Package org.apache.spark.graphx

Class Graph<VD,ED>

Object
org.apache.spark.graphx.Graph<VD,ED>
Type Parameters:
VD - the vertex attribute type
ED - the edge attribute type
All Implemented Interfaces:
Serializable, scala.Serializable
Direct Known Subclasses:
GraphImpl
public abstract class Graph<VD,ED> extends Object implements scala.Serializable
The Graph abstractly represents a graph with arbitrary objects associated with vertices and edges. The graph provides basic operations to access and manipulate the data associated with vertices and edges as well as the underlying structure. Like Spark RDDs, the graph is a functional data-structure in which mutating operations return new graphs.

See Also:
Note:
GraphOps contains additional convenience operations and graph algorithms.

  • Method Summary

    Modifier and Type
    Method
    Description
    <A> VertexRDD<A>
    aggregateMessages(scala.Function1<EdgeContext<VD,ED,A>,scala.runtime.BoxedUnit> sendMsg, scala.Function2<A,A,A> mergeMsg, TripletFields tripletFields, scala.reflect.ClassTag<A> evidence$11)
    Aggregates values from the neighboring edges and vertices of each vertex.
    static <VD, ED> Graph<VD,ED>
    apply(RDD<scala.Tuple2<Object,VD>> vertices, RDD<Edge<ED>> edges, VD defaultVertexAttr, StorageLevel edgeStorageLevel, StorageLevel vertexStorageLevel, scala.reflect.ClassTag<VD> evidence$18, scala.reflect.ClassTag<ED> evidence$19)
    Construct a graph from a collection of vertices and edges with attributes.
    abstract Graph<VD,ED>
    cache()
    Caches the vertices and edges associated with this graph at the previously-specified target storage levels, which default to MEMORY_ONLY.
    abstract void
    checkpoint()
    Mark this Graph for checkpointing.
    abstract EdgeRDD<ED>
    edges()
    An RDD containing the edges and their associated attributes.
    static <VD, ED> Graph<VD,ED>
    fromEdges(RDD<Edge<ED>> edges, VD defaultValue, StorageLevel edgeStorageLevel, StorageLevel vertexStorageLevel, scala.reflect.ClassTag<VD> evidence$16, scala.reflect.ClassTag<ED> evidence$17)
    Construct a graph from a collection of edges.
    static <VD> Graph<VD,Object>
    fromEdgeTuples(RDD<scala.Tuple2<Object,Object>> rawEdges, VD defaultValue, scala.Option<PartitionStrategy> uniqueEdges, StorageLevel edgeStorageLevel, StorageLevel vertexStorageLevel, scala.reflect.ClassTag<VD> evidence$15)
    Construct a graph from a collection of edges encoded as vertex id pairs.
    abstract scala.collection.Seq<String>
    getCheckpointFiles()
    Gets the name of the files to which this Graph was checkpointed.
    static <VD, ED> GraphOps<VD,ED>
    graphToGraphOps(Graph<VD,ED> g, scala.reflect.ClassTag<VD> evidence$20, scala.reflect.ClassTag<ED> evidence$21)
    Implicitly extracts the GraphOps member from a graph.
    abstract Graph<VD,ED>
    groupEdges(scala.Function2<ED,ED,ED> merge)
    Merges multiple edges between two vertices into a single edge.
    abstract boolean
    isCheckpointed()
    Return whether this Graph has been checkpointed or not.
    <ED2> Graph<VD,ED2>
    mapEdges(scala.Function1<Edge<ED>,ED2> map, scala.reflect.ClassTag<ED2> evidence$4)
    Transforms each edge attribute in the graph using the map function.
    abstract <ED2> Graph<VD,ED2>
