Flow - Family of functions - pgRouting Manual (3.2)

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

Inner queries

For pgr_pushRelabel , pgr_edmondsKarp , pgr_boykovKolmogorov :

Edges SQL :

an SQL query of a directed graph of capacities, which should return a set of rows with the following columns:

Where:

ANY-INTEGER :

SMALLINT, INTEGER, BIGINT

For pgr_maxFlowMinCost - Experimental and pgr_maxFlowMinCost_Cost - Experimental :

Edges SQL :

an SQL query of a directed graph of capacities, which should return a set of rows with the following columns:

Where:

ANY-INTEGER :

SMALLINT, INTEGER, BIGINT

ANY-NUMERICAL :

smallint, int, bigint, real, float

For pgr_pushRelabel , pgr_edmondsKarp , pgr_boykovKolmogorov , pgr_edgeDisjointPaths , pgr_maxFlowMinCost and pgr_maxFlowMinCost_Cost :

Combinations SQL :

an SQL query which should return a set of rows with the following columns:

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

For pgr_pushRelabel , pgr_edmondsKarp , pgr_boykovKolmogorov :

For pgr_maxFlowMinCost - Experimental

Adcanced Documentation

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

• \(residual\_capacity_i = capacity_i - flow_i\)

See Also

• https://en.wikipedia.org/wiki/Maximum_flow_problem

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