• 3/18/2013
  • 2


  • Data Cube Computation: Preliminary Concepts 
  • Data Cube Computation Methods
  • Processing Advanced Queries by Exploring Data Cube Technology
  • Multidimensional Data Analysis in Cube Space
  • Summary

Data Cube: A Lattice of Cuboids

Data Cube: A Lattice of Cuboids

Cube Materialization: Full Cube vs. Iceberg Cube

  • Full cube vs. iceberg cube
      • compute cube sales iceberg as
      • select month, city, customer group, count(*)
      • from salesInfo
      • cube by month, city, customer group
      • iceberg condition: having count(*) >= min support
  • Computing only the cuboid cells whose measure satisfies the iceberg condition
  • Only a small portion of cells may be “above the water’’ in a sparse cube
  • Avoid explosive growth: A cube with 100 dimensions
    • 2 base cells: (a1, a2, …., a100), (b1, b2, …, b100)
    • How many aggregate cells if “having count >= 1”?
    • What about “having count >= 2”?

Iceberg Cube, Closed Cube & Cube Shell

  • Is iceberg cube good enough?
    • 2 base cells: {(a1, a2, a3 . . . , a100):10, (a1, a2, b3, . . . , b100):10}
    • How many cells will the iceberg cube have if having count(*) >= 10? Hint: A huge but tricky number!
  • Close cube:
    • Closed cell c: if there exists no cell d, s.t. d is a descendant of c, and d has the same measure value as c.
    • Closed cube: a cube consisting of only closed cells
    • What is the closed cube of the above base cuboid? Hint: only 3 cells
  • Cube Shell
    • Precompute only the cuboids involving a small # of dimensions, e.g., 3
    • More dimension combinations will need to be computed on the fly
  • For (A1, A2, … A10), how many combinations to compute?

Roadmap for Efficient Computation

  • General cube computation heuristics (Agarwal et al.’96)
  • Computing full/iceberg cubes: 3 methodologies
    • Bottom-Up: Multi-Way array aggregation (Zhao, Deshpande & Naughton, SIGMOD’97)
    • Top-down:
      • BUC (Beyer & Ramarkrishnan, SIGMOD’99)
      • H-cubing technique (Han, Pei, Dong & Wang: SIGMOD’01)
    • Integrating Top-Down and Bottom-Up:
      • Star-cubing algorithm (Xin, Han, Li & Wah: VLDB’03)
  • High-dimensional OLAP: A Minimal Cubing Approach (Li, et al. VLDB’04)
  • Computing alternative kinds of cubes:
    • Partial cube, closed cube, approximate cube, etc.

General Heuristics (Agarwal et al. VLDB’96)

  • Sorting, hashing, and grouping operations are applied to the dimension attributes in order to reorder and cluster related tuples
  • Aggregates may be computed from previously computed aggregates, rather than from the base fact table
    • Smallest-child: computing a cuboid from the smallest, previously computed cuboid
    • Cache-results: caching results of a cuboid from which other cuboids are computed to reduce disk I/Os
    • Amortize-scans: computing as many as possible cuboids at the same time to amortize disk reads
    • Share-sorts: sharing sorting costs cross multiple cuboids when sort-based method is used
    • Share-partitions: sharing the partitioning cost across multiple cuboids when hash-based algorithms are used

Data Cube Computation Methods

  • Multi-Way Array Aggregation
  • BUC
  • Star-Cubing
  • High-Dimensional OLAP

Multi-Way Array Aggregation

  • Array-based “bottom-up” algorithm
  • Using multi-dimensional chunks
  • No direct tuple comparisons
  • Simultaneous aggregation on multiple dimensions
  • Intermediate aggregate values are re-used for computing ancestor cuboids
  • Cannot do Apriori pruning: No iceberg optimization

Multi-way Array Aggregation for Cube Computation (MOLAP)

  • Partition arrays into chunks (a small subcube which fits in memory).
  • Compressed sparse array addressing: (chunk_id, offset)
  • Compute aggregates in “multiway” by visiting cube cells in the order which minimizes the # of times to visit each cell, and reduces memory access and storage cost.

