the direction (source, target).

agg_cost

FLOAT

The aggregate cost.

A flow network is a directed graph where each edge has a capacity and a flow. The flow through an edge must not exceed the capacity of the edge. Additionally, the incoming and outgoing flow of a node must be equal except for source which only has outgoing flow, and the destination(sink) which only has incoming flow.

Maximum flow algorithms calculate the maximum flow through the graph and the flow of each edge.

The maximum flow through the graph is guaranteed to be the same with all implementations, but the actual flow through each edge may vary. Given the following query:

pgr_maxFlow $$(edges\_sql, source\_vertex, sink\_vertex)$$

where $$edges\_sql = \{(id_i, source_i, target_i, capacity_i, reverse\_capacity_i)\}$$

Graph definition

The weighted directed graph, $$G(V,E)$$, is defined as:

• the set of vertices $$V$$

• $$source\_vertex \cup sink\_vertex \bigcup source_i \bigcup target_i$$

• the set of edges $$E$$

• $$E = \begin{cases} \text{ } \{(source_i, target_i, capacity_i) \text{ when } capacity > 0 \} & \quad \text{ if } reverse\_capacity = \varnothing \\ \text{ } & \quad \text{ } \\ \{(source_i, target_i, capacity_i) \text{ when } capacity > 0 \} & \text{ } \\ \cup \{(target_i, source_i, reverse\_capacity_i) \text{ when } reverse\_capacity_i > 0)\} & \quad \text{ if } reverse\_capacity \neq \varnothing \\ \end{cases}$$

Maximum flow problem

Given:

• $$G(V,E)$$

• $$source\_vertex \in V$$ the source vertex

• $$sink\_vertex \in V$$ the sink vertex

Then:

• $$pgr\_maxFlow(edges\_sql, source, sink) = \boldsymbol{\Phi}$$

• $$\boldsymbol{\Phi} = {(id_i, edge\_id_i, source_i, target_i, flow_i, residual\_capacity_i)}$$

Where:

$$\boldsymbol{\Phi}$$ is a subset of the original edges with their residual capacity and flow. The maximum flow through the graph can be obtained by aggregating on the source or sink and summing the flow from/to it. In particular:

• $$id_i = i$$

• $$edge\_id = id_i$$ in edges_sql

• $$residual\_capacity_i = capacity_i - flow_i$$