Minimum Weight Spanning Tree

The Minimum Weight Spanning Tree (MST) starts from a given node, and finds all its reachable nodes and the set of relationships that connect the nodes together with the minimum possible weight. Prim’s algorithm is one of the simplest and best-known minimum spanning tree algorithms. The K-Means variant of this algorithm can be used to detect clusters in the graph.

This algorithm is in the alpha tier. For more information on algorithm tiers, see Graph algorithms.

1. History and explanation

The first known algorithm for finding a minimum spanning tree was developed by the Czech scientist Otakar Borůvka in 1926, while trying to find an efficient electricity network for Moravia. Prim’s algorithm was invented by Jarnik in 1930 and rediscovered by Prim in 1957. It is similar to Dijkstra’s shortest path algorithm but, rather than minimizing the total length of a path ending at each relationship, it minimizes the length of each relationship individually. Unlike Dijkstra’s, Prim’s can tolerate negative-weight relationships.

The algorithm operates as follows:

Start with a tree containing only one node (and no relationships).
Select the minimal-weight relationship coming from that node, and add it to our tree.
Repeatedly choose a minimal-weight relationship that joins any node in the tree to one that is not in the tree, adding the new relationship and node to our tree.
When there are no more nodes to add, the tree we have built is a minimum spanning tree.

2. Use-cases - when to use the Minimum Weight Spanning Tree algorithm

Minimum spanning tree was applied to analyze airline and sea connections of Papua New Guinea, and minimize the travel cost of exploring the country. It could be used to help design low-cost tours that visit many destinations across the country. The research mentioned can be found in "An Application of Minimum Spanning Trees to Travel Planning".
Minimum spanning tree has been used to analyze and visualize correlations in a network of currencies, based on the correlation between currency returns. This is described in "Minimum Spanning Tree Application in the Currency Market".
Minimum spanning tree has been shown to be a useful tool to trace the history of transmission of infection, in an outbreak supported by exhaustive clinical research. For more information, see Use of the Minimum Spanning Tree Model for Molecular Epidemiological Investigation of a Nosocomial Outbreak of Hepatitis C Virus Infection.

3. Constraints - when not to use the Minimum Weight Spanning Tree algorithm

The MST algorithm only gives meaningful results when run on a graph, where the relationships have different weights. If the graph has no weights, or all relationships have the same weight, then any spanning tree is a minimum spanning tree.

4. Syntax

The following will run the algorithm and write back results:

CALL gds.alpha.spanningTree.write(
  graphName: string,
  configuration: map
)
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount

The following will compute the minimum weight spanning tree and write the results:

CALL gds.alpha.spanningTree.minimum.write(
  graphName: string,
  configuration: map
)
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount

The following will compute the maximum weight spanning tree and write the results:

CALL gds.alpha.spanningTree.maximum.write(
  graphName: string,
  configuration: map
)
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount

Table 1. Configuration
Name	Type	Default	Optional	Description
startNodeId	Integer	null	no	The start node ID
relationshipWeightProperty	String	null	yes	Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted.
writeProperty	String	'mst'	yes	The relationship type written back as result
weightWriteProperty	String	n/a	no	The weight property of the `writeProperty` relationship type written back

Table 2. 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
writeMillis	Integer	Milliseconds for writing result data back

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

CALL gds.alpha.spanningTree.kmin.write(
  graphName: string,
  configuration: map
)
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount

CALL gds.alpha.spanningTree.kmax.write(
  graphName: string,
  configuration: map
)
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount

Table 3. Configuration
Name	Type	Default	Optional	Description
k	Integer	null	no	The result is a tree with `k` nodes and `k − 1` relationships
startNodeId	Integer	null	no	The start node ID
relationshipWeightProperty	String	null	yes	Name of the relationship property to use as weights. If unspecified, the algorithm runs unweighted.
writeProperty	String	'MST'	yes	The relationship type written back as result
weightWriteProperty	String	n/a	no	The weight property of the `writeProperty` relationship type written back