    mapEdges(scala.Function2<Object,scala.collection.Iterator<Edge<ED>>,scala.collection.Iterator<ED2>> map, scala.reflect.ClassTag<ED2> evidence$5)
    Transforms each edge attribute using the map function, passing it a whole partition at a time.
    <ED2> Graph<VD,ED2>
    mapTriplets(scala.Function1<EdgeTriplet<VD,ED>,ED2> map, TripletFields tripletFields, scala.reflect.ClassTag<ED2> evidence$7)
    Transforms each edge attribute using the map function, passing it the adjacent vertex attributes as well.
    <ED2> Graph<VD,ED2>
    mapTriplets(scala.Function1<EdgeTriplet<VD,ED>,ED2> map, scala.reflect.ClassTag<ED2> evidence$6)
    Transforms each edge attribute using the map function, passing it the adjacent vertex attributes as well.
    abstract <ED2> Graph<VD,ED2>
    mapTriplets(scala.Function2<Object,scala.collection.Iterator<EdgeTriplet<VD,ED>>,scala.collection.Iterator<ED2>> map, TripletFields tripletFields, scala.reflect.ClassTag<ED2> evidence$8)
    Transforms each edge attribute a partition at a time using the map function, passing it the adjacent vertex attributes as well.
    abstract <VD2> Graph<VD2,ED>
    mapVertices(scala.Function2<Object,VD,VD2> map, scala.reflect.ClassTag<VD2> evidence$3, scala.Predef.$eq$colon$eq<VD,VD2> eq)
    Transforms each vertex attribute in the graph using the map function.
    abstract <VD2, ED2> Graph<VD,ED>
    mask(Graph<VD2,ED2> other, scala.reflect.ClassTag<VD2> evidence$9, scala.reflect.ClassTag<ED2> evidence$10)
    Restricts the graph to only the vertices and edges that are also in other, but keeps the attributes from this graph.
    GraphOps<VD,ED>
    ops()
    The associated GraphOps object.
    abstract <U, VD2> Graph<VD2,ED>
    outerJoinVertices(RDD<scala.Tuple2<Object,U>> other, scala.Function3<Object,VD,scala.Option<U>,VD2> mapFunc, scala.reflect.ClassTag<U> evidence$13, scala.reflect.ClassTag<VD2> evidence$14, scala.Predef.$eq$colon$eq<VD,VD2> eq)
    Joins the vertices with entries in the table RDD and merges the results using mapFunc.
    abstract Graph<VD,ED>
    partitionBy(PartitionStrategy partitionStrategy)
    Repartitions the edges in the graph according to partitionStrategy.
    abstract Graph<VD,ED>
    partitionBy(PartitionStrategy partitionStrategy, int numPartitions)
    Repartitions the edges in the graph according to partitionStrategy.
    abstract Graph<VD,ED>
    persist(StorageLevel newLevel)
    Caches the vertices and edges associated with this graph at the specified storage level, ignoring any target storage levels previously set.
    abstract Graph<VD,ED>
    reverse()
    Reverses all edges in the graph.
    abstract Graph<VD,ED>
    subgraph(scala.Function1<EdgeTriplet<VD,ED>,Object> epred, scala.Function2<Object,VD,Object> vpred)
    Restricts the graph to only the vertices and edges satisfying the predicates.
    abstract RDD<EdgeTriplet<VD,ED>>
    triplets()
    An RDD containing the edge triplets, which are edges along with the vertex data associated with the adjacent vertices.
    abstract Graph<VD,ED>
    unpersist(boolean blocking)
    Uncaches both vertices and edges of this graph.
    abstract Graph<VD,ED>
    unpersistVertices(boolean blocking)
    Uncaches only the vertices of this graph, leaving the edges alone.
    abstract VertexRDD<VD>
    vertices()
    An RDD containing the vertices and their associated attributes.

