STHoles: A Multidimensional Workload-Aware Histogram

1
STHoles A Multidimensional Workload-Aware
Histogram

• Nicolas BrunoColumbia University

Luis GravanoColumbia University
Surajit ChaudhuriMicrosoft Research
Work done in part while the authors were
visiting Microsoft Research.
SIGMOD 2001
2
Histograms as Succinct Data Set Summaries

• Used for selectivity estimation and approximate
  query processing.
• Data set partitioned into buckets, each
  approximated by aggregate statistics.

3
Histograms

• Each bucket consists of a bounding box and a
  tuple frequency value.
• Uniformity is assumed inside buckets.
• Histograms should partition data set in buckets
  with uniform tuple density.
• Multi-dimensional data makes partitioning even
  more challenging.

4
Outline

• Overview of existing multidimensional histogram
  techniques.
• Introduction to STHoles histograms.
• System architecture and STHoles construction
  algorithm.
• Experimental evaluation.

5
Histograms Techniques EquiDepth
Gaussian Data Set
EquiDepth Histogram Muralikrishna and DeWitt
1988

• Correctly identifies core of densest clusters.
• Partitioning uses equi-count instead of
  equi-density

6
Histogram Techniques MHist
Gaussian Data Set
MHist Histogram Poosala and Ioannidis 1997

• Works well for highly skewed data distributions.
• Devotes too many buckets to the densest clusters.
• Bad initial choices are amplified in later
  steps.

7
Histogram Techniques GenHist
GenHist Histogram Gunopulos et al. 2000
Gaussian Data Set

• More robust than previous techniques (based on
  multidimensional information).
• Difficult to choose right values of various
  parameters.
• Requires at least 5-10 passes over the data.

8
Histogram Techniques STGrid
Gaussian Data Set
STGrid Histogram Aboulnaga and Chaudhuri 1999

• Incorporates feedback from query execution.
• Grid partitioning strategy is sometimes too
  rigid.
• Focuses on efficiency rather than accuracy.

9
Our New Histogram Technique STHoles

• Flexible bucket partitioning.
• Exploits workload information to allocate
  buckets.
• Query feedback captures uniformly dense regions.
• Does not examine actual data set.

10
STHoles Histograms

• Tree structure among buckets.
• Buckets with holes relaxes rectangular regions
  while using rectangular bucket structures.

11
System Architecture for STHoles
Range Query
12
STHoles Construction Algorithm

• Initialize histogram H as an empty histogram.
• For each query q in workload
• 1- Gather simple statistics from query results.
• 2- Identify candidate holes and drill (add) them
  as new buckets in H.
• 3- Merge superfluous buckets in H.

13
Drilling New Candidate Buckets
For each query q in workload and bucket b in
histogram

• Count how many tuples in result stream lie inside
  q?b.
• Drill q?b as a new bucket (child of b).

q
14
Shrinking Candidate Buckets

• Partition constraint Bounding boxes must be
  rectangular.
• Apply greedy technique to shrink a candidate hole
  to a rectangle.

15
Merging Buckets

• To avoid exceeding available space.
• Merge most similar buckets in terms of tuple
  density.

16
Parent-Child Merges

• Eliminate buckets too similar to their parents.
• Example The interesting region in bc is covered
  by its child b1.

17
Sibling-Sibling Merges

• Consolidate buckets with similar densities that
  cover close regions.
• Extrapolate frequency distributions to yet unseen
  regions.

18
An Example STHoles Histogram
Gaussian Data Set
STHoles Histogram
19
Experimental Setting

• Data Sets
• Real (UCI Repository)
• Sample of Census data set (200K tuples)
• Cover data set (500K tuples)
• Synthetic Variations of Gaussian and
  Zipfian(Array) distributions.
• 200K to 500K tuples, 2 to 4 dimensions.
• Histograms
• 1024 available bytes per histogram.
• EquiDept, MHist, GenHist, STGrid, STHoles.

20
Experimental Setting (cont.)

• Workloads Pagel et al. 1993
• 1,000 queries.
• Query centers follow different distributions
  Uniform, Biased, Gaussian.
• Query boundaries follow different constraints
  area covered, tuples covered.

21
Comparison with Other Approaches Biased Workload
22
Comparison with Other Approaches Uniform Workload
23
Convergence with Workload
24
Handling Data Set Updates
From Gaussian to Zipfian data distributions.
25
Other Experiments

• Varying
• data skew.
• data dimensionality.
• histogram size.
• workload generation parameters.
• number of attributes in queries.
• Overhead for intercepting query results in
  Microsoft SQL Server 2000 is less than 8.
• STHoles lead to robust selectivity estimates
  across data distributions and workloads.
• See full paper for details!

26
Summary STHoles, a Multidimensional
Workload-Aware Histogram

• Exploits query feedback.
• Built without examining data set.
• Allows bucket nesting to capture complex shapes
  using only rectangular bucket structures.
• Results in robust and accurate selectivity
  estimations.
• In many cases, outperforms the best techniques
  that access full data sets.

27
Related Work (Histograms)

• Unidimensional
• EquiDepth Piatetsky-Shapiro and Connell 1984
• MaxDiff Poosala et al. 1996
• V-Optimal Jagadish et al. 1998
• Many more!
• Multidimensional
• EquiDepth Muralikrishna and DeWitt 1988
• MHist Poosala and Ioannidis 1997
• GenHist Gunopulos et al. 2000
• STGrid Aboulnaga and Chaudhuri 1999

28
Related Work (Other Techniques)

• Sampling Olken and Rotem 1990
• Wavelets Matias et al. 1997
• Discrete transformations Lee et al. 1999
• Parametric Curve Fitting Chen and Roussopoulos
  1994

29
Evaluation Metric

• Absolute Error
• Normalized Absolute Error

30
Overhead Evaluation over Microsoft SQL Server 2000
31
Varying Histogram Size
32
Varying Spatial Selectivity