BellmanFord SingleSource Shortest Path
Glossary
 Directed

Directed trait. The algorithm is welldefined 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 welldefined 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.
1. Introduction
The BellmanFord Shortest Path algorithm computes the shortest path between nodes.
In contrast to the Dijkstra algorithm which works only for graphs with nonnegative relationship weights, BellmanFord can also handle graphs with negative weights provided that the source cannot reach any node involved in a negative cycle. A cycle in a graph is a path starting and ending at the same node. A negative cycle is a cycle for which the sum of the relationship weights is negative. When negative cycles exist, shortest paths cannot easily be defined. That is so because we can traverse a negative cycle multiple times to get smaller and smaller costs each time.
When the BellmanFord algorithm detects negative cycles, it will return negative cycles instead of shortest paths. As the full set of negative cycles can be too large to enumerate, each node will be included in at most one returned negative cycle.
The ability to handle negative weights makes BellmanFord more versatile than Dijkstra, but also slower in practice.
The Neo4j GDS Library provides an adaptation of the original BellmanFord algorithm called ShortestPath Faster Algorithm (SPFA). SPFA significantly reduces the computational time of BellmanFord by working only on a subset of the nodes rather than iterating over the set of nodes at each step. In addition, the computations are parallelized to further speedup computations.
2. Syntax
This section covers the syntax used to execute the BellmanFord 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.bellmanFord.stream(
graphName: String,
configuration: Map
)
YIELD
index: Integer,
sourceNode: Integer,
targetNode: Integer,
totalCost: Float,
nodeIds: List of Integer,
costs: List of Float,
route: Path,
isNegativeCycle: Boolean
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. 

Boolean 

yes 
If disabled the progress percentage will not be logged. 

sourceNode 
Integer 

no 
The Neo4j source node or node id. 
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 route. 
sourceNode 
Integer 
Source node of the route. 
targetNode 
Integer 
Target node of the route. 
totalCost 
Float 
Total cost from source to target. 
nodeIds 
List of Integer 
Node ids on the route in traversal order. 
costs 
List of Float 
Accumulated costs for each node on the route. 
route 
Path 
The route represented as Cypher entity. 
isNegativeCycle 
Boolean 
If true, the discovered route is a negative cycle. Otherwise it is a shortest path. 
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.bellmanFord.mutate(
graphName: String,
configuration: Map
)
YIELD
relationshipsWritten: Integer,
preProcessingMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
mutateMillis: Integer,
containsNegativeCycle: Boolean,
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. 
mutateNegativeCycles 
Boolean 

yes 
If set to true, any discovered negative cycles will be added in the inmemory graph. Otherwise they will be skipped. 
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. 
containsNegativeCycle 
Boolean 
True if negative cycles were discovered. 
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.bellmanFord.write(
graphName: String,
configuration: Map
)
YIELD
relationshipsWritten: Integer,
preProcessingMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
writeMillis: Integer,
containsNegativeCycle: Boolean,
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 

sourceNode 
Integer 

no 
The Neo4j source node or node id. 
writeNegativeCycles 
Boolean 

yes 
If set to true, any discovered negative cycles will be written back to the Neo4j graph. Otherwise they will be skipped. 
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. 
containsNegativeCycle 
Boolean 
True if negative cycles were discovered. 
configuration 
Map 
The configuration used for running the algorithm. 
CALL gds.bellmanFord.stats(
graphName: String,
configuration: Map
)
YIELD
preProcessingMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
containsNegativeCycle: Boolean,
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 

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. 
containsNegativeCycle 
Boolean 
True if negative cycles were discovered. 
configuration 
Map 
The configuration used for running the algorithm. 
3. Examples
In this section we will show examples of running the BellmanFord 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 example network graph of a handful nodes connected in a particular pattern. The example graph looks like this:
CREATE (a:Node {name: 'A'}),
(b:Node {name: 'B'}),
(c:Node {name: 'C'}),
(d:Node {name: 'D'}),
(e:Node {name: 'E'}),
(f:Node {name: 'F'}),
(g:Node {name: 'G'}),
(h:Node {name: 'H'}),
(i:Node {name: 'I'}),
(a)[:REL {cost: 50}]>(b),
(a)[:REL {cost: 50}]>(c),
(a)[:REL {cost: 100}]>(d),
(b)[:REL {cost: 40}]>(d),
(c)[:REL {cost: 40}]>(d),
(c)[:REL {cost: 80}]>(e),
(d)[:REL {cost: 30}]>(e),
(d)[:REL {cost: 80}]>(f),
(e)[:REL {cost: 40}]>(f),
(g)[:REL {cost: 40}]>(h),
(h)[:REL {cost: 60}]>(i),
(i)[:REL {cost: 10}]>(g)
This graph builds an example network with relationships between nodes having both negative and positive weights.
These weights 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',
'Node',
'REL',
{
relationshipProperties: 'cost'
}
)
In the following example we will demonstrate the use of the BellmanFord 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:Node {name: 'A'})
CALL gds.bellmanFord.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 