    Methods inherited from class java.lang.Object

    equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

  • Method Details

    • fromEdgeTuples

      public static <VD> Graph<VD,Object> fromEdgeTuples(RDD<scala.Tuple2<Object,Object>> rawEdges, VD defaultValue, scala.Option<PartitionStrategy> uniqueEdges, StorageLevel edgeStorageLevel, StorageLevel vertexStorageLevel, scala.reflect.ClassTag<VD> evidence$15)
      Construct a graph from a collection of edges encoded as vertex id pairs.

      Parameters:
      rawEdges - a collection of edges in (src, dst) form
      defaultValue - the vertex attributes with which to create vertices referenced by the edges
      uniqueEdges - if multiple identical edges are found they are combined and the edge attribute is set to the sum. Otherwise duplicate edges are treated as separate. To enable uniqueEdges, a PartitionStrategy must be provided.
      edgeStorageLevel - the desired storage level at which to cache the edges if necessary
      vertexStorageLevel - the desired storage level at which to cache the vertices if necessary

      evidence$15 - (undocumented)
      Returns:
      a graph with edge attributes containing either the count of duplicate edges or 1 (if uniqueEdges is None) and vertex attributes containing the total degree of each vertex.

    • fromEdges

      public static <VD, ED> Graph<VD,ED> fromEdges(RDD<Edge<ED>> edges, VD defaultValue, StorageLevel edgeStorageLevel, StorageLevel vertexStorageLevel, scala.reflect.ClassTag<VD> evidence$16, scala.reflect.ClassTag<ED> evidence$17)
      Construct a graph from a collection of edges.

      Parameters:
      edges - the RDD containing the set of edges in the graph
      defaultValue - the default vertex attribute to use for each vertex
      edgeStorageLevel - the desired storage level at which to cache the edges if necessary
      vertexStorageLevel - the desired storage level at which to cache the vertices if necessary

      evidence$16 - (undocumented)
      evidence$17 - (undocumented)
      Returns:
      a graph with edge attributes described by edges and vertices given by all vertices in edges with value defaultValue

    • apply

      public static <VD, ED> Graph<VD,ED> apply(RDD<scala.Tuple2<Object,VD>> vertices, RDD<Edge<ED>> edges, VD defaultVertexAttr, StorageLevel edgeStorageLevel, StorageLevel vertexStorageLevel, scala.reflect.ClassTag<VD> evidence$18, scala.reflect.ClassTag<ED> evidence$19)
      Construct a graph from a collection of vertices and edges with attributes. Duplicate vertices are picked arbitrarily and vertices found in the edge collection but not in the input vertices are assigned the default attribute.

      Parameters:
      vertices - the "set" of vertices and their attributes
      edges - the collection of edges in the graph
      defaultVertexAttr - the default vertex attribute to use for vertices that are mentioned in edges but not in vertices
      edgeStorageLevel - the desired storage level at which to cache the edges if necessary
      vertexStorageLevel - the desired storage level at which to cache the vertices if necessary
      evidence$18 - (undocumented)
      evidence$19 - (undocumented)
      Returns:
      (undocumented)

    • graphToGraphOps

      public static <VD, ED> GraphOps<VD,ED> graphToGraphOps(Graph<VD,ED> g, scala.reflect.ClassTag<VD> evidence$20, scala.reflect.ClassTag<ED> evidence$21)
      Implicitly extracts the GraphOps member from a graph.

      To improve modularity the Graph type only contains a small set of basic operations. All the convenience operations are defined in the GraphOps class which may be shared across multiple graph implementations.

      Parameters:
      g - (undocumented)
      evidence$20 - (undocumented)
      evidence$21 - (undocumented)
      Returns:
      (undocumented)

    • vertices

      public abstract VertexRDD<VD> vertices()
      An RDD containing the vertices and their associated attributes.

      Returns:
      an RDD containing the vertices in this graph
      Note:
      vertex ids are unique.

    • edges

      public abstract EdgeRDD<ED> edges()
      An RDD containing the edges and their associated attributes. The entries in the RDD contain just the source id and target id along with the edge data.

      Returns:
      an RDD containing the edges in this graph

      See Also:
      • Edge for the edge type.
      • Graph#triplets to get an RDD which contains all the edges along with their vertex data.

    • triplets

      public abstract RDD<EdgeTriplet<VD,ED>> triplets()
      An RDD containing the edge triplets, which are edges along with the vertex data associated with the adjacent vertices. The caller should use edges() if the vertex data are not needed, i.e. if only the edge data and adjacent vertex ids are needed.

      Returns:
      an RDD containing edge triplets

      Example:
      This operation might be used to evaluate a graph coloring where we would like to check that both vertices are a different color. 
      
 type Color = Int
 val graph: Graph[Color, Int] = GraphLoader.edgeListFile("hdfs://file.tsv")
 val numInvalid = graph.triplets.map(e => if (e.src.data == e.dst.data) 1 else 0).sum

    • persist

      public abstract Graph<VD,ED> persist(StorageLevel newLevel)
      Caches the vertices and edges associated with this graph at the specified storage level, ignoring any target storage levels previously set.

      Parameters:
      newLevel - the level at which to cache the graph.

      Returns:
      A reference to this graph for convenience.

