A*

This section describes the A* Shortest Path algorithm in the Neo4j Graph Data Science library.

1. Introduction

The A* (pronounced "A-Star") Shortest Path algorithm computes the shortest path between two nodes. A* is an informed search algorithm as it uses a heuristic function to guide the graph traversal. The algorithm supports weighted graphs with positive relationship weights.

Unlike Dijkstra’s shortest path algorithm, the next node to search from is not solely picked on the already computed distance. Instead, the algorithm combines the already computed distance with the result of a heuristic function. That function takes a node as input and returns a value that corresponds to the cost to reach the target node from that node. In each iteration, the graph traversal is continued from the node with the lowest combined cost.

In GDS, the A* algorithm is based on the Dijkstra’s shortest path algorithm. The heuristic function is the haversine distance, which defines the distance between two points on a sphere. Here, the sphere is the earth and the points are geo-coordinates stored on the nodes in the graph.

The algorithm implementation is executed using a single thread. Altering the concurrency configuration has no effect.

2. Requirements

In GDS, the heuristic function used to guide the search is the haversine formula. The formula computes the distance between two points on a sphere given their longitudes and latitudes. The distance is computed in nautical miles.

In order to guarantee finding the optimal solution, i.e., the shortest path between two points, the heuristic must be admissible. To be admissible, the function must not overestimate the distance to the target, i.e., the lowest possible cost of a path must always be greater or equal to the heuristic.

This leads to a requirement on the relationship weights of the input graph. Relationship weights must represent the distance between two nodes and ideally scaled to nautical miles. Kilometers or miles also work, but the heuristic works best for nautical miles.

3. Syntax

This section covers the syntax used to execute the A* algorithm in each of its execution modes. We are describing the named graph variant of the syntax. To learn more about general syntax variants, see Syntax overview.

Example 1. A* syntax per mode

Run A* in stream mode on a named graph.

CALL gds.shortestPath.astar.stream(
  graphName: String,
  configuration: Map
)
YIELD
  index: Integer,
  sourceNode: Integer,
  targetNode: Integer,
  totalCost: Float,
  nodeIds: List of Integer,
  costs: List of Float,
  path: Path

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. General configuration for algorithm execution on a named graph.
Name	Type	Default	Optional	Description
nodeLabels	String[]	`['*']`	yes	Filter the named graph using the given node labels.
relationshipTypes	String[]	`['*']`	yes	Filter the named graph using the given relationship types.
concurrency	Integer	`4`	yes	The number of concurrent threads used for running the algorithm.

Table 3. Algorithm specific configuration
Name	Type	Default	Optional	Description
sourceNode	Integer	`n/a`	no	The Neo4j source node or node id.
targetNode	Integer	`n/a`	no	The Neo4j target node or node id.
latitudeProperty	Float	`n/a`	no	The node property that stores the latitude value.
longitudeProperty	Float	`n/a`	no	The node property that stores the longitude value.

Table 4. Results
Name	Type	Description
index	Integer	0-based index of the found path.
sourceNode	Integer	Source node of the path.
targetNode	Integer	Target node of the path.
totalCost	Float	Total cost from source to target.
nodeIds	List of Integer	Node ids on the path in traversal order.
costs	List of Float	Accumulated costs for each node on the path.
path	Path	The path represented as Cypher entity.

The mutate mode creates new relationships in the in-memory graph. Each relationship represents a path from the source node to the target node. The total cost of a path is stored via the totalCost relationship property.

Run A* in mutate mode on a named graph.

CALL gds.shortestPath.astar.mutate(
  graphName: String,
  configuration: Map
)
YIELD
  relationshipsWritten: Integer,
  createMillis: Integer,
  computeMillis: Integer,
  postProcessingMillis: Integer,
  mutateMillis: Integer,
  configuration: Map

Table 5. 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 6. General configuration for algorithm execution on a named graph.
Name	Type	Default	Optional	Description
nodeLabels	String[]	`['*']`	yes	Filter the named graph using the given node labels.
relationshipTypes	String[]	`['*']`	yes	Filter the named graph using the given relationship types.
concurrency	Integer	`4`	yes	The number of concurrent threads used for running the algorithm.
mutateRelationshipType	String	`n/a`	no	The relationship type used for the new relationships written to the in-memory graph.

