Partial Materialization in Data Cubes

The two main approaches to data cube materialization are: 1) No materialization - don't precompute any of the non-base cuboids, leading to slow multidimensional aggregation on the fly. 2) Full materialization - precompute all the cubes, which leads to very fast query running but requires huge memory.

The text states that full precomputation of the entire data cube requires an excessive amount of memory, and that this depends on the number of dimensions and the cardinality of the dimensions.

The main drawback of not materializing any cuboids (no materialization) is that it leads to slow multidimensional aggregation on the fly during online analytical processing.

The main advantage of fully materializing all cuboids in the data cube is that it leads to very fast query running times, as all the necessary aggregates are precomputed.

Many cells in a data cube cuboid have a measure value of zero and are of little or no interest, meaning the cuboids are often sparse.

The purpose of data cube materialization or precomputation is to enable fast response times during online analytical processing by precomputing some of the cuboids in advance, avoiding redundant computations.

The main purpose of partial materialization is to selectively compute a proper subset of the cuboids, which contains only those cells that satisfy some user specified criterion.

A base cell is a cell that belongs to a base cuboid, while an aggregate cell is a cell that belongs to a non-base cuboid. Each aggregate dimension is indicated by a "*".

An Iceberg Cube is a data cube that only materializes the cells that satisfy a user-specified Iceberg Condition, which means that only the cells with measure values above a certain threshold are computed. This is in contrast to a Full Cube, which materializes all possible cells in the cube.

Multiway Array Aggregation is a technique for efficiently computing data cubes by using a multidimensional array data structure to store and aggregate the data. This approach allows for parallel processing and avoids the need to materialize the entire cube, leading to significant performance improvements.

The BUC (Bottom-Up Cube) algorithm is a method for efficiently computing data cubes by starting with the base cuboid and progressively building up the higher-level cuboids. This approach is different from other algorithms that may start with the apex cuboid and work downwards. BUC is designed to reduce the computational cost and storage requirements of data cube construction.

A Closed Cube is a data cube where no ancestor cell is created if its measure is equal to that of its descendant cell. This is in contrast to a Full Cube, which materializes all possible cells, and an Iceberg Cube, which only materializes cells that satisfy a user-specified Iceberg Condition.

Full materialization refers to computing and storing every possible cuboid (group-by) in the data cube lattice. This maximizes query performance as any group-by can be directly retrieved, but requires immense storage space, especially for high-dimensional cubes.

The formula is: $\sum_{i=1}^{n} \binom{n}{i}L^i$. This represents the sum of combinations of choosing i dimensions (from n dimensions) multiplied by the number of possible group-bys for those i dimensions (L^i).

The base cuboid contains all the finest-granularity data, i.e. no aggregation. It has the maximum number of cells/tuples. The apex cuboid contains just one cell with the grand total aggregated across all dimensions. Base is useful for detailed queries, apex for total values.

One approach is to use a Pipe-Sort-Pipe sequence: 1) Partition/distribute base data 2) Sort each partition on cuboid's dimensions 3) Aggregate/merge sorted partitions to compute cuboid. This avoids redundant computation by sorting once and merging efficiently.

Descriptive data mining techniques summarize and describe the data concisely to highlight general properties. Concept data mining abstracts higher-level concepts/knowledge by generalizing and inferring patterns from the low-level data.

Partial materialization reduces storage requirements compared to full, but requires computing the non-materialized cuboids at query time. Cuboid selection is based on size, sharing, access frequency, etc. to balance storage and query performance goals.