    • cache

      public abstract Graph<VD,ED> cache()
      Caches the vertices and edges associated with this graph at the previously-specified target storage levels, which default to MEMORY_ONLY. This is used to pin a graph in memory enabling multiple queries to reuse the same construction process.
      Returns:
      (undocumented)

    • checkpoint

      public abstract void checkpoint()
      Mark this Graph for checkpointing. It will be saved to a file inside the checkpoint directory set with SparkContext.setCheckpointDir() and all references to its parent RDDs will be removed. It is strongly recommended that this Graph is persisted in memory, otherwise saving it on a file will require recomputation.

    • isCheckpointed

      public abstract boolean isCheckpointed()
      Return whether this Graph has been checkpointed or not. This returns true iff both the vertices RDD and edges RDD have been checkpointed.
      Returns:
      (undocumented)

    • getCheckpointFiles

      public abstract scala.collection.Seq<String> getCheckpointFiles()
      Gets the name of the files to which this Graph was checkpointed. (The vertices RDD and edges RDD are checkpointed separately.)
      Returns:
      (undocumented)

    • unpersist

      public abstract Graph<VD,ED> unpersist(boolean blocking)
      Uncaches both vertices and edges of this graph. This is useful in iterative algorithms that build a new graph in each iteration.

      Parameters:
      blocking - Whether to block until all data is unpersisted (default: false)
      Returns:
      (undocumented)

    • unpersistVertices

      public abstract Graph<VD,ED> unpersistVertices(boolean blocking)
      Uncaches only the vertices of this graph, leaving the edges alone. This is useful in iterative algorithms that modify the vertex attributes but reuse the edges. This method can be used to uncache the vertex attributes of previous iterations once they are no longer needed, improving GC performance.

      Parameters:
      blocking - Whether to block until all data is unpersisted (default: false)
      Returns:
      (undocumented)

    • partitionBy

      public abstract Graph<VD,ED> partitionBy(PartitionStrategy partitionStrategy)
      Repartitions the edges in the graph according to partitionStrategy.

      Parameters:
      partitionStrategy - the partitioning strategy to use when partitioning the edges in the graph.
      Returns:
      (undocumented)

    • partitionBy

      public abstract Graph<VD,ED> partitionBy(PartitionStrategy partitionStrategy, int numPartitions)
      Repartitions the edges in the graph according to partitionStrategy.

      Parameters:
      partitionStrategy - the partitioning strategy to use when partitioning the edges in the graph.
      numPartitions - the number of edge partitions in the new graph.
      Returns:
      (undocumented)

    • mapVertices

      public abstract <VD2> Graph<VD2,ED> mapVertices(scala.Function2<Object,VD,VD2> map, scala.reflect.ClassTag<VD2> evidence$3, scala.Predef.$eq$colon$eq<VD,VD2> eq)
      Transforms each vertex attribute in the graph using the map function.

      Parameters:
      map - the function from a vertex object to a new vertex value

      evidence$3 - (undocumented)
      eq - (undocumented)
      Returns:
      (undocumented)
      Example:
      We might use this operation to change the vertex values from one type to another to initialize an algorithm. 
      
 val rawGraph: Graph[(), ()] = Graph.textFile("hdfs://file")
 val root = 42
 var bfsGraph = rawGraph.mapVertices[Int]((vid, data) => if (vid == root) 0 else Math.MaxValue)

      Note:
      The new graph has the same structure. As a consequence the underlying index structures can be reused.

    • mapEdges

      public <ED2> Graph<VD,ED2> mapEdges(scala.Function1<Edge<ED>,ED2> map, scala.reflect.ClassTag<ED2> evidence$4)
      Transforms each edge attribute in the graph using the map function. The map function is not passed the vertex value for the vertices adjacent to the edge. If vertex values are desired, use mapTriplets.

      Parameters:
      map - the function from an edge object to a new edge value.

      evidence$4 - (undocumented)
      Returns:
      (undocumented)
      Example:
      This function might be used to initialize edge attributes.

      Note:
      This graph is not changed and that the new graph has the same structure. As a consequence the underlying index structures can be reused.

    • mapEdges

      public abstract <ED2> Graph<VD,ED2> mapEdges(scala.Function2<Object,scala.collection.Iterator<Edge<ED>>,scala.collection.Iterator<ED2>> map, scala.reflect.ClassTag<ED2> evidence$5)
      Transforms each edge attribute using the map function, passing it a whole partition at a time. The map function is given an iterator over edges within a logical partition as well as the partition's ID, and it should return a new iterator over the new values of each edge. The new iterator's elements must correspond one-to-one with the old iterator's elements. If adjacent vertex values are desired, use mapTriplets.