Table 7. Algorithm specific configuration
Name	Type	Default	Optional	Description
sourceNode	Integer	`n/a`	no	The Neo4j source node or node id.
targetNode	Integer	`n/a`	no	The Neo4j target node or node id.
latitudeProperty	Float	`n/a`	no	The node property that stores the latitude value.
longitudeProperty	Float	`n/a`	no	The node property that stores the longitude value.

Table 8. Results
Name	Type	Description
createMillis	Integer	Milliseconds for creating the graph.
computeMillis	Integer	Milliseconds for running the algorithm.
postProcessingMillis	Integer	Unused.
mutateMillis	Integer	Milliseconds for adding relationships to the in-memory graph.
relationshipsWritten	Integer	The number of relationships that were added.
configuration	Map	The configuration used for running the algorithm.

The write mode creates new relationships in the Neo4j database. Each relationship represents a path from the source node to the target node. Additional path information is stored using relationship properties. By default, the write mode stores a totalCost property. Optionally, one can also store nodeIds and costs of intermediate nodes on the path.

Run A* in write mode on a named graph.

CALL gds.shortestPath.astar.write(
  graphName: String,
  configuration: Map
)
YIELD
  relationshipsWritten: Integer,
  createMillis: Integer,
  computeMillis: Integer,
  postProcessingMillis: Integer,
  writeMillis: Integer,
  configuration: Map

Table 9. 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 10. General configuration for algorithm execution on a named graph.
Name	Type	Default	Optional	Description
nodeLabels	String[]	`['*']`	yes	Filter the named graph using the given node labels.
relationshipTypes	String[]	`['*']`	yes	Filter the named graph using the given relationship types.
concurrency	Integer	`4`	yes	The number of concurrent threads used for running the algorithm. Also provides the default value for 'writeConcurrency'.
writeConcurrency	Integer	`value of 'concurrency'`	yes	The number of concurrent threads used for writing the result to Neo4j.
writeRelationshipType	String	`n/a`	no	The relationship type used to persist the computed relationships in the Neo4j database.

Table 11. Algorithm specific configuration
Name	Type	Default	Optional	Description
sourceNode	Integer	`n/a`	no	The Neo4j source node or node id.
targetNode	Integer	`n/a`	no	The Neo4j target node or node id.
latitudeProperty	Float	`n/a`	no	The node property that stores the latitude value.
longitudeProperty	Float	`n/a`	no	The node property that stores the longitude value.
writeNodeIds	Boolean	`false`	yes	If true, the written relationship has a nodeIds list property.
writeCosts	Boolean	`false`	yes	If true, the written relationship has a costs list property.

Table 12. Results
Name	Type	Description
createMillis	Integer	Milliseconds for creating the graph.
computeMillis	Integer	Milliseconds for running the algorithm.
postProcessingMillis	Integer	Unused.
writeMillis	Integer	Milliseconds for writing relationships to Neo4j.
relationshipsWritten	Integer	The number of relationships that were written.
configuration	Map	The configuration used for running the algorithm.

3.1. Anonymous graphs

It is also possible to execute the algorithm on a graph that is projected in conjunction with the algorithm execution. In this case, the graph does not have a name, and we call it anonymous. When executing over an anonymous graph the configuration map contains a graph projection configuration as well as an algorithm configuration. All execution modes support execution on anonymous graphs, although we only show syntax and mode-specific configuration for the write mode for brevity.

For more information on syntax variants, see Syntax overview.

Run A* in write mode on an anonymous graph:

CALL gds.shortestPath.astar.write(
  configuration: Map
)
YIELD
  relationshipsWritten: Integer,
  ranIterations: Integer,
  didConverge: Boolean,
  createMillis: Integer,
  computeMillis: Integer,
  writeMillis: Integer,
  configuration: Map

