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