A* Shortest Path
1. Introduction
The A* (pronounced "AStar") 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 geocoordinates 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.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 algorithmspecifics and/or graph filtering. 
Name  Type  Default  Optional  Description 

List of String 

yes 
Filter the named graph using the given node labels. 

List of String 

yes 
Filter the named graph using the given relationship types. 

Integer 

yes 
The number of concurrent threads used for running the algorithm. 

String 

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

sourceNode 
Integer 

no 
The Neo4j source node or node id. 
targetNode 
Integer 

no 
The Neo4j target node or node id. 
latitudeProperty 
Float 

no 
The node property that stores the latitude value. 
longitudeProperty 
Float 

no 
The node property that stores the longitude value. 
String 

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

index 
Integer 
0based 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 projected 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.shortestPath.astar.mutate(
graphName: String,
configuration: Map
)
YIELD
relationshipsWritten: Integer,
preProcessingMillis: 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 algorithmspecifics and/or graph filtering. 
Name  Type  Default  Optional  Description 

mutateRelationshipType 
String 

no 
The relationship type used for the new relationships written to the projected graph. 
List of String 

yes 
Filter the named graph using the given node labels. 

List of String 

yes 
Filter the named graph using the given relationship types. 

Integer 

yes 
The number of concurrent threads used for running the algorithm. 

String 

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

sourceNode 
Integer 

no 
The Neo4j source node or node id. 
targetNode 
Integer 

no 
The Neo4j target node or node id. 
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 

preProcessingMillis 
Integer 
Milliseconds for preprocessing the graph. 
computeMillis 
Integer 
Milliseconds for running the algorithm. 
postProcessingMillis 
Integer 
Unused. 
mutateMillis 
Integer 
Milliseconds for adding relationships to the projected 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.shortestPath.astar.write(
graphName: String,
configuration: Map
)
YIELD
relationshipsWritten: Integer,
preProcessingMillis: 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 algorithmspecifics and/or graph filtering. 
Name  Type  Default  Optional  Description 

List of String 

yes 
Filter the named graph using the given node labels. 

List of String 

yes 
Filter the named graph using the given relationship types. 

Integer 

yes 
The number of concurrent threads used for running the algorithm. 

String 

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

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 

Optional 
Description 
sourceNode 
Integer 

no 
The Neo4j source node or node id. 
targetNode 
Integer 

no 
The Neo4j target node or node id. 
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 

preProcessingMillis 
Integer 
Milliseconds for preprocessing 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. 
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 geocoordinate, 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, Cypher projections can also be used. 
CALL gds.graph.project(
'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.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
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 sourcetargetpair.
This allows us to inspect the results directly or postprocess 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.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
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.
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
relationshipsWritten 

1 
After executing the above query, the inmemory graph will be updated with new relationships of type PATH
.
The new relationships will store a single property totalCost
.
The relationship produced is always directed, even if the input graph is undirected. 
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.shortestPath.astar.write('myGraph', {
sourceNode: source,
targetNode: 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
.
The relationship written is always directed, even if the input graph is undirected. 
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