Minimum Weight k-Spanning Tree

This feature is in the alpha tier. For more information on feature tiers, see API Tiers.

Directed

Undirected

Heterogeneous nodes

Heterogeneous relationships

Weighted relationships

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

Sometimes, we might require a spanning tree(a tree where its nodes are connected with each via a single path) that does not necessarily span all nodes in the graph. The K-Spanning tree heuristic algorithm returns a tree with k nodes and k − 1 relationships. Our heuristic processes the result found by Prim’s algorithm for the Minimum Weight Spanning Tree problem. Like Prim, it starts from a given source node, finds a spanning tree for all nodes and then removes nodes using heuristics to produce a tree with 'k' nodes. Note that the source node will not be necessarily included in the final output as the heuristic tries to find a globally good tree.

Considerations

The Minimum weight k-Spanning Tree is NP-Hard. The algorithm in the Neo4j GDS Library is therefore not guaranteed to find the optimal answer, but should hopefully return a good approximation in practice.

Like Prim algorithm, the algorithm focuses only on the component of the source node. If that component has fewer than k nodes, it will not look into other components, but will instead return the component.

Syntax

This section covers the syntax used to execute the k-Spanning Tree heuristic 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.

K Spanning Tree syntax per mode

The following will run the k-spanning tree algorithms and write back results:

CALL gds.kSpanningTree.write(
  graphName: String,
  configuration: Map
)
YIELD effectiveNodeCount: Integer,
      preProcessingMillis: Integer,
      computeMillis: Integer,
      postProcessingMillis: Integer,
      writeMillis: Integer,
      configuration: Map

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	`1`	yes	The algorithm is single-threaded and changing the concurrency parameter has no effect on the runtime.
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.
writeProperty	String	`n/a`	no	The node property in the Neo4j database to which the spanning tree is written.
k	Number	`n/a`	no	The size of the tree to be returned
sourceNode	Integer	`null`	n/a	The starting source node ID.
relationshipWeightProperty	String	`null`	yes	Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted.
objective	String	`'minimum'`	yes	If specified, the parameter dictates whether to seek a minimum or the maximum weight k-spanning tree. By default, the procedure looks for a minimum weight k-spanning tree. Permitted values are 'minimum' and 'maximum'.

Table 3. Results
Name	Type	Description
effectiveNodeCount	Integer	The number of visited nodes.
preProcessingMillis	Integer	Milliseconds for preprocessing the data.
computeMillis	Integer	Milliseconds for running the algorithm.
postProcessingMillis	Integer	Milliseconds for postprocessing results of the algorithm.
writeMillis	Integer	Milliseconds for writing result data back.
configuration	Map	The configuration used for running the algorithm.

Minimum Weight k-Spanning Tree algorithm examples

In this section we will show examples of running the k-Spanning Tree heuristic 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 road network graph of a handful nodes connected in a particular pattern. The example graph looks like this:

The following will create the sample graph depicted in the figure:

CREATE (a:Place {id: 'A'}),
       (b:Place {id: 'B'}),
       (c:Place {id: 'C'}),
       (d:Place {id: 'D'}),
       (e:Place {id: 'E'}),
       (f:Place {id: 'F'}),
       (g:Place {id: 'G'}),
       (d)-[:LINK {cost:4}]->(b),
       (d)-[:LINK {cost:6}]->(e),
       (b)-[:LINK {cost:1}]->(a),
       (b)-[:LINK {cost:3}]->(c),
       (a)-[:LINK {cost:2}]->(c),
       (c)-[:LINK {cost:5}]->(e),
       (f)-[:LINK {cost:1}]->(g);

The following will project and store a named graph:

MATCH (source:Place)
OPTIONAL MATCH (source)-[r:LINK]->(target:Place)
RETURN gds.graph.project(
  'graph',
  source,
  target,
  { relationshipProperties: r { .cost } },
  { undirectedRelationshipTypes: ['*'] }
)

K-Spanning tree examples

Minimum K-Spanning Tree example

In our sample graph we have 7 nodes. By setting the k=3, we define that we want to find a 3-minimum spanning tree that covers 3 nodes and has 2 relationships.

The following will run the k-minimum spanning tree algorithm and write back results:

MATCH (n:Place{id: 'A'})
CALL gds.kSpanningTree.write('graph', {
  k: 3,
  sourceNode: n,
  relationshipWeightProperty: 'cost',
  writeProperty:'kmin'
})
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount
RETURN preProcessingMillis,computeMillis,writeMillis, effectiveNodeCount;

The following will find the nodes that belong to our k-spanning tree result:

MATCH (n)
WITH n.kmin AS p, count(n) AS c
WHERE c = 3
MATCH (n)
WHERE n.kmin = p
RETURN n.id As Place, p as Partition

Table 4. Results
Place	Partition
"A"	0
"B"	0
"C"	0

Nodes A, B, and C form the discovered 3-minimum spanning tree of our graph.

Maximum K-Spanning Tree example

The following will run the k-maximum spanning tree algorithm and write back results:

MATCH (n:Place{id: 'D'})
CALL gds.kSpanningTree.write('graph', {
  k: 3,
  sourceNode: n,
  relationshipWeightProperty: 'cost',
  writeProperty:'kmax',
  objective: 'maximum'
})
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount
RETURN preProcessingMillis,computeMillis,writeMillis, effectiveNodeCount;

Find nodes that belong to our k-spanning tree result:

MATCH (n)
WITH n.kmax AS p, count(n) AS c
WHERE c = 3
MATCH (n)
WHERE n.kmax = p
RETURN n.id As Place, p as Partition

Table 5. Results
Place	Partition
"C"	4
"D"	4
"E"	4

Nodes C, D, and E form a 3-maximum spanning tree of our graph.