Table 13. General configuration for algorithm execution on an anonymous graph.
Name	Type	Default	Optional	Description
nodeProjection	String, String[] or Map	`null`	yes	The node projection used for anonymous graph creation via a Native projection.
relationshipProjection	String, String[] or Map	`null`	yes	The relationship projection used for anonymous graph creation a Native projection.
nodeQuery	String	`null`	yes	The Cypher query used to select the nodes for anonymous graph creation via a Cypher projection.
relationshipQuery	String	`null`	yes	The Cypher query used to select the relationships for anonymous graph creation via a Cypher projection.
nodeProperties	String, String[] or Map	`null`	yes	The node properties to project during anonymous graph creation.
relationshipProperties	String, String[] or Map	`null`	yes	The relationship properties to project during anonymous graph creation.
concurrency	Integer	`4`	yes	The number of concurrent threads used for running the algorithm. Also provides the default value for 'readConcurrency' and 'writeConcurrency'.
readConcurrency	Integer	`value of 'concurrency'`	yes	The number of concurrent threads used for creating the graph.
writeConcurrency	Integer	`value of 'concurrency'`	yes	The number of concurrent threads used for writing the result to Neo4j.
writeRelationshipType	String	`n/a`	no	The relationship type used to persist the computed relationships in the Neo4j database.

Table 14. Algorithm specific configuration
Name	Type	Default	Optional	Description
sourceNode	Integer	`n/a`	no	The Neo4j node id of the source node.
targetNode	Integer	`n/a`	no	The Neo4j node id of the target node.
latitudeProperty	Float	`n/a`	no	The node property that stores the latitude value.
longitudeProperty	Float	`n/a`	no	The node property that stores the longitude value.
writeNodeIds	Boolean	`false`	yes	Iff true, the written relationship has a nodeIds list property.
writeCosts	Boolean	`false`	yes	Iff true, the written relationship has a costs list property.

The results are the same as for running write mode with a named graph, see the write mode syntax above.

4. Examples

In this section we will show examples of running the A* algorithm on a concrete graph. The intention is to illustrate what the results look like and to provide a guide in how to make use of the algorithm in a real setting. We will do this on a small transport network graph of a handful nodes connected in a particular pattern. The example graph looks like this:

The following Cypher statement will create the example graph in the Neo4j database:

CREATE (a:Station {name: 'Kings Cross',         latitude: 51.5308, longitude: -0.1238}),
       (b:Station {name: 'Euston',              latitude: 51.5282, longitude: -0.1337}),
       (c:Station {name: 'Camden Town',         latitude: 51.5392, longitude: -0.1426}),
       (d:Station {name: 'Mornington Crescent', latitude: 51.5342, longitude: -0.1387}),
       (e:Station {name: 'Kentish Town',        latitude: 51.5507, longitude: -0.1402}),
       (a)-[:CONNECTION {distance: 0.7}]->(b),
       (b)-[:CONNECTION {distance: 1.3}]->(c),
       (b)-[:CONNECTION {distance: 0.7}]->(d),
       (d)-[:CONNECTION {distance: 0.6}]->(c),
       (c)-[:CONNECTION {distance: 1.3}]->(e)

The graph represents a transport network of stations. Each station has a geo-coordinate, expressed by latitude and longitude properties. Stations are connected via connections. We use the distance property as relationship weight which represents the distance between stations in kilometers. The algorithm will pick the next node in the search based on the already traveled distance and the distance to the target station.

In the examples below we will use named graphs and native projections as the norm. However, anonymous graphs and/or Cypher projections can also be used.

The following statement will create a graph using a native projection and store it in the graph catalog under the name 'myGraph'.

CALL gds.graph.create(
    'myGraph',
    'Station',
    'CONNECTION',
    {
        nodeProperties: ['latitude', 'longitude'],
        relationshipProperties: 'distance'
    }
)

In the following example we will demonstrate the use of the A* Shortest Path algorithm using this graph.

4.1. Memory Estimation

First off, we will estimate the cost of running the algorithm using the estimate procedure. This can be done with any execution mode. We will use the write mode in this example. Estimating the algorithm is useful to understand the memory impact that running the algorithm on your graph will have. When you later actually run the algorithm in one of the execution modes the system will perform an estimation. If the estimation shows that there is a very high probability of the execution going over its memory limitations, the execution is prohibited. To read more about this, see Automatic estimation and execution blocking.

For more details on estimate in general, see Memory Estimation.