9 
12 
1336 
1336 
"1336 Bytes" 
The algorithm supports writing (or mutating) negative cycles if they exist in the graph, this is controlled by the writeNegativeCycles
(mutateNegativeCycles
) configuration parameter.
This requires additional memory because the negative cycles have to be tracked.
MATCH (source:Node {name: 'A'})
CALL gds.bellmanFord.write.estimate('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost',
writeRelationshipType: 'PATH',
writeNegativeCycles: true
})
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
RETURN nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
nodeCount  relationshipCount  bytesMin  bytesMax  requiredMemory 

9 
12 
1448 
1448 
"1448 Bytes" 
3.2. Stream
In the stream
execution mode, the algorithm returns the shortest path for each sourcetargetpair or negative cycles.
This allows us to inspect the results directly or postprocess them in Cypher without any side effects.
3.2.1. Stream without negative cycles
MATCH (source:Node {name: 'A'})
CALL gds.bellmanFord.stream('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost'
})
YIELD index, sourceNode, targetNode, totalCost, nodeIds, costs, route, isNegativeCycle
RETURN
index,
gds.util.asNode(sourceNode).name AS sourceNode,
gds.util.asNode(targetNode).name AS targetNode,
totalCost,
[nodeId IN nodeIds  gds.util.asNode(nodeId).name] AS nodeNames,
costs,
nodes(route) as route,
isNegativeCycle as isNegativeCycle
ORDER BY index
index  sourceNode  targetNode  totalCost  nodeNames  costs  route  isNegativeCycle 

0 
"A" 
"A" 
0.0 
[A] 
[0] 
[Node[0]] 
false 
1 
"A" 
"B" 
50.0 
[A, B] 
[0, 50] 
[Node[0], Node[1]] 
false 
2 
"A" 
"C" 
50.0 
[A, C] 
[0, 50] 
[Node[0], Node[2]] 
false 
3 
"A" 
"D" 
10.0 
[A, C, D] 
[0, 50, 10] 
[Node[0], Node[2], Node[3]] 
false 
4 
"A" 
"E" 
20.0 
[A, C, D, E] 
[0, 50, 10, 20] 
[Node[0], Node[2], Node[3], Node[4]] 
false 
5 
"A" 
"F" 
60.0 
[A, C, D, E, F] 
[0, 50, 10.0, 20, 60] 
[Node[0], Node[2], Node[3], Node[4], Node[5]] 
false 
Since the component of A
does not contain any negative cycles, the results depict the shortest paths from A
to all of its reachable nodes.
The ordered lists of node ids for each path as well as their accumulated costs are also returned.
The Cypher Path objects are returned by the path
return field, they contain node objects and virtual relationships which have a cost
property.
3.2.2. Stream with negative cycles
MATCH (source:Node {name: 'G'})
CALL gds.bellmanFord.stream('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost'
})
YIELD index, sourceNode, targetNode, totalCost, nodeIds, costs, route, isNegativeCycle
RETURN
index,
gds.util.asNode(sourceNode).name AS sourceNode,
gds.util.asNode(targetNode).name AS targetNode,
totalCost,
[nodeId IN nodeIds  gds.util.asNode(nodeId).name] AS nodeNames,
costs,
nodes(route) as route,
isNegativeCycle as isNegativeCycle
ORDER BY index
index  sourceNode  targetNode  totalCost  nodeNames  costs  route  isNegativeCycle 