Multi-way Array Aggregation for Cube Computation (3-D to 2-D)

  • The best order is the one that minimizes the memory requirement and reduced I/Os  

Multi-way Array Aggregation for Cube Computation (2-D to 1-D)

Multi-Way Array Aggregation for Cube Computation (Method Summary)

  • Method: the planes should be sorted and computed according to their size in ascending order
    • Idea: keep the smallest plane in the main memory, fetch and compute only one chunk at a time for the largest plane
  • Limitation of the method: computing well only for a small number of dimensions
    • If there are a large number of dimensions, “top-down” computation and iceberg cube computation methods can be explored

Bottom-Up Computation (BUC)

    • BUC (Beyer & Ramakrishnan, SIGMOD’99)
    • Bottom-up cube computation
      • (Note: top-down in our view!)
    • Divides dimensions into partitions and facilitates iceberg pruning
      • If a partition does not satisfy min_sup , its descendants can be pruned
      • If minsup = 1 →compute full CUBE!
    • No simultaneous aggregation

      BUC: Partitioning

      • Usually, entire data set can’t fit in main memory
      • Sort distinct values
        • partition into blocks that fit
      • Continue processing
      • Optimizations
        • Partitioning
          • External Sorting, Hashing, Counting Sort
        • Ordering dimensions to encourage pruning
          • Cardinality, Skew, Correlation
        • Collapsing duplicates
          • Can’t do holistic aggregates anymore!

      Star-Cubing: An Integrating Method

      • D. Xin, J. Han, X. Li, B. W. Wah, Star-Cubing: Computing Iceberg Cubes by Top-Down and Bottom-Up Integration, VLDB'03
      • Explore shared dimensions
        • E.g., dimension A is the shared dimension of ACD and AD
        • ABD/AB means cuboid ABD has shared dimensions AB
      • Allows for shared computations
        • e.g., cuboid AB is computed simultaneously as ABD
      • Aggregate in a top-down manner but with the bottom-up sub-layer underneath which will allow Apriori pruning
      • Shared dimensions grow in bottom-up fashion

      Iceberg Pruning in Shared Dimensions

      • Anti-monotonic property of shared dimensions
        • If the measure is anti-monotonic , and if the aggregate value on a shared dimension does not satisfy the iceberg condition , then all the cells extended from this shared dimension cannot satisfy the condition either
      • Intuition: if we can compute the shared dimensions before the actual cuboid, we can use them to do Apriori pruning
      • Problem: how to prune while still aggregate simultaneously on multiple dimensions?

      Cell Trees

      • Use a tree structure similar to H-tree to represent cuboids
      • Collapses common prefixes to save memory
      • Keep count at node
      • Traverse the tree to retrieve a particular tuple

      Star Attributes and Star Nodes

      • Intuition: If a single-dimensional aggregate on an attribute value p does not satisfy the iceberg condition, it is useless to distinguish them during the iceberg computation  
        • E.g., b2, b3, b4, c1, c2, c4, d1, d2, d3  
      • Solution: Replace such attributes by a *. Such attributes are star attributes, and the corresponding nodes in the cell tree are star nodes

      Example: Star Reduction

        • Suppose minsup = 2
        • Perform one-dimensional aggregation. Replace attribute values whose count < 2 with *. And collapse all *’s together
        • Resulting table has all such attributes replaced with the star-attribute
        • With regards to the iceberg computation, this new table is a lossless compression of the original table

        Star Tree

        • Given the new compressed table, it is possible to construct the corresponding cell tree—called star tree
        • Keep a star table at the side for easy lookup of star attributes
        • The star tree is a lossless compression of the original cell tree

        Star-Cubing Algorithm—DFS on Lattice Tree

        Multi-Way Aggregation

        Star-Cubing Algorithm—DFS on Star-Tree

        Multi-Way Star-Tree Aggregation

        • Start depth-first search at the root of the base star tree
        • At each new node in the DFS, create corresponding star tree that are descendents of the current tree according to the integrated traversal ordering
          • E.g., in the base tree, when DFS reaches a1, the ACD/A tree is created
          • When DFS reaches b*, the ABD/AD tree is created
        • The counts in the base tree are carried over to the new trees

        Multi-Way Aggregation (2)

        • When DFS reaches a leaf node (e., d*), start backtracking
        • On every backtracking branch, the count in the corresponding trees are output, the tree is destroyed, and the node in the base tree is destroyed
        • Example
          • When traversing from d* back to c*, the a1b*c*/a1b*c* tree is output and destroyed
          • When traversing from c* back to b*, the a1b*D/a1b* tree is output and destroyed
          • When at b*, jump to b1 and repeat similar process

        The Curse of Dimensionality

        • None of the previous cubing method can handle high dimensionality!
        • A database of 600k tuples. Each dimension has cardinality of 100 and zipf of 2.

