Selectivity Estimation of Twig Queries on Cyclic Graphs Department of Computer Science Hong Kong...
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![Page 1: Selectivity Estimation of Twig Queries on Cyclic Graphs Department of Computer Science Hong Kong Baptist University Speaker: Byron Choi Joint work with.](https://reader038.fdocuments.in/reader038/viewer/2022110207/56649d695503460f94a479ce/html5/thumbnails/1.jpg)
Selectivity Estimation of Twig Queries on Cyclic Graphs
Department of Computer Science
Hong Kong Baptist University
Speaker: Byron Choi
Joint work with *Yun Peng and Jianliang Xu
(to appear ICDE 2011)
March 21 2011 @ COMP630Q
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Agenda
BackgroundProblem StatementOverview of our FrameworkMatrix and its TransformationsHistograms and EstimationExperimental EvaluationFuture Works
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Graph Data is Ubiquitous
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Navigational Queries
SELECT a set of nodes via a user-specified path◦ //person[//open auction//person]◦ //
ancestor-descendant axes CONNECT in logic (reachability tests)
there are evidently many other query formalisms
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Selectivity Estimation
A classical problem: Given a query, estimate the count of the results efficiently
Requirements◦ accurate◦ efficient estimation time◦ small overhead in terms of size
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XMark, used in this Presentation
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Selectivity Estimation (cont’)
Query optimizers rely on the counts to evaluate the costs of query plans
Example:◦ XMark 1.0 (> 180,000 nodes)◦ Query: //person[//open auction//person]◦ 25,500 person’s◦ 12,000 open_auction’s◦ 13,192 open_auction//person’s◦ //open_auction → //person → ↑↑person
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Problem StatementData: A rooted directed labeled graph (i.e.,
possibly cyclic)Query: Twig queries (i.e., parent-child and
ancestor-descendant axes and branches)Problem statement: given a cyclic graph G and
a twig Q, estimate the result count of Q on G.department
facul ty facul ty facul ty
name RA TA RA TA TA TAname name
Graph G Twig Query Q
department
f acul ty
RA TA
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Our Position Relative to the Current State-of-the-Art
Graph Complexity
Que
ry C
ompl
exit
y
Tree (XML) Cyclic Graph
Path Query
Twig Query
XSketch ’06
Xseed ’06TreeSketch ’06
CST ’04XPathLearner ’02
DataGuide ’99
Our Work
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Related Work (Graph-based approaches) Dataguide – Automata theories
◦ J. McHugh and J. Widom. Query optimization for xml. In VLDB, pages 315–326, 1999
TreeSketch and XSketch -- Bisimulation◦ N. Polyzotis and M. Garofalakis. Xsketch synopses for xml
data graphs. ACM Trans. Database Syst., 31(3):1014–1063, 2006.
◦ N. Polyzotis, M. Garofalakis, and Y. Ioannidis. Approximate xml query answers. In SIGMOD, pages 263–274, 2004.
Correlated Subpath Tree (CST)◦ Z. Chen, H. V. Jagadish, F. Korn, N. Koudas, S.
Muthukrishnan, R. Ng, and D. Srivastava. Counting twig matches in a tree. In ICDE, pages 595–604, 2001
Straight-Line Grammar◦ D. K. Fisher and S. Maneth. Structural selectivity
estimation for xml documents. In ICDE, pages 626–635, 2007
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2-dimensional histograms on TREEs◦ Y. Wu, J. M. Patel, and H. V. Jagadish. Using
histograms to estimate answer sizes for xml queries. Inf. Syst., 28(1-2):33–59, 2003.
Hidden Markov Model◦ A. Aboulnaga, A. R. Alameldeen, and J. F. Naughton.
Estimating the selectivity of xml path expressions for internet scale applications. InVLDB, pages 591–600, 2001.
A novel bloom filter – two 1-dimensional histograms◦ W. Wang, H. Jiang, H. Lu, and J. X. Yu. Bloom
histogram: Path selectivity estimation for xml data with updates. In VLDB, pages 240–251, 2004.
