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Cube Tree. Dimension: number of group-by values Relation tuples map to a point in the space Aggregates: projection of all data points on all the subspaces. Intersection between a subspace and the orthogonal hyper-plane stores the aggregates. Origin represents aggregate with no grouping

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Cube tree l.jpg
Cube Tree

  • Dimension: number of group-by values

  • Relation tuples map to a point in the space

  • Aggregates: projection of all data points on all the subspaces.

  • Intersection between a subspace and the orthogonal hyper-plane stores the aggregates.

  • Origin represents aggregate with no grouping

  • Query a group-by aggregate on the corresponding hyper-planes


Packed r tree l.jpg
Packed R-Tree

  • Sort-pack: (for multi-dimension data)

    • Achieves excellent clustering

    • Significantly reduces the overlap and dead space

  • A preferred structure for Datcubes storage

  • Representation of Datacube only provide good clustering for half of the total group-bys

  • Degradation due to strong interleaving between points of these group-bys.


Dataless reduced cubetree l.jpg
Dataless & Reduced Cubetree

  • Dataless Cubtree: Only contains aggregate values but no data values

  • Better clustering than a full tree in a R-Tree

    • Projection points are not interleaved

  • Reduced Cubetree: Each hyper-plane which containing aggregates will form a R-Tree independently

  • The dimension of R-Tree reduced by one.

  • Better clustering and query performance


Allocating of goupbys to r trees l.jpg
Allocating of goupbys to R-Trees

  • A set of group-bys are compatible if there exist a sort order that guarantees no dispersion

  • Allocate a group-by to one of the N R-Trees

    • the set of group-bys for this R-Tree is compatible

    • if a group-by cannot find a compatible set

      • assign it to a set that contain all of its gorup-by attributes. (false allocation)

  • Selection of sort order for Packed R-Tree is also an import parameter for favoring some prefered group-bys



Iceberg cube l.jpg
Iceberg Cube

  • Selectively compute only those partitions that satisfy an aggregate condition

  • Aggregate with low support reveal little meaning & make the cube sparse

  • Conditions like

    • Minimum support of a partition

    • Required Range


Bottom up cube l.jpg
Bottom-Up Cube

Parent to compu the child


Bottom up cube 2 l.jpg
Bottom-Up Cube (2)

  • Starting from a bottom single dimension groupby

  • If current inputs can be pruned return

  • Partition the data in this group-by

  • If a partition is greater than the minsup

    • recursive call on BUC with the partition as inputs

  • Loop until all dimensions is done


Bottom up cube 3 l.jpg
Bottom-Up Cube (3)

  • Similar idea of Apriori-gen

  • Apriori will generate all the candidates at the same level first (breadth first)

  • BUC is in depth first manner.

    • To reduce memory requirement

  • Dimension ordering: provide better pruning

    • Cardinality, Skew & Correlation


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