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.

Directed

Undirected

Heterogeneous nodes

Heterogeneous relationships

Weighted relationships

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

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);

Using native projection

The following will project and store a graph using native projection:

CALL gds.graph.project(
  'nativeGraph',
  'Loc',
  {
    ROAD: {
      properties: 'cost'
    }
  }
)
YIELD graphName

The following will run the algorithm and stream results:

CALL gds.allShortestPaths.stream('nativeGraph', {
  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
"A"	"E"	120
"B"	"F"	110
"C"	"F"	110
"A"	"D"	90
"B"	"E"	70
"C"	"E"	70
"D"	"F"	70
"A"	"B"	50
"A"	"C"	50

This query returned the top 10 pairs of nodes that are the furthest away from each other. F and E appear to be quite distant from the others.

Using Cypher projection

The following will project and store an undirected graph using cypher projection:

MATCH (src:Loc)-[r:ROAD]->(trg:Loc)
RETURN gds.graph.project(
  'cypherGraph',
  src,
  trg,
  {relationshipType: type(r), relationshipProperties: {cost: 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 5. Results
source	target	distance
"A"	"F"	160
"F"	"A"	160
"A"	"E"	120
"E"	"A"	120
"B"	"F"	110
"C"	"F"	110
"F"	"B"	110
"F"	"C"	110
"A"	"D"	90
"D"	"A"	90

source	target	distance
"A"	"F"	160
"A"	"E"	120
"B"	"F"	110
"C"	"F"	110
"A"	"D"	90
"B"	"E"	70
"C"	"E"	70
"D"	"F"	70
"A"	"B"	50
"A"	"C"	50

source	target	distance
"A"	"F"	160
"F"	"A"	160
"A"	"E"	120
"E"	"A"	120
"B"	"F"	110
"C"	"F"	110
"F"	"B"	110
"F"	"C"	110
"A"	"D"	90
"D"	"A"	90

source	target	distance
"A"	"F"	160
"A"	"E"	120
"B"	"F"	110
"C"	"F"	110
"A"	"D"	90
"B"	"E"	70
"C"	"E"	70
"D"	"F"	70
"A"	"B"	50
"A"	"C"	50

source	target	distance
"A"	"F"	160
"F"	"A"	160
"A"	"E"	120
"E"	"A"	120
"B"	"F"	110
"C"	"F"	110
"F"	"B"	110
"F"	"C"	110
"A"	"D"	90
"D"	"A"	90

source	target	distance
"A"	"F"	160
"A"	"E"	120
"B"	"F"	110
"C"	"F"	110
"A"	"D"	90
"B"	"E"	70
"C"	"E"	70
"D"	"F"	70
"A"	"B"	50
"A"	"C"	50

source	target	distance
"A"	"F"	160
"F"	"A"	160
"A"	"E"	120
"E"	"A"	120
"B"	"F"	110
"C"	"F"	110
"F"	"B"	110
"F"	"C"	110
"A"	"D"	90
"D"	"A"	90