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STHoles: A Multidimensional Workload-Aware Histogram Nicolas Bruno* Columbia University Luis Gravano* Columbia University Surajit Chaudhuri Microsoft Research SIGMOD 2001 * Work done in part while the authors were visiting Microsoft Research.

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2 Histograms as Succinct Data Set Summaries n Used for selectivity estimation and approximate query processing. n Data set partitioned into buckets, each approximated by aggregate statistics.

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

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

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5 Gaussian Data Set Histograms Techniques: EquiDepth EquiDepth Histogram [Muralikrishna and DeWitt 1988] n Correctly identifies core of densest clusters. n Partitioning uses “equi-count” instead of “equi- density”

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6 Gaussian Data SetMHist Histogram [Poosala and Ioannidis 1997] Histogram Techniques: MHist n Works well for highly skewed data distributions. n Devotes too many buckets to the densest clusters. n Bad initial “choices” are amplified in later steps.

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7 Gaussian Data Set GenHist Histogram [Gunopulos et al. 2000] Histogram Techniques: GenHist n More robust than previous techniques (based on multidimensional information). n Difficult to choose right values of various parameters. n Requires at least 5-10 passes over the data.

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8 Gaussian Data SetSTGrid Histogram [Aboulnaga and Chaudhuri 1999] Histogram Techniques: STGrid n Incorporates feedback from query execution. n Grid partitioning strategy is sometimes too rigid. n Focuses on efficiency rather than accuracy.

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9 Our New Histogram Technique: STHoles n Flexible bucket partitioning. n Exploits workload information to allocate buckets. n Query feedback captures uniformly dense regions. n Does not examine actual data set.

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10 STHoles Histograms n Tree structure among buckets. n Buckets with holes: relaxes rectangular regions while using rectangular bucket structures. Non rectangular region

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11 System Architecture for STHoles Range Query

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12 STHoles Construction Algorithm n Initialize histogram H as an empty histogram. n 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.

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13 ? Drilling New Candidate Buckets Count how many tuples in result stream lie inside q b. n Drill q b as a new bucket (child of b). q For each query q in workload and bucket b in histogram:

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14 Shrinking Candidate Buckets n Partition constraint: Bounding boxes must be rectangular. n Apply greedy technique to shrink a candidate hole to a rectangle.

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15 Merging Buckets n To avoid exceeding available space. n Merge most “similar” buckets in terms of tuple density.

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16 Parent-Child Merges Eliminate buckets too similar to their parents. Example: The interesting region in bc is covered by its child b1.

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17 Sibling-Sibling Merges n Consolidate buckets with similar densities that cover close regions. n Extrapolate frequency distributions to yet unseen regions.

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18 Gaussian Data SetSTHoles Histogram An Example STHoles Histogram

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19 Experimental Setting n 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. n Histograms: –1024 available bytes per histogram. –EquiDept, MHist, GenHist, STGrid, STHoles.

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20 Experimental Setting (cont.) n 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. Census data setBiased (tuples) workloadGaussian (area) workload n Accuracy Metric: Absolute Error. (with some normalization; details in paper)

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21 Comparison with Other Approaches: Biased Workload Biased workload, query boundaries cover around 1% of the data domain

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22 Comparison with Other Approaches: Uniform Workload Uniform workload, query boundaries cover around 1% of the data set tuples.

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23 Convergence with Workload Biased workload

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24 Handling Data Set Updates From Gaussian to Zipfian data distributions.

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

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26 Summary: STHoles, a Multidimensional Workload-Aware Histogram n Exploits query feedback. n Built without examining data set. n Allows bucket nesting to capture complex shapes using only rectangular bucket structures. n Results in robust and accurate selectivity estimations. n In many cases, outperforms the best techniques that access full data sets.

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27 Related Work (Histograms) n Unidimensional: –EquiDepth [Piatetsky-Shapiro and Connell 1984] –MaxDiff [Poosala et al. 1996] –V-Optimal [Jagadish et al. 1998] –Many more! n Multidimensional: –EquiDepth [Muralikrishna and DeWitt 1988] –MHist [Poosala and Ioannidis 1997] –GenHist [Gunopulos et al. 2000] –STGrid [Aboulnaga and Chaudhuri 1999]

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28 Related Work (Other Techniques) n Sampling [Olken and Rotem 1990] n Wavelets [Matias et al. 1997] n Discrete transformations [Lee et al. 1999] n Parametric Curve Fitting [Chen and Roussopoulos 1994]

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29 Evaluation Metric n Absolute Error: n Normalized Absolute Error:

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30 Overhead Evaluation over Microsoft SQL Server 2000

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31 Varying Histogram Size Gaussian Data Set Zipfian Data Set Census Data Set

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32 Varying Spatial Selectivity Gaussian Data Set Zipfian Data Set Census Data Set

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