0 
"G" 
"G" 
10 
[G, H, I, G] 
[0, 40, 20, 10] 
[Node[6], Node[7], Node[8], Node[6]] 
true 
For this example, BellmanFord did not yield any shortest paths as it detected negative cycles.
A negative cycle for G
of 10 total cost is emitted as the output, with the isNegativeCycle
field set to true.
3.3. Stats
In the stats
execution mode, the algorithm returns a single row containing a summary of the algorithm result.
This execution mode does not have any side effects.
It can be useful for evaluating algorithm performance by inspecting the computeMillis
return item.
In the examples below we will omit returning the timings.
The full signature of the procedure can be found in the syntax section.
For more details on the stats
mode in general, see Stats.
stats
modeMATCH (source:Node {name: 'A'})
CALL gds.bellmanFord.stats('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost'
})
YIELD containsNegativeCycle
RETURN containsNegativeCycle
containsNegativeCycle 

false 
Running stats mode can be useful if we want to discover if the graph has any negative cycles, but we do not to have them computed or stored.
For this example, we can see that the containsNegativeCycle
field is false as A
cannot reach any negative cycles.
3.4. Mutate
The mutate
execution mode updates the named graph with new relationships.
Each new relationship represents a path from source node to target node or a negative cycle.
The relationship type is configured using the mutateRelationshipType option.
The total route cost is stored using the totalCost property.
3.4.1. Mutate without negative cycles
MATCH (source:Node {name: 'A'})
CALL gds.bellmanFord.mutate('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost',
mutateRelationshipType: 'ROUTE'
})
YIELD relationshipsWritten, containsNegativeCycle
RETURN relationshipsWritten, containsNegativeCycle
relationshipsWritten  containsNegativeCycle 

6 
false 
After executing the above query, the inmemory graph will be updated with new relationships of type ROUTE
.
Since containsNegativeCycle
is false, these relationships represent shortest paths.
The new relationships will store a single property totalCost
, corresponding to the shortest path cost from source to target.
3.4.2. Mutate with negative cycles
MATCH (source:Node {name: 'G'})
CALL gds.bellmanFord.mutate('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost',
mutateRelationshipType: 'ROUTE',
mutateNegativeCycles: true
})
YIELD relationshipsWritten, containsNegativeCycle
RETURN relationshipsWritten, containsNegativeCycle
relationshipsWritten  containsNegativeCycle 

1 
true 
After executing the above query, the inmemory graph will be updated with a single relationship of type ROUTE
.
Since containsNegativeCycle
is true, this relationship represents the discovered negative cycle.
The new relationship stores a single property totalCost
, corresponding to the weight of the negative cycle.
Note that by default, when negative cycles are detected during mutate mode, they will not be written back to the inmemory graph.
This can be bypassed by setting the mutateNegativeCycles
to true as showcased in the above example.
The relationships produced are always directed, even if the input graph is undirected. 
3.5. Write
The write
execution mode updates the Neo4j database with new relationships.
Each new relationship represents a path from source node to target node or a negative cycle.
The relationship type is configured using the writeRelationshipType option.
The total 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.
3.5.1. Write without negative cycles
MATCH (source:Node {name: 'A'})
CALL gds.bellmanFord.write('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost',
writeRelationshipType: 'ROUTE',
writeNodeIds: true,
writeCosts: true
})
YIELD relationshipsWritten, containsNegativeCycle
RETURN relationshipsWritten, containsNegativeCycle
relationshipsWritten  containsNegativeCycle 

6 
false 
The above query will write 6 relationships of type ROUTE
back to Neo4j.
The relationships store three properties describing the path: totalCost
, nodeIds
and costs
.
3.5.2. Write with negative cycles
MATCH (source:Node {name: 'G'})
CALL gds.bellmanFord.write('myGraph', {
sourceNode: source,
relationshipWeightProperty: 'cost',
writeRelationshipType: 'ROUTE',
writeNodeIds: true,
writeCosts: true,
writeNegativeCycles:true
})
YIELD relationshipsWritten, containsNegativeCycle
RETURN relationshipsWritten, containsNegativeCycle
relationshipsWritten  containsNegativeCycle 

1 
true 
After executing the above query, one relationship of type ROUTE
is written back to the Neo4j graph.
Since containsNegativeCycle
is true, the relationship represents a negative cycle.
Similar to the mutate
mode, the default behavior when encountering negative cycles is to not write them back in tne Neo4j database.
We can set writeNegativeCycles
to true as in the example to override this setting.
The relationships written are always directed, even if the input graph is undirected. 
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