Related Work (Relational approaches)
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Technical Challenges
Interactions between cyclic graphs and recursions (i.e. //) in twig queries
Branches of twigs
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A Typical Framework
Graph Representati onGraph
Summari zati on of Graph’ s
Representati onSel ecti vi ty
Query
Summari zati on techni que
Sel ecti vi ty esti mati on techni que
• Previous research differs from each other in one or more steps
• We also follow this general framework
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Framework – with Our Solution Now
Graph Representation
Summarization of Graph Rep.
Selectivity Estimation
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Summary of Contributions
1. Cyclic graph representation◦ Prime labeling (vs. other representations)
◦ Matrix representation of prime labeling
◦ Matrix transformation to C1P matrix
2. Summarization of graph’s representation◦ 2-dimensional histogram for cyclic graph
3. Algorithms for selectivity estimation
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Characteristics of our ContributionsMatrix representation of cyclic graphs
◦ Reuse some research from matrices
Histogram-based selectivity estimation◦ No uniform distribution assumption
One data node/vertex – one 2-dimensional pointOne query step (child or descendant) – multiple 2-
dimensional points
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Agenda
BackgroundProblem StatementOverview of our FrameworkMatrix and its TransformationsHistograms and EstimationExperimental EvaluationFuture Works
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Alternative Representations for Cyclic Graphs
Adjacency matrix/list◦ Easy to construct
◦ Inefficient in determining ancestors/descendants
Transitive closure◦ Efficient in ancestors/descendants;
◦ Inefficient in terms of space
Prime labeling◦ Smaller than transitive closure but larger than adjacency matrix
◦ Query efficiency better than adjacency matrix but worse than transitive closure
…
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Prime Labeling
Originally proposed for tree data [X. Wu, ICDE’04]◦ To address update-friendly XML index for reachability tests
Later extended to DAGs [G. Wu, DASFAA’06]◦ Each vertex is assigned a prime number
Our extension to cyclic graphs◦ Applied to cyclic graphs
◦ Reduced labeling size further Not each vertex is labeled with a unique prime number →
smaller than G. Wu et al.
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Prime Labeling (con’t)
Large prime numbers near the root of the graph
• assign each leaf vertex a prime number
• assign an intermediate vertex production of label of its children
• label the root
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Prime Labeling (our Def.)
G. Wu et al
Yun Peng
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Querying with Prime LabelingReachability ≡ Divisibility
◦ c → d: 7 * 11 * 3 / 3 = 7 * 11
◦ c → e: 7 * 11 * 3 / 5 = 46.2
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Matrix Representation
3 5 7 11
1 1 1 1
1 1 0 0
1 0 1 1
1 0 0 0
0 1 0 0
0 0 1 0
1 0 0 1
a
b
c
d
e
f
g
Columns:Prime numbers
Rows:Vertices
Reachability: Divisibility ≡ Logic op.s
Experiments: Often just a constant factor smaller than the adjacency matrix!
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Where are we?Experiments from XMark
◦ It is just a constant factor smaller than adjacency matrix
◦ How on earth would this be summarized?
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Consecutive Ones Property (C1P)
A Consecutive Ones Matrix (C1P matrix) is a 0/1 matrix, in which 1s of each row are consecutive.
Since 1s are consecutive, each row of a C1P matrix can be summarized by an interval: [start column id of 1s, end column id of 1s]
One row → One vertex → One interval
1
2
1 1 1 0 1 1
0 0 1 1 1 1
r
r
1
2
0 1 1 1 1 1
0 0 1 1 1 0
r
r
non-consecutive ones matrix
consecutive ones matrix
[1,5]
[2,4]
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What do we get from C1P?
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What do we get from C1P? (cont’)
Adopting a property of intervals◦ Vertex w is reachable from vertex v, if w locates
within the right-bottom field of v on the plane◦ For example, dot (2,4) is at right bottom part of dot
(1,5), so r2 is reachable from r1
1
2
0 1 1 1 1 1
0 0 1 1 1 0
r
r
[1,5]
[2,4]
r1(1,5)
r2(2,4)
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Complexities related to C1P
C1P matrix detection◦ Linear time solvable [Hsu, Algorithms’02]
Transform a non-C1P matrix to a C1P matrix◦ NP-hard [Tan, Algorithmica’07]◦ No polynomial time approximation [Tan, Algorithmica’07]
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Our Heuristic Algorithm
Main Idea: given any m*n matrix with r 1s, extract C1P sub matrixes (by the C1P matrix detection algorithm) and then concatenate them one by one
Time complexity: 2( ( ))O m m n r
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Pseudo-code of the Matrix Transformation
Extract a submatrix for this iteration
Adding one row at a time
C1P detection – linear time
Transform to C1P – linear time
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Optimizations for C1P Trans.