Table 4. 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
writeMillis	Integer	Milliseconds for writing result data back

5. Minimum Weight Spanning Tree algorithm sample

The following will create a sample graph:

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:

CALL gds.graph.project(
  'graph',
  'Place',
  {
    LINK: {
      type: 'LINK',
      properties: 'cost',
      orientation: 'UNDIRECTED'
    }
  }
)

Minimum weight spanning tree visits all nodes that are in the same connected component as the starting node, and returns a spanning tree of all nodes in the component where the total weight of the relationships is minimized.

The following will run the Minimum Weight Spanning Tree algorithm and write back results:

MATCH (n:Place {id: 'D'})
CALL gds.alpha.spanningTree.minimum.write('graph', {
  startNodeId: id(n),
  relationshipWeightProperty: 'cost',
  writeProperty: 'MINST',
  weightWriteProperty: 'writeCost'
})
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount
RETURN preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount;

To find all pairs of nodes included in our minimum spanning tree, run the following query:

MATCH path = (n:Place {id: 'D'})-[:MINST*]-()
WITH relationships(path) AS rels
UNWIND rels AS rel
WITH DISTINCT rel AS rel
RETURN startNode(rel).id AS source, endNode(rel).id AS destination, rel.writeCost AS cost

Figure 1. Results

Table 5. Results
Source	Destination	Cost
D	B	4
B	A	1
A	C	2
C	E	5

The minimum spanning tree excludes the relationship with cost 6 from D to E, and the one with cost 3 from B to C. Nodes F and G aren’t included because they’re unreachable from D.

Maximum weighted tree spanning algorithm is similar to the minimum one, except that it returns a spanning tree of all nodes in the component where the total weight of the relationships is maximized.

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

MATCH (n:Place{id: 'D'})
CALL gds.alpha.spanningTree.maximum.write('graph', {
  startNodeId: id(n),
  relationshipWeightProperty: 'cost',
  writeProperty: 'MAXST',
  weightWriteProperty: 'writeCost'
})
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount
RETURN preProcessingMillis,computeMillis, writeMillis, effectiveNodeCount;

Figure 2. Results

5.1. K-Spanning tree

Sometimes we want to limit the size of our spanning tree result, as we are only interested in finding a smaller tree within our graph that does not span across all nodes. K-Spanning tree algorithm returns a tree with k nodes and k − 1 relationships.

In our sample graph we have 5 nodes. When we ran MST above, we got a 5-minimum spanning tree returned, that covered all five nodes. By setting the k=3, we define that we want to get returned 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: 'D'})
CALL gds.alpha.spanningTree.kmin.write('graph', {
  k: 3,
  startNodeId: id(n),
  relationshipWeightProperty: 'cost',
  writeProperty:'kminst'
})
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount
RETURN preProcessingMillis,computeMillis,writeMillis, effectiveNodeCount;

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

MATCH (n:Place)
WITH n.id AS Place, n.kminst AS Partition, count(*) AS count
WHERE count = 3
RETURN Place, Partition

Table 6. Results
Place	Partition
A	1
B	1
C	1
D	3
E	4

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

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

MATCH (n:Place{id: 'D'})
CALL gds.alpha.spanningTree.kmax.write('graph', {
  k: 3,
  startNodeId: id(n),
  relationshipWeightProperty: 'cost',
  writeProperty:'kmaxst'
})
YIELD preProcessingMillis, computeMillis, writeMillis, effectiveNodeCount
RETURN preProcessingMillis,computeMillis,writeMillis, effectiveNodeCount;

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

MATCH (n:Place)
WITH n.id AS Place, n.kmaxst AS Partition, count(*) AS count
WHERE count = 3
RETURN Place, Partition

Table 7. Results
Place	Partition
A	0
B	1
C	3
D	3
E	3

Nodes C, D, and E are the result 3-maximum spanning tree of our graph.

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