DeltaStepping SingleSource Shortest Path
Glossary
 Directed

Directed trait. The algorithm is welldefined on a directed graph.
 Undirected

Undirected trait. The algorithm is welldefined on an undirected graph.
 Homogeneous

Homogeneous trait. The algorithm will treat all nodes and relationships in its input graph(s) similarly, as if they were all of the same type. If multiple types of nodes or relationships exist in the graph, this must be taken into account when analysing the results of the algorithm.
 Heterogeneous

Heterogeneous trait. The algorithm has the ability to distinguish between nodes and/or relationships of different types.
 Weighted

Weighted trait. The algorithm supports configuration to set node and/or relationship properties to use as weights. These values can represent cost, time, capacity or some other domainspecific properties, specified via the nodeWeightProperty, nodeProperties and relationshipWeightProperty configuration parameters. The algorithm will by default consider each node and/or relationship as equally important.
1. Introduction
The DeltaStepping Shortest Path algorithm computes all shortest paths between a source node and all reachable nodes in the graph. The algorithm supports weighted graphs with positive relationship weights. To compute the shortest path between a source and a single target node, Dijkstra SourceTarget can be used.
In contrast to Dijkstra SingleSource, the DeltaStepping algorithm is a distance correcting algorithm. This property allows it to traverse the graph in parallel. The algorithm is guaranteed to always find the shortest path between a source node and a target node. However, if multiple shortest paths exist between two nodes, the algorithm is not guaranteed to return the same path in each computation.
The GDS implementation is based on [1] and incorporates the bucket fusion optimization discussed in [2]. The algorithm implementation is executed using multiple threads which can be defined in the procedure configuration.
For more information on this algorithm, see:
2. Syntax
This section covers the syntax used to execute the DeltaStepping 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.allShortestPaths.delta.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. 
delta 
Float 

yes 
The bucket width for grouping nodes with the same tentative distance to the source node. 
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.allShortestPaths.delta.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. 
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.allShortestPaths.delta.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. 
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. 
CALL gds.allShortestPaths.delta.stats(
graphName: String,
configuration: Map
)
YIELD
preProcessingMillis: Integer,
computeMillis: Integer,
postProcessingMillis: 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. 

sourceNode 
Integer 

no 
The Neo4j source node or node id. 
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. 
configuration 
Map 
The configuration used for running the algorithm. 
2.1. Delta
The delta
parameter defines a range which is used to group nodes with the same tentative distance to the start node.
The ranges are also called buckets.
In each iteration of the algorithm, the nonempty bucket with the smallest tentative distance is processed in parallel.
The delta
parameter is the main tuning knob for the algorithm and controls the workload that can be processed in parallel.
Generally, for powerlaw graphs, where many nodes can be reached within a few hops, a small delta (e.g. 2
) is recommended.
For highdiameter graphs, e.g. transport networks, a high delta value (e.g. 10000
) is recommended.
Note, that the value might vary depending on the graph topology and the value range of relationship properties.
3. Examples
In this section we will show examples of running the DeltaStepping 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:Location {name: 'A'}),
(b:Location {name: 'B'}),
(c:Location {name: 'C'}),
(d:Location {name: 'D'}),
(e:Location {name: 'E'}),
(f:Location {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);
This graph builds a transportation network with roads between locations.
Like in the real world, the roads in the graph have different lengths.
These lengths are represented by the cost
relationship property.
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',
'Location',
'ROAD',
{
relationshipProperties: 'cost'
}
)
In the following example we will demonstrate the use of the DeltaStepping Shortest Path algorithm using this graph.
3.1. 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:Location {name: 'A'})
CALL gds.allShortestPaths.delta.write.estimate('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost',
writeRelationshipType: 'PATH'
})
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
RETURN nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
nodeCount  relationshipCount  bytesMin  bytesMax  requiredMemory 

6 
9 
368 
576 
"[368 Bytes ... 576 Bytes]" 
3.2. 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:Location {name: 'A'})
CALL gds.allShortestPaths.delta.stream('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost',
delta: 3.0
})
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 
"A" 
"A" 
0.0 
[A] 
[0.0] 
[Node[0]] 
1 
"A" 
"B" 
50.0 
[A, B] 
[0.0, 50.0] 
[Node[0], Node[1]] 
2 
"A" 
"C" 
50.0 
[A, C] 
[0.0, 50.0] 
[Node[0], Node[2]] 
3 
"A" 
"D" 
90.0 
[A, B, D] 
[0.0, 50.0, 90.0] 
[Node[0], Node[1], Node[3]] 
4 
"A" 
"E" 
120.0 
[A, B, D, E] 
[0.0, 50.0, 90.0, 120.0] 
[Node[0], Node[1], Node[3], Node[4]] 
5 
"A" 
"F" 
160.0 
[A, B, D, E, F] 
[0.0, 50.0, 90.0, 120.0, 160.0] 
[Node[0], Node[1], Node[3], Node[4], Node[5]] 
The result shows the total cost of the shortest path between node A
and all other reachable nodes 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.
3.3. 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:Location {name: 'A'})
CALL gds.allShortestPaths.delta.mutate('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost',
mutateRelationshipType: 'PATH'
})
YIELD relationshipsWritten
RETURN relationshipsWritten
relationshipsWritten 

6 
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 relationships produced are always directed, even if the input graph is undirected. 
3.4. 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:Location {name: 'A'})
CALL gds.allShortestPaths.delta.write('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost',
writeRelationshipType: 'PATH',
writeNodeIds: true,
writeCosts: true
})
YIELD relationshipsWritten
RETURN relationshipsWritten
relationshipsWritten 

6 
The above query will write 6 relationships of type PATH
back to Neo4j.
The relationships store three properties describing the path: totalCost
, nodeIds
and costs
.
The relationships written are always directed, even if the input graph is undirected. 
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