Horizontal matrix decomposition prior to C1P heuristics◦ ◦ Use the 3 sigmas rule on the number of 1’s in rows
Common pattern extraction◦ Done by an intersection of the rows
Compressed (extensible) hash mappings◦ One column in the original matrix may be mapped to
multiple positions in a C1P matrix◦ Support mapping ops in the compressed domain
))(( 2 rnmmO
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What do we get from C1P? (Recall)
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Agenda
BackgroundProblem StatementOverview of our FrameworkMatrix and its TransformationsHistograms and EstimationExperimental EvaluationFuture Works
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2-Dimensional Histogram Recall we summarize rows of a C1P
matrix by intervals and then dot them on the 2-d plane
The plane is divided into cells. For each cell, we record the number of dots located within it.
Given a vertex v, the set of vertices reachable from v must be located in the right bottom part of v
Sum up the size of cells located at right-bottom part of v as 1+2 = 3
1
2
1
1
We build a 2-dimensional histogram for each kind of nodes
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2-Dimensional Histogram -- Observations
Data dots are always on top of the diagonal lineData dots are often skewed towards the diagonal
line◦ This is consistent to an observation from an XML
researchThere are different types of cells w.r.t a query →
there should be different estimation rules
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Our 2-Dimensional HistogramsMore histogram/structure in a cell
Different estimation rules for different classes of cells
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Schematics of Our Estimation
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Estimation Details that have been Skipped in this TalkA top-down recursive estimation algorithm
based on the (syntactic) structure of twigsDetails on handling branches
◦ A bottom-up recursive algorithmestimate_intermediate: generating next
query dotsestimate_count: generating count from a
query dot
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top_down (very briefly)
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A rule in estimate_intermediate
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Illustration of estimation rules in estimate_intermediate
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A rule in estimate_count
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Query-dot generation
Compress f and f^-1Generate query dots in the
compressed domain in one scan
1.They can be large, sometimes2. Many query dots have 0 count
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Agenda
BackgroundProblem StatementOverview of our FrameworkMatrix and its TransformationsHistograms and EstimationExperimental EvaluationFuture Works
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ExperimentsDatasets
◦ XMark; DBLP; Treebank.05
Queries◦ Skewed queries based on the tags’ popularities
Optimizations◦ Used all optimizations unless specified otherwise
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Error Metrics
◦ Relative error: from XSketch/TreeSketch
◦ Root Mean Square Error (RMSE): from XSeed
◦ Normalized RMSE: from XSeed
| . |est realrealn
2( . )est real
n
RMSEreal
n
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Our Est. Error (relative error)
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Our Est. Error (RMSE & NRMSE)
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Ours vs. XSeed
RMSE NRMSE
XMark 7.1 times better 6.9 times better
Treebank.05 6.8 times better 6.8 times better
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Ours vs. XSketch/TreeSketch (indirect)
XSketch focuses on path queries on cyclic graph, which controls error under 10%
TreeSketch focuses on twig queries on tree, which control error under 5%
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Our Est. Time on XMark Graph
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Performance of C1P Matrix Transformation Optimization
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Query Dot Gen. Optimization
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ConclusionsWe are the first work on selectivity estimation of twig
queries on cyclic graphsWe propose a new graph representation technique
◦ Extend prime labeling to cyclic graphs◦ Transform prime labeling to a C1P matrix for summarization
We extend 2-dimensional histogram selectivity estimation technique to cyclic graphs
Experiment results shows that we outperform previous works◦ Our ~1.3% error vs. XSketch/TreeSketch’s 5% error◦ Errors are at least 6.8 times smaller than XSeed
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Future Works
Incorporating this technique with estimation on◦ Data values◦ Queries with negations
External implementation◦ For quick implementation, we put almost all data
structures in main memoryEstimation performance guarantees
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