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.
CALL gds.beta.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
Name | Type | Default | Optional | Description |
---|---|---|---|---|
graphName |
String |
|
no |
The name of a graph stored in the catalog. |
configuration |
Map |
|
yes |
Configuration for algorithm-specifics and/or graph filtering. |
Name | Type | Default | Optional | Description |
---|---|---|---|---|
nodeLabels |
String[] |
|
yes |
Filter the named graph using the given node labels. |
String[] |
|
yes |
Filter the named graph using the given relationship types. |
|
Integer |
|
yes |
The number of concurrent threads used for running the algorithm. |
Name | Type | Default | Optional | Description |
---|---|---|---|---|
sourceNode |
Integer |
|
no |
The Neo4j node id of the source node. |
targetNode |
Integer |
|
no |
The Neo4j node id of the source node. |
latitudeProperty |
Float |
|
no |
The node property that stores the latitude value. |
longitudeProperty |
Float |
|
no |
The node property that stores the longitude value. |
path |
Boolean |
|
yes |
Iff true, the result contains a Cypher Path object. |
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.
CALL gds.beta.shortestPath.astar.mutate(
graphName: String,
configuration: Map
)
YIELD
relationshipsWritten: Integer,
createMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
mutateMillis: Integer,
configuration: Map
Name | Type | Default | Optional | Description |
---|---|---|---|---|
graphName |
String |
|
no |
The name of a graph stored in the catalog. |
configuration |
Map |
|
yes |
Configuration for algorithm-specifics and/or graph filtering. |
Name | Type | Default | Optional | Description |
---|---|---|---|---|
nodeLabels |
String[] |
|
yes |
Filter the named graph using the given node labels. |
String[] |
|
yes |
Filter the named graph using the given relationship types. |
|
Integer |
|
yes |
The number of concurrent threads used for running the algorithm. |
|
mutateRelationshipType |
String |
|
no |
The relationship type used for the new relationships written to the in-memory graph. |
Name | Type | Default | Optional | Description |
---|---|---|---|---|
sourceNode |
Integer |
|
no |
The Neo4j node id of the source node. |
targetNode |
Integer |
|
no |
The Neo4j node id of the source node. |
latitudeProperty |
Float |
|
no |
The node property that stores the latitude value. |
longitudeProperty |
Float |
|
no |
The node property that stores the longitude value. |
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.
CALL gds.beta.shortestPath.astar.write(
graphName: String,
configuration: Map
)
YIELD
relationshipsWritten: Integer,
createMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
writeMillis: Integer,
configuration: Map
Name | Type | Default | Optional | Description |
---|---|---|---|---|
graphName |
String |
|
no |
The name of a graph stored in the catalog. |
configuration |
Map |
|
yes |
Configuration for algorithm-specifics and/or graph filtering. |
Name | Type | Default | Optional | Description |
---|---|---|---|---|
nodeLabels |
String[] |
|
yes |
Filter the named graph using the given node labels. |
String[] |
|
yes |
Filter the named graph using the given relationship types. |
|
Integer |
|
yes |
The number of concurrent threads used for running the algorithm. Also provides the default value for 'writeConcurrency'. |
|
Integer |
|
yes |
The number of concurrent threads used for writing the result to Neo4j. |
|
writeRelationshipType |
String |
|
no |
The relationship type used to persist the computed relationships in the Neo4j database. |
Name | Type | Default | Optional | Description |
---|---|---|---|---|
sourceNode |
Integer |
|
no |
The Neo4j node id of the source node. |
targetNode |
Integer |
|
no |
The Neo4j node id of the source node. |
latitudeProperty |
Float |
|
no |
The node property that stores the latitude value. |
longitudeProperty |
Float |
|
no |
The node property that stores the longitude value. |
writeNodeIds |
Boolean |
|
yes |
If true, the written relationship has a nodeIds list property. |
writeCosts |
Boolean |
|
yes |
If true, the written relationship has a costs list property. |
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.
