All Pairs Shortest Path

The All Pairs Shortest Path (APSP) calculates the shortest (weighted) path between all pairs of nodes. This algorithm has optimizations that make it quicker than calling the Single Source Shortest Path algorithm for every pair of nodes in the graph.

This feature is in the alpha tier. For more information on feature tiers, see API Tiers.

Glossary

Directed

Directed trait. The algorithm is well-defined on a directed graph.

Directed

Directed trait. The algorithm ignores the direction of the graph.

Directed

Directed trait. The algorithm does not run on a directed graph.

Undirected

Undirected trait. The algorithm is well-defined on an undirected graph.

Undirected

Undirected trait. The algorithm ignores the undirectedness of the graph.

Heterogeneous nodes

Heterogeneous nodes fully supported. The algorithm has the ability to distinguish between nodes of different types.

Heterogeneous nodes

Heterogeneous nodes allowed. The algorithm treats all selected nodes similarly regardless of their label.

Heterogeneous relationships

Heterogeneous relationships fully supported. The algorithm has the ability to distinguish between relationships of different types.

Heterogeneous relationships

Heterogeneous relationships allowed. The algorithm treats all selected relationships similarly regardless of their type.

Weighted relationships

Weighted trait. The algorithm supports a relationship property to be used as weight, specified via the relationshipWeightProperty configuration parameter.

Weighted relationships

Weighted trait. The algorithm treats each relationship as equally important, discarding the value of any relationship weight.

History and explanation

Some pairs of nodes might not be reachable between each other, so no shortest path exists between these pairs. In this scenario, the algorithm will return Infinity value as a result between these pairs of nodes.

GDS includes functions such as gds.util.isFinite to help filter infinity values from results. Starting with Neo4j 5, the Infinity literal is now included in Cypher too.

Use-cases - when to use the All Pairs Shortest Path algorithm

  • The All Pairs Shortest Path algorithm is used in urban service system problems, such as the location of urban facilities or the distribution or delivery of goods. One example of this is determining the traffic load expected on different segments of a transportation grid. For more information, see Urban Operations Research.

  • All pairs shortest path is used as part of the REWIRE data center design algorithm that finds a network with maximum bandwidth and minimal latency. There are more details about this approach in "REWIRE: An Optimization-based Framework for Data Center Network Design"

Syntax

The following will run the algorithm and stream results:
CALL gds.allShortestPaths.stream(
  graphName: string,
  configuration: map
)
YIELD sourceNodeId, targetNodeId, distance
Table 1. Parameters
Name Type Default Optional Description

graphName

String

n/a

no

The name of a graph stored in the catalog.

configuration

Map

{}

yes

Configuration for algorithm-specifics and/or graph filtering.

Table 2. Configuration
Name Type Default Optional Description

nodeLabels

List of String

['*']

yes

Filter the named graph using the given node labels. Nodes with any of the given labels will be included.

relationshipTypes

List of String

['*']

yes

Filter the named graph using the given relationship types. Relationships with any of the given types will be included.

concurrency

Integer

4

yes

The number of concurrent threads used for running the algorithm.

jobId

String

Generated internally

yes

An ID that can be provided to more easily track the algorithm’s progress.

logProgress

Boolean

true

yes

If disabled the progress percentage will not be logged.

relationshipWeightProperty

String

null

yes

Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted.

Table 3. Results
Name Type Description

sourceNodeId

Integer

The source node.

targetNodeId

Integer

The target node.

distance

Float

The distance of the shortest path from source to target.

All Pairs Shortest Path algorithm sample

shortest path graph
The following will create a sample graph:
CREATE (a:Loc {name: 'A'}),
       (b:Loc {name: 'B'}),
       (c:Loc {name: 'C'}),
       (d:Loc {name: 'D'}),
       (e:Loc {name: 'E'}),
       (f:Loc {name: 'F'}),
       (a)-[:ROAD {cost: 50}]->(b),
       (a)-[:ROAD {cost: 50}]->(c),
       (a)-[:ROAD {cost: 100}]->(d),
       (b)-[:ROAD {cost: 40}]->(d),
       (c)-[:ROAD {cost: 40}]->(d),
       (c)-[:ROAD {cost: 80}]->(e),
       (d)-[:ROAD {cost: 30}]->(e),
       (d)-[:ROAD {cost: 80}]->(f),
       (e)-[:ROAD {cost: 40}]->(f);
The following will project and store an undirected graph using a Cypher projection:
MATCH (src:Loc)-[r:ROAD]->(trg:Loc)
RETURN gds.graph.project(
  'cypherGraph',
  src,
  trg,
  {
    relationshipType: type(r),
    relationshipProperties: r { .cost }
  },
  { undirectedRelationshipTypes: ['ROAD'] }
)
The following will run the algorithm, treating the graph as undirected:
CALL gds.allShortestPaths.stream('cypherGraph', {
  relationshipWeightProperty: 'cost'
})
YIELD sourceNodeId, targetNodeId, distance
WITH sourceNodeId, targetNodeId, distance
WHERE gds.util.isFinite(distance) = true
WITH gds.util.asNode(sourceNodeId) AS source, gds.util.asNode(targetNodeId) AS target, distance WHERE source <> target

RETURN source.name AS source, target.name AS target, distance
ORDER BY distance DESC, source ASC, target ASC
LIMIT 10
Table 4. Results
source target distance

"A"

"F"

160.0

"F"

"A"

160.0

"A"

"E"

120.0

"E"

"A"

120.0

"B"

"F"

110.0

"C"

"F"

110.0

"F"

"B"

110.0

"F"

"C"

110.0

"A"

"D"

90.0

"D"

"A"

90.0