        Motivation of High-D OLAP

        • X. Li, J. Han, and H. Gonzalez, High-Dimensional OLAP: A Minimal Cubing Approach, VLDB'04
        • Challenge to current cubing methods:
          • The “curse of dimensionality’’ problem
          • Iceberg cube and compressed cubes: only delay the inevitable explosion
          • Full materialization: still significant overhead in accessing results on disk
        • High-D OLAP is needed in applications
          • Science and engineering analysis
          • Bio-data analysis: thousands of genes
          • Statistical surveys: hundreds of variables

        Fast High-D OLAP with Minimal Cubing

        • Observation: OLAP occurs only on a small subset of dimensions at a time
        • Semi-Online Computational Model
          • Partition the set of dimensions into shell fragments
          • Compute data cubes for each shell fragment while retaining inverted indices or value-list indices
          • Given the pre-computed fragment cubes , dynamically compute cube cells of the high-dimensional data cube online

        Properties of Proposed Method

        • Partitions the data vertically
        • Reduces high-dimensional cube into a set of lower dimensional cubes
        • Online re-construction of original high-dimensional space
        • Lossless reduction
        • Offers tradeoffs between the amount of pre-processing and the speed of online computation

        Example Computation

        • Let the cube aggregation function be count
        • Divide the 5 dimensions into 2 shell fragments:
          (A, B, C) and (D, E)

        1-D Inverted Indices

        • Build traditional invert index or RID list

        Shell Fragment Cubes: Ideas

        • Generalize the 1-D inverted indices to multi-dimensional ones in the data cube sense
        • Compute all cuboids for data cubes ABC and DE while retaining the inverted indices
        • For example, shell fragment cube ABC contains 7 cuboids:
          • A, B, C
          • AB, AC, BC
          • ABC
        • This completes the offline computation stage

        Shell Fragment Cubes: Size and Design

        • Given a database of T tuples, D dimensions, and F shell fragment size, the fragment cubes’ space requirement is:
          \[ O\left ( T\left \lceil \frac{D}{F} \right \rceil(2^{F}-1) \right ) \]  
          • For F < 5, the growth is sub-linear
        • Shell fragments do not have to be disjoint
        • Fragment groupings can be arbitrary to allow for maximum online performance
          • Known common combinations (e.g., ) should be grouped together.
        • Shell fragment sizes can be adjusted for optimal balance between offline and online computation 

        ID_Measure Table

        • If measures other than count are present, store in ID_measure table separate from the shell fragments  

        The Frag-Shells Algorithm

        • Partition set of dimension (A1,…,An) into a set of k fragments (P1,…,Pk).
        • Scan base table once and do the following
          • insert into ID_measure table.
          • for each attribute value ai of each dimension Ai
            • build inverted index entry
        • For each fragment partition Pi
          • build local fragment cube Si by intersecting tid-lists in bottom- up fashion.

        Frag-Shells (cont')


        Online Query Computation: Query

        • A query has the general form <a1,a2,...an: M>
        • Each ai has 3 possible values
          • Instantiated value
          • Aggregate * function
          • Inquire ? function
        • For example, <3 ? ? * 1: count> returns a 2-D data cube.

        Online Query Computation: Method

        • Given the fragment cubes, process a query as follows
          • Divide the query into fragment, same as the shell
          • Fetch the corresponding TID list for each fragment from the fragment cube
          • Intersect the TID lists from each fragment to construct instantiated base table
          • Compute the data cube using the base table with any cubing algorithm

        Online Query Computation: Sketch

        Experiment: Size vs. Dimensionality (50 and 100 cardinality)

          • (50-C): 106 tuples, 0 skew, 50 cardinality, fragment size 3.
          • (100-C): 106 tuples, 2 skew, 100 cardinality, fragment size 2.

          Experiments on Real World Data

          • UCI Forest CoverType data set
            • 54 dimensions, 581K tuples
            • Shell fragments of size 2 took 33 seconds and 325MB to compute
            • 3-D subquery with 1 instantiate D: 85ms~1.4 sec.
          • Longitudinal Study of Vocational Rehab. Data
            • 24 dimensions, 8818 tuples
            • Shell fragments of size 3 took 0.9 seconds and 60MB to compute
            • 5-D query with 0 instantiated D: 227ms~2.6 sec.