The following will estimate the memory requirements for running the algorithm in write mode:

MATCH (source:Station {name: 'Kings Cross'}), (target:Station {name: 'Kentish Town'})
CALL gds.shortestPath.astar.write.estimate('myGraph', {
    sourceNode: source,
    targetNode: target,
    latitudeProperty: 'latitude',
    longitudeProperty: 'longitude',
    writeRelationshipType: 'PATH'
})
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
RETURN nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory

Table 15. Results
nodeCount	relationshipCount	bytesMin	bytesMax	requiredMemory
5	5	984	984	"984 Bytes"

4.2. Stream

In the stream execution mode, the algorithm returns the shortest path for each source-target-pair. This allows us to inspect the results directly or post-process them in Cypher without any side effects.

For more details on the stream mode in general, see Stream.

The following will run the algorithm and stream results:

MATCH (source:Station {name: 'Kings Cross'}), (target:Station {name: 'Kentish Town'})
CALL gds.shortestPath.astar.stream('myGraph', {
    sourceNode: source,
    targetNode: target,
    latitudeProperty: 'latitude',
    longitudeProperty: 'longitude',
    relationshipWeightProperty: 'distance'
})
YIELD index, sourceNode, targetNode, totalCost, nodeIds, costs, path
RETURN
    index,
    gds.util.asNode(sourceNode).name AS sourceNodeName,
    gds.util.asNode(targetNode).name AS targetNodeName,
    totalCost,
    [nodeId IN nodeIds | gds.util.asNode(nodeId).name] AS nodeNames,
    costs,
    nodes(path) as path
ORDER BY index

Table 16. Results
index	sourceNodeName	targetNodeName	totalCost	nodeNames	costs	path
0	"Kings Cross"	"Kentish Town"	3.3	[Kings Cross, Euston, Camden Town, Kentish Town]	[0.0, 0.7, 2.0, 3.3]	[Node[0], Node[1], Node[2], Node[4]]

The result shows the total cost of the shortest path between node King’s Cross and Kentish Town in the graph. It also shows ordered lists of node ids that were traversed to find the shortest paths as well as the accumulated costs of the visited nodes. This can be verified in the example graph. Cypher Path objects can be returned by the path return field. The Path objects contain the node objects and virtual relationships which have a cost property.

4.3. Mutate

The mutate execution mode updates the named graph with new relationships. Each new relationship represents a path from source node to target node. The relationship type is configured using the mutateRelationshipType option. The total path cost is stored using the totalCost property.

The mutate mode is especially useful when multiple algorithms are used in conjunction.

For more details on the mutate mode in general, see Mutate.

The following will run the algorithm in mutate mode:

MATCH (source:Station {name: 'Kings Cross'}), (target:Station {name: 'Kentish Town'})
CALL gds.shortestPath.astar.mutate('myGraph', {
    sourceNode: source,
    targetNode: target,
    latitudeProperty: 'latitude',
    longitudeProperty: 'longitude',
    relationshipWeightProperty: 'distance',
    mutateRelationshipType: 'PATH'
})
YIELD relationshipsWritten
RETURN relationshipsWritten

Table 17. Results
relationshipsWritten
1

After executing the above query, the in-memory graph will be updated with new relationships of type PATH. The new relationships will store a single property totalCost.

4.4. Write

The write execution mode updates the Neo4j database with new relationships. Each new relationship represents a path from source node to target node. The relationship type is configured using the writeRelationshipType option. The total path cost is stored using the totalCost property. The intermediate node ids are stored using the nodeIds property. The accumulated costs to reach an intermediate node are stored using the costs property.

For more details on the write mode in general, see Write.

The following will run the algorithm in write mode:

MATCH (source:Station {name: 'Kings Cross'}), (target:Station {name: 'Kentish Town'})
CALL gds.shortestPath.astar.write('myGraph', {
    sourceNode: source,
    targetNode: target,
    latitudeProperty: 'latitude',
    longitudeProperty: 'longitude',
    relationshipWeightProperty: 'distance',
    writeRelationshipType: 'PATH',
    writeNodeIds: true,
    writeCosts: true
})
YIELD relationshipsWritten
RETURN relationshipsWritten

Table 18. Results
relationshipsWritten
1

The above query will write one relationship of type PATH back to Neo4j. The relationship stores three properties describing the path: totalCost, nodeIds and costs.

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