CALL gds.beta.shortestPath.astar.write(
configuration: Map
)
YIELD
relationshipsWritten: Integer,
ranIterations: Integer,
didConverge: Boolean,
createMillis: Integer,
computeMillis: Integer,
writeMillis: Integer,
configuration: Map
Name | Type | Default | Optional | Description |
---|---|---|---|---|
nodeProjection |
String, String[] or Map |
|
yes |
The node projection used for anonymous graph creation via a Native projection. |
relationshipProjection |
String, String[] or Map |
|
yes |
The relationship projection used for anonymous graph creation a Native projection. |
nodeQuery |
String |
|
yes |
The Cypher query used to select the nodes for anonymous graph creation via a Cypher projection. |
relationshipQuery |
String |
|
yes |
The Cypher query used to select the relationships for anonymous graph creation via a Cypher projection. |
nodeProperties |
String, String[] or Map |
|
yes |
The node properties to project during anonymous graph creation. |
relationshipProperties |
String, String[] or Map |
|
yes |
The relationship properties to project during anonymous graph creation. |
Integer |
|
yes |
The number of concurrent threads used for running the algorithm. Also provides the default value for 'readConcurrency' and 'writeConcurrency'. |
|
readConcurrency |
Integer |
|
yes |
The number of concurrent threads used for creating the graph. |
Integer |
|
yes |
The number of concurrent threads used for writing the result to Neo4j. |
|
writeRelationshipType |
String |
|
no |
The relationship type used to persist the computed relationships in the Neo4j database. |
Name | Type | Default | Optional | Description |
---|---|---|---|---|
sourceNode |
Integer |
|
no |
The Neo4j node id of the source node. |
targetNode |
Integer |
|
no |
The Neo4j node id of the target node. |
latitudeProperty |
Float |
|
no |
The node property that stores the latitude value. |
longitudeProperty |
Float |
|
no |
The node property that stores the longitude value. |
writeNodeIds |
Boolean |
|
yes |
Iff true, the written relationship has a nodeIds list property. |
writeCosts |
Boolean |
|
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:
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. |
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.
MATCH (source:Station {name: 'Kings Cross'}), (target:Station {name: 'Kentish Town'})
CALL gds.beta.shortestPath.astar.write.estimate('myGraph', {
sourceNode: id(source),
targetNode: id(target),
latitudeProperty: 'latitude',
longitudeProperty: 'longitude',
writeRelationshipType: 'PATH'
})
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
RETURN nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
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.
MATCH (source:Station {name: 'Kings Cross'}), (target:Station {name: 'Kentish Town'})
CALL gds.beta.shortestPath.astar.stream('myGraph', {
sourceNode: id(source),
targetNode: id(target),
latitudeProperty: 'latitude',
longitudeProperty: 'longitude',
relationshipWeightProperty: 'distance'
})
YIELD index, sourceNode, targetNode, totalCost, nodeIds, costs
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
ORDER BY index
index | sourceNodeName | targetNodeName | totalCost | nodeNames | costs |
---|---|---|---|---|---|
0 |
"Kings Cross" |
"Kentish Town" |
3.3 |
[Kings Cross, Euston, Camden Town, Kentish Town] |
[0.0, 0.7, 2.0, 3.3] |
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.
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.
mutate
mode:MATCH (source:Station {name: 'Kings Cross'}), (target:Station {name: 'Kentish Town'})
CALL gds.beta.shortestPath.astar.mutate('myGraph', {
sourceNode: id(source),
targetNode: id(target),
latitudeProperty: 'latitude',
longitudeProperty: 'longitude',
relationshipWeightProperty: 'distance',
mutateRelationshipType: 'PATH'
})
YIELD relationshipsWritten
RETURN relationshipsWritten
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.
write
mode:MATCH (source:Station {name: 'Kings Cross'}), (target:Station {name: 'Kentish Town'})
CALL gds.beta.shortestPath.astar.write('myGraph', {
sourceNode: id(source),
targetNode: id(target),
latitudeProperty: 'latitude',
longitudeProperty: 'longitude',
relationshipWeightProperty: 'distance',
writeRelationshipType: 'PATH',
writeNodeIds: true,
writeCosts: true
})
YIELD relationshipsWritten
RETURN relationshipsWritten
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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