          Processing Advanced Queries by Exploring Data Cube Technology

          • Sampling Cube
            • X. Li, J. Han, Z. Yin, J.-G. Lee, Y. Sun, “Sampling Cube: A Framework for Statistical OLAP over Sampling Data”, SIGMOD’08
          • Ranking Cube
            • D. Xin, J. Han, H. Cheng, and X. Li. Answering top-k queries with multi-dimensional selections: The ranking cube approach. VLDB’06
          • Other advanced cubes for processing data and queries
            • Stream cube, spatial cube, multimedia cube, text cube, RFID cube, etc. — to be studied in volume 2

          Statistical Surveys and OLAP

          • Statistical survey: A popular tool to collect information about a population based on a sample
            • Ex.: TV ratings, US Census, election polls
          • A common tool in politics, health, market research, science, and many more
          • An efficient way of collecting information (Data collection is expensive)
          • Many statistical tools available, to determine validity
            • Confidence intervals
            • Hypothesis tests
          • OLAP (multidimensional analysis) on survey data
            • highly desirable but can it be done well?

          Surveys: Sample vs. Whole Population

          Data is only a sample of population

          Problems for Drilling in Multidim. Space

          • Data is only a sample of population but samples could be small when drilling to certain multidimensional space

          OLAP on Survey (i.e., Sampling) Data

          • Semantics of query is unchanged
          • Input data has changed

          Challenges for OLAP on Sampling Data

          • Computing confidence intervals in OLAP context
          • No data?
            • Not exactly. No data in subspaces in cube
            • Sparse data
            • Causes include sampling bias and query selection bias
          • Curse of dimensionality
            • Survey data can be high dimensional
            • Over 600 dimensions in real world example
            • Impossible to fully materialize

          Example 1: Confidence Interval

          • What is the average income of 19-year-old high-school students?
          • Return not only query result but also confidence interval

          Confidence Interval

          • Confidence interval at x':
            \[ \bar{x}\pm t_{c}\hat{\sigma}_{\bar{x}} \]
            • x is a sample of data set;x' is the mean of sample
              tc is the critical t-value, calculated by a look-up  
              \[ \hat{\sigma}_{\bar{x}}=\frac{s}{\sqrt{l}} \]
              is the estimated standard error of the mean
          • Example : $50,000 ± $3,000 with 95% confidence
            • Treat points in cube cell as samples
            • Compute confidence interval as traditional sample set 
          • Return answer in the form of confidence interval 
            • Indicates quality of query answer  
            • User selects desired confidence interval  

          Efficient Computing Confidence Interval Measures

          • Efficient computation in all cells in data cube
              • Both mean and confidence interval are algebraic
              • Why confidence interval measure is algebraic?

          \[ \bar{x}\pm t_{c}\hat{\sigma}_{\bar{x}} \]

          x' is algebraic

          \[ \hat{\sigma}_{\bar{x}}=\frac{s}{\sqrt{l}} \]

          where both s and l (count) are algebraic

          • Thus one can calculate cells efficiently at more general cuboids without having to start at the base cuboid each time

          Example 2: Query Expansion

          • What is the average income of 19-year-old college students?

          Boosting Confidence by Query Expansion

          • From the example: The queried cell “19-year-old college students” contains only 2 samples
          • Confidence interval is large (i.e., low confidence). why?
            • Small sample size
            • High standard deviation with samples
          • Small sample sizes can occur at relatively low dimensional selections
            • Collect more data?― expensive!
            • Use data in other cells? Maybe, but have to be careful

          Intra-Cuboid Expansion: Choice 1

          •  Expand query to include 18 and 20 year olds?

          Intra-Cuboid Expansion: Choice 2

          •  Expand query to include high-school and graduate students?

          Query Expansion

          Intra-Cuboid Expansion

          • Combine other cells’ data into own to “boost” confidence
            • If share semantic and cube similarity
            • Use only if necessary
            • Bigger sample size will decrease confidence interval
          • Cell segment similarity
            • Some dimensions are clear: Age
            • Some are fuzzy: Occupation
            • May need domain knowledge
          • Cell value similarity
            • How to determine if two cells’ samples come from the same population?
            • Two-sample t-test (confidence-based)

          Inter-Cuboid Expansion

          • If a query dimension is
              • Not correlated with cube value
              • But is causing small sample size by drilling down too much
          • Remove dimension (i.e., generalize to *) and move to a more general cuboid
          • Can use two-sample t-test to determine similarity between two cells across cuboids
          • Can also use a different method to be shown later

          Query Expansion Experiments

          • Real world sample data: 600 dimensions and 750,000 tuples
          • 0.05% to simulate “sample” (allows error checking)

