# Yen’s Shortest Path algorithm

## Glossary

Directed

Directed trait. The algorithm is well-defined 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 well-defined 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.

## Introduction

Yen’s Shortest Path algorithm computes a number of shortest paths between two nodes. The algorithm is often referred to as Yen’s k-Shortest Path algorithm, where k is the number of shortest paths to compute. The algorithm supports weighted graphs with positive relationship weights. It also respects parallel relationships between the same two nodes when computing multiple shortest paths.

For k = 1, the algorithm behaves exactly like Dijkstra’s shortest path algorithm and returns the shortest path. For k = 2, the algorithm returns the shortest path and the second shortest path between the same source and target node. Generally, for k = n, the algorithm computes at most n paths which are discovered in the order of their total cost.

The GDS implementation is based on the original description. For the actual path computation, Yen’s algorithm uses Dijkstra’s shortest path algorithm. The algorithm makes sure that an already discovered shortest path will not be traversed again.

The algorithm implementation is parallelized, but limited by the number of nodes in source-target paths. If these paths are expected to have small length (i.e., a few new nodes) setting a high value for concurrency is discouraged as some of the cores might be left unitilized.

## Syntax

This section covers the syntax used to execute the Yen’s 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.

Yen’s syntax per mode
Run Yen’s in stream mode on a named graph.
CALL gds.shortestPath.yens.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. Configuration
Name Type Default Optional Description

nodeLabels

List of String

['*']

yes

Filter the named graph using the given node labels. Nodes with any of the given labels will be included.

relationshipTypes

List of String

['*']

yes

Filter the named graph using the given relationship types. Relationships with any of the given types will be included.

concurrency

Integer

4

yes

The number of concurrent threads used for running the algorithm.

jobId

String

Generated internally

yes

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

logProgress

Boolean

true

yes

If disabled the progress percentage will not be logged.

sourceNode

Integer

n/a

no

The Neo4j source node or node id.

targetNode

Integer

n/a

no

The Neo4j target node or node id.

k

Integer

1

yes

The number of shortest paths to compute between source and target node.

relationshipWeightProperty

String

null

yes

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

Table 3. 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 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.

Run Yen’s in mutate mode on a named graph.
CALL gds.shortestPath.yens.mutate(
graphName: String,
configuration: Map
)
YIELD
relationshipsWritten: Integer,
preProcessingMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
mutateMillis: Integer,
configuration: Map
Table 4. 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 5. Configuration
Name Type Default Optional Description

mutateRelationshipType

String

n/a

no

The relationship type used for the new relationships written to the projected graph.

nodeLabels

List of String

['*']

yes

Filter the named graph using the given node labels.

relationshipTypes

List of 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.

jobId

String

Generated internally

yes

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

sourceNode

Integer

n/a

no

The Neo4j source node or node id.

targetNode

Integer

n/a

no

The Neo4j target node or node id.

k

Integer

1

yes

The number of shortest paths to compute between source and target node.

relationshipWeightProperty

String

null

yes

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

Table 6. Results
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.

Run Yen’s in write mode on a named graph.
CALL gds.shortestPath.yens.write(
graphName: String,
configuration: Map
)
YIELD
relationshipsWritten: Integer,
preProcessingMillis: Integer,
computeMillis: Integer,
postProcessingMillis: Integer,
writeMillis: Integer,
configuration: Map
Table 7. 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 8. Configuration
Name Type Default Optional Description

nodeLabels

List of String

['*']

yes

Filter the named graph using the given node labels. Nodes with any of the given labels will be included.

relationshipTypes

List of String

['*']

yes

Filter the named graph using the given relationship types. Relationships with any of the given types will be included.

concurrency

Integer

4

yes

The number of concurrent threads used for running the algorithm.

jobId

String

Generated internally

yes

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

logProgress

Boolean

true

yes

If disabled the progress percentage will not be logged.

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.

sourceNode

Integer

n/a

no

The Neo4j source node or node id.

targetNode

Integer

n/a

no

The Neo4j target node or node id.

k

Integer

1

yes

The number of shortest paths to compute between source and target node.

relationshipWeightProperty

String

null

yes

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

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 9. Results
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.

## Examples

 All the examples below should be run in an empty database. The examples use native projections as the norm, although Cypher projections can be used as well.

In this section we will show examples of running the Yen’s 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:Location {name: 'A'}),
(b:Location {name: 'B'}),
(c:Location {name: 'C'}),
(d:Location {name: 'D'}),
(e:Location {name: 'E'}),
(f:Location {name: '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.

The following statement will project a graph using a native projection and store it in the graph catalog under the name 'myGraph'.
CALL gds.graph.project(
'myGraph',
'Location',
{
relationshipProperties: 'cost'
}
)

In the following example we will demonstrate the use of the Yen’s Shortest Path algorithm using this graph.

### 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:Location {name: 'A'}), (target:Location {name: 'F'})
CALL gds.shortestPath.yens.write.estimate('myGraph', {
sourceNode: source,
targetNode: target,
k: 3,
relationshipWeightProperty: 'cost',
writeRelationshipType: 'PATH'
})
YIELD nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
RETURN nodeCount, relationshipCount, bytesMin, bytesMax, requiredMemory
Table 10. Results
nodeCount relationshipCount bytesMin bytesMax requiredMemory

6

9

4608

4608

"4608 Bytes"

### 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:Location {name: 'A'}), (target:Location {name: 'F'})
CALL gds.shortestPath.yens.stream('myGraph', {
sourceNode: source,
targetNode: target,
k: 3,
relationshipWeightProperty: 'cost'
})
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 11. Results
index sourceNodeName targetNodeName totalCost nodeNames costs path

0

"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]]

1

"A"

"F"

160.0

[A, C, D, E, F]

[0.0, 50.0, 90.0, 120.0, 160.0]

[Node[0], Node[2], Node[3], Node[4], Node[5]]

2

"A"

"F"

170.0

[A, B, D, F]

[0.0, 50.0, 90.0, 170.0]

[Node[0], Node[1], Node[3], Node[5]]

The result shows the three shortest paths between node A and node F. The first two paths have the same total cost, however the first one traversed from A to D via the B node, while the second traversed via the C node. The third path has a higher total cost as it goes directly from D to F using the relationship with a cost of 80, whereas the detour via E for the first two paths costs 70. 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.

### 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:Location {name: 'A'}), (target:Location {name: 'F'})
CALL gds.shortestPath.yens.mutate('myGraph', {
sourceNode: source,
targetNode: target,
k: 3,
relationshipWeightProperty: 'cost',
mutateRelationshipType: 'PATH'
})
YIELD relationshipsWritten
RETURN relationshipsWritten
Table 12. Results
relationshipsWritten

3

After executing the above query, the projected graph will be updated with a new relationship of type PATH. The new relationship will store a single property totalCost.

 The relationships produced are always directed, even if the input graph is undirected.

### 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:Location {name: 'A'}), (target:Location {name: 'F'})
CALL gds.shortestPath.yens.write('myGraph', {
sourceNode: source,
targetNode: target,
k: 3,
relationshipWeightProperty: 'cost',
writeRelationshipType: 'PATH',
writeNodeIds: true,
writeCosts: true
})
YIELD relationshipsWritten
RETURN relationshipsWritten
Table 13. Results
relationshipsWritten

3

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

 The relationships written are always directed, even if the input graph is undirected.