      Parameters:
      map - a function that takes a partition id and an iterator over all the edges in the partition, and must return an iterator over the new values for each edge in the order of the input iterator

      evidence$5 - (undocumented)
      Returns:
      (undocumented)
      Note:
      This does not change the structure of the graph or modify the values of this graph. As a consequence the underlying index structures can be reused.

    • mapTriplets

      public <ED2> Graph<VD,ED2> mapTriplets(scala.Function1<EdgeTriplet<VD,ED>,ED2> map, scala.reflect.ClassTag<ED2> evidence$6)
      Transforms each edge attribute using the map function, passing it the adjacent vertex attributes as well. If adjacent vertex values are not required, consider using mapEdges instead.

      Parameters:
      map - the function from an edge object to a new edge value.

      evidence$6 - (undocumented)
      Returns:
      (undocumented)
      Example:
      This function might be used to initialize edge attributes based on the attributes associated with each vertex. 
      
 val rawGraph: Graph[Int, Int] = someLoadFunction()
 val graph = rawGraph.mapTriplets[Int]( edge =>
   edge.src.data - edge.dst.data)

      Note:
      This does not change the structure of the graph or modify the values of this graph. As a consequence the underlying index structures can be reused.

    • mapTriplets

      public <ED2> Graph<VD,ED2> mapTriplets(scala.Function1<EdgeTriplet<VD,ED>,ED2> map, TripletFields tripletFields, scala.reflect.ClassTag<ED2> evidence$7)
      Transforms each edge attribute using the map function, passing it the adjacent vertex attributes as well. If adjacent vertex values are not required, consider using mapEdges instead.

      Parameters:
      map - the function from an edge object to a new edge value.
      tripletFields - which fields should be included in the edge triplet passed to the map function. If not all fields are needed, specifying this can improve performance.

      evidence$7 - (undocumented)
      Returns:
      (undocumented)
      Example:
      This function might be used to initialize edge attributes based on the attributes associated with each vertex. 
      
 val rawGraph: Graph[Int, Int] = someLoadFunction()
 val graph = rawGraph.mapTriplets[Int]( edge =>
   edge.src.data - edge.dst.data)

      Note:
      This does not change the structure of the graph or modify the values of this graph. As a consequence the underlying index structures can be reused.

    • mapTriplets

      public abstract <ED2> Graph<VD,ED2> mapTriplets(scala.Function2<Object,scala.collection.Iterator<EdgeTriplet<VD,ED>>,scala.collection.Iterator<ED2>> map, TripletFields tripletFields, scala.reflect.ClassTag<ED2> evidence$8)
      Transforms each edge attribute a partition at a time using the map function, passing it the adjacent vertex attributes as well. The map function is given an iterator over edge triplets within a logical partition and should yield a new iterator over the new values of each edge in the order in which they are provided. If adjacent vertex values are not required, consider using mapEdges instead.

      Parameters:
      map - the iterator transform
      tripletFields - which fields should be included in the edge triplet passed to the map function. If not all fields are needed, specifying this can improve performance.

      evidence$8 - (undocumented)
      Returns:
      (undocumented)
      Note:
      This does not change the structure of the graph or modify the values of this graph. As a consequence the underlying index structures can be reused.

    • reverse

      public abstract Graph<VD,ED> reverse()
      Reverses all edges in the graph. If this graph contains an edge from a to b then the returned graph contains an edge from b to a.
      Returns:
      (undocumented)

    • subgraph

      public abstract Graph<VD,ED> subgraph(scala.Function1<EdgeTriplet<VD,ED>,Object> epred, scala.Function2<Object,VD,Object> vpred)
      Restricts the graph to only the vertices and edges satisfying the predicates. The resulting subgraph satisfies 

      
 V' = {v : for all v in V where vpred(v)}
 E' = {(u,v): for all (u,v) in E where epred((u,v)) && vpred(u) && vpred(v)}

      Parameters:
      epred - the edge predicate, which takes a triplet and evaluates to true if the edge is to remain in the subgraph. Note that only edges where both vertices satisfy the vertex predicate are considered.