          Ranking Cubes – Efficient Computation of Ranking queries

          • Data cube helps not only OLAP but also ranked search
          • (top-k) ranking query: only returns the best k results according to a user-specified preference, consisting of (1) a selection condition and (2) a ranking function
          • Ex.: Search for apartments with expected price 1000 and expected square feet 800
              • Select top 1 from Apartment
              • where City = “LA” and Num_Bedroom = 2
              • order by [price – 1000]^2 + [sq feet - 800]^2 asc
          • Efficiency question: Can we only search what we need?
            • Build a ranking cube on both selection dimensions and ranking dimensions

          Ranking Cube: Partition Data on Both Selection and Ranking Dimensions

          Materialize Ranking-Cube

          Search with Ranking-Cube: Simultaneously Push Selection and Ranking

          Processing Ranking Query: Execution Trace

          Ranking Cube: Methodology and Extension

          • Ranking cube methodology
            • Push selection and ranking simultaneously
            • It works for many sophisticated ranking functions
          • How to support high-dimensional data?
            • Materialize only those atomic cuboids that contain single selection dimensions
              • Uses the idea similar to high-dimensional OLAP
              • Achieves low space overhead and high performance in answering ranking queries with a high number of selection dimensions

          Data Mining in Cube Space

          • Data cube greatly increases the analysis bandwidth
          • Four ways to interact OLAP-styled analysis and data mining
            • Using cube space to define data space for mining
            • Using OLAP queries to generate features and targets for mining, e.g., multi-feature cube
            • Using data-mining models as building blocks in a multi-step mining process, e.g., prediction cube
            • Using data-cube computation techniques to speed up repeated model construction
              • Cube-space data mining may require building a model for each candidate data space
              • Sharing computation across model-construction for different candidates may lead to efficient mining

          Prediction Cubes

          • Prediction cube: A cube structure that stores prediction models in multidimensional data space and supports prediction in OLAP manner
          • Prediction models are used as building blocks to define the interestingness of subsets of data, i.e., to answer which subsets of data indicate better prediction

          How to Determine the Prediction Power of an Attribute?

          • Ex. A customer table D:
            • Two dimensions Z: Time (Month, Year ) and Location (State, Country)
            • Two features X: Gender and Salary
            • One class-label attribute Y: Valued Customer
          • Q: “Are there times and locations in which the value of a customer depended greatly on the customers gender (i.e., Gender: predictiveness attribute V)?”
          • Idea:
            • Compute the difference between the model built on that using X to predict Y and that built on using X – V to predict Y
            • If the difference is large, V must play an important role at predicting Y

          Efficient Computation of Prediction Cubes

          • Naïve method: Fully materialize the prediction cube, i.e., exhaustively build models and evaluate them for each cell and for each granularity
          • Better approach: Explore score function decomposition that reduces prediction cube computation to data cube computation

          Complex Aggregation at Multiple Granularities: Multi-Feature Cubes

          • Multi-feature cubes (Ross, et al. 1998): Compute complex queries involving multiple dependent aggregates at multiple granularities
          • Ex. Grouping by all subsets of {item, region, month}, find the maximum price in 2010 for each group, and the total sales among all maximum price tuples

            select item, region, month, max(price), sum(R.sales)
            from purchases
            where year = 2010
            cube by item, region, month: R
            such that R. price = max(price)
          • Continuing the last example, among the max price tuples, find the min and max shelf live, and find the fraction of the total sales due to tuple that have min shelf life within the set of all max price tuples

          Discovery-Driven Exploration of Data Cubes

          • Hypothesis-driven
            • exploration by user, huge search space
          • Discovery-driven (Sarawagi, et al.’98)
            • Effective navigation of large OLAP data cubes
            • pre-compute measures indicating exceptions, guide user in the data analysis, at all levels of aggregation
            • Exception: significantly different from the value anticipated, based on a statistical model
            • Visual cues such as background color are used to reflect the degree of exception of each cell

          Kinds of Exceptions and their Computation

          • Parameters
            • SelfExp: surprise of cell relative to other cells at same level of aggregation
            • InExp: surprise beneath the cell
            • PathExp: surprise beneath cell for each drill-down path
          • Computation of exception indicator (modeling fitting and computing SelfExp, InExp, and PathExp values) can be overlapped with cube construction
          • Exception themselves can be stored, indexed and retrieved like precomputed aggregates

          Examples: Discovery-Driven Data Cubes

          H-Cubing: Using H-Tree Structure

          • Bottom-up computation
          • Exploring an H-tree structure
          • If the current computation of an H-tree cannot pass min_sup, do not proceed further (pruning)
          • No simultaneous aggregation

          H-tree: A Prefix Hyper-tree

          Computing Cells Involving “City”

          Computing Cells Involving Month But No City

          Computing Cells Involving Only Cust_grp