Approximate XML Query Answers

1
Approximate XML Query Answers

• Alkis Polyzotis (UC Santa Cruz)
• Minos Garofalakis (Bell Labs)
• Yannis Ioannidis (U. of Athens, Hellas)

2
Motivation

• XML de-facto standard for data exchange
• Development of the XML Warehouse
• Conflict between on-line and query execution
  cost
• Increased query response times
• Users might wait for un-interesting results

Q
Warehouse
R
3
Approximate Query Answers

• Evaluate query over a concise data synopsis and
  obtain an approximation R of the true result
• Use approximate result as timely feedback
• User can assess the value of the query
• Goal reduce number of evaluated queries

R
Q
Warehouse
R
4
Contributions

• TreeSketch Synopses
• Structural summaries for XML data
• Approximate answers for complex twig queries
• Summarization model ? Structural clustering of
  elements
• Efficient processing and construction
• Element Simulation Distance
• Novel distance metric for XML data
• Captures approximate similarity between two XML
  trees
• Experimental Results
• Accurate approximate answers for low space
  budgets
• Low-error selectivity estimates
• Efficient construction algorithm

5
Outline

• Preliminaries
• TreeSketches
• Synopsis model
• Computing approximate answers
• Summary construction
• Element Simulation Distance
• Experimental Study
• Conclusions

6
Data and Query Model
XML Document
7
Problem Definition
Approximate Nesting Tree
True Nesting Tree

• Process twig query over a synopsis
• Compute approximation of nesting tree

8
TreeSketch Model
9
Graph Synopsis
XML Document
Graph Synopsis

• Synopsis node ? Set of elements of the same tag
• Synopsis edge ? Document edge(s)

10
TreeSketch Synopsis
XML Document
TreeSketch

• Augment graph-synopsis with edge counts
• countu,v mean children in v per element in u

11
TreeSketch Synopsis
XML Document
TreeSketch

• Is there a lossless synopsis?
• What is the quality of a lossy synopsis?

12
Count Stability
XML Document
TreeSketch
1
2
1
1
1
1
1

• (u,v) count-stable all elements in u have the
  same child-count in v

13
Count-Stable TreeSketch
XML Document
TreeSketch
R(1)
1
P(1)
1
1
S(1)
S(1)
2
2
F(2)
F(2)
1
1
1
C(4)
E(2)

• A count-stable synopsis can recover the input
  tree
• Efficient one-pass construction
• Stable summary can be too large for practical use!

14
Lossy TreeSketch
XML Document
TreeSketch
15
TreeSketches and Clustering

• TreeSketch ? Element clustering
• All elements in a node are mapped to a centroid
• Tight clusters ? Accurate synopsis
• Synopsis quality ? Clustering error
• Options Manhattan Distance, Squared Error,
• Quality can be measured independent of a workload
• Key for effective construction

16
Computing Approximate Answers
TreeSketch
Query
Approximate Nesting Tree
R
q0
//section
q1
.//caption
.//equation
q2
q3

• Compute TreeSketch of approximate answer
• Accuracy depends on quality of clustering

17
TreeSketch Construction

• Given an XML tree T, build a TreeSketch of size B
• Difficult clustering problem
• Space dimensionality depends on the clustering
  itself
• Construction based on bottom-up clustering
• Compress perfect synopsis by merging clusters
• Best merge determined by marginal gains
• Heuristic to reduce number of candidate merges

Space Budget
Perfect
18
Element Simulation Distance
19
Error of Approximation

• Error ? Distance between R and R
• Popular metric Tree-edit distance
• Min-cost sequence of operations that transform R
  to R
• Measures syntactic differences between R and R
• Not intuitive for approximate answers!

Different counts Similar Trait
Same counts Opposite Trait
T1
T
T2
20
Element Simulation Distance

• Capture approximate similarity between R and R
• u simulates v u and v have identical structure
• ESD(u,v) degree of simulation between u,v
• How well the structure of u matches the structure
  of v
• Modeled as the distance between multi-sets
• Efficient computation using perfect summaries

21
Experimental Results
22
Methodology

• Data Sets XMark, DBLP, IMDB, SwissProt
• Workload 1000 random twig queries
• Evaluation metrics
• Average ESD for approximate answers
• Mean absolute relative error for selectivity
  estimation

23
Approximate Answers - IMDB
IMDB (102K Elements) Avg. Result Size 3,477
tuples
24
Selectivity Estimation - SwissProt
SwissProt (182K Elements) Avg. Result Size
104,592 tuples
25
Selectivity Estimation - ALL
26
Conclusions

• Approximate query answering for XML databases
• TreeSketch Synopses
• Structural summaries for tree-structured XML
• Approximate answers for twig-queries
• Model Graph Synopsis Edge-counts
• Efficient processing and construction
• Element Simulation Distance
• Capture approximate similarity between XML trees
• Experimental Results
• High accuracy for low space budgets
• Efficient construction

27
Questions?
28
TreeSketch Model (2/2)
XML Document
TreeSketch
r
R
1
p1
P(1)
2
S(2)
s2
s3
1
1
F(2)
F(2)
f7
f9
f9
f5
1
1
1
C(4)
E(2)
c14
c17
e11
c12
e13
c17

• Average number of children Edge count

29
XML
XML Document
r
p1
p paper s section c caption t title f
figure e equation
s2
s3
f7
f9
f9
f5
c14
c17
e11
c12
e13
c17
30
TreeSketch Synopsis
XML Document
TreeSketch
R(1)
1
P(1)
2
S(2)
F
2
F(4)
1
0.5
C(4)
E(2)

• Augment graph-synopsis with edge counts
• countu,v mean children in v per element in u

31
Depth-Guided Merging

• Key observation Two elements have similar
  structure, if their children have similar
  structure
• Bottom-up merging, based on depth
• Depth distance from the leaves of the tree
• Build a pool of candidate merges by increasing
  depth
• Replenish the pool when it falls below a given
  threshold
• Reduced construction time - Accurate synopses

32
Depth-Guided Merging

• Observation Two elements have similar structure,
  if their children have similar structure
• Heuristic If a merge of two clusters is good,
  then merges of the child clusters are likely to
  have been good as well
• Bottom-up merging strategy
• Savings in construction time - Accurate synopses