      vpred - the vertex predicate, which takes a vertex object and evaluates to true if the vertex is to be included in the subgraph

      Returns:
      the subgraph containing only the vertices and edges that satisfy the predicates

    • mask

      public abstract <VD2, ED2> Graph<VD,ED> mask(Graph<VD2,ED2> other, scala.reflect.ClassTag<VD2> evidence$9, scala.reflect.ClassTag<ED2> evidence$10)
      Restricts the graph to only the vertices and edges that are also in other, but keeps the attributes from this graph.
      Parameters:
      other - the graph to project this graph onto
      evidence$9 - (undocumented)
      evidence$10 - (undocumented)
      Returns:
      a graph with vertices and edges that exist in both the current graph and other, with vertex and edge data from the current graph

    • groupEdges

      public abstract Graph<VD,ED> groupEdges(scala.Function2<ED,ED,ED> merge)
      Merges multiple edges between two vertices into a single edge. For correct results, the graph must have been partitioned using partitionBy.

      Parameters:
      merge - the user-supplied commutative associative function to merge edge attributes for duplicate edges.

      Returns:
      The resulting graph with a single edge for each (source, dest) vertex pair.

    • aggregateMessages

      public <A> VertexRDD<A> aggregateMessages(scala.Function1<EdgeContext<VD,ED,A>,scala.runtime.BoxedUnit> sendMsg, scala.Function2<A,A,A> mergeMsg, TripletFields tripletFields, scala.reflect.ClassTag<A> evidence$11)
      Aggregates values from the neighboring edges and vertices of each vertex. The user-supplied sendMsg function is invoked on each edge of the graph, generating 0 or more messages to be sent to either vertex in the edge. The mergeMsg function is then used to combine all messages destined to the same vertex.

      Parameters:
      sendMsg - runs on each edge, sending messages to neighboring vertices using the EdgeContext.
      mergeMsg - used to combine messages from sendMsg destined to the same vertex. This combiner should be commutative and associative.
      tripletFields - which fields should be included in the EdgeContext passed to the sendMsg function. If not all fields are needed, specifying this can improve performance.

      evidence$11 - (undocumented)
      Returns:
      (undocumented)
      Example:
      We can use this function to compute the in-degree of each vertex 
      
 val rawGraph: Graph[_, _] = Graph.textFile("twittergraph")
 val inDeg: RDD[(VertexId, Int)] =
   rawGraph.aggregateMessages[Int](ctx => ctx.sendToDst(1), _ + _)

      Note:
      By expressing computation at the edge level we achieve maximum parallelism. This is one of the core functions in the Graph API that enables neighborhood level computation. For example this function can be used to count neighbors satisfying a predicate or implement PageRank.

    • outerJoinVertices

      public abstract <U, VD2> Graph<VD2,ED> outerJoinVertices(RDD<scala.Tuple2<Object,U>> other, scala.Function3<Object,VD,scala.Option<U>,VD2> mapFunc, scala.reflect.ClassTag<U> evidence$13, scala.reflect.ClassTag<VD2> evidence$14, scala.Predef.$eq$colon$eq<VD,VD2> eq)
      Joins the vertices with entries in the table RDD and merges the results using mapFunc. The input table should contain at most one entry for each vertex. If no entry in other is provided for a particular vertex in the graph, the map function receives None.

      Parameters:
      other - the table to join with the vertices in the graph. The table should contain at most one entry for each vertex.
      mapFunc - the function used to compute the new vertex values. The map function is invoked for all vertices, even those that do not have a corresponding entry in the table.

      evidence$13 - (undocumented)
      evidence$14 - (undocumented)
      eq - (undocumented)
      Returns:
      (undocumented)
      Example:
      This function is used to update the vertices with new values based on external data. For example we could add the out-degree to each vertex record: 

      
 val rawGraph: Graph[_, _] = Graph.textFile("webgraph")
 val outDeg: RDD[(VertexId, Int)] = rawGraph.outDegrees
 val graph = rawGraph.outerJoinVertices(outDeg) {
   (vid, data, optDeg) => optDeg.getOrElse(0)
 }

    • ops

      public GraphOps<VD,ED> ops()
      The associated GraphOps object.
      Returns:
      (undocumented)