1. Integrating XML Data Sources Using Approximate Joins
Author
Sudipto Guha, H. V. Jagadish, Nick Koudas,
Divesh Srivastava, Ting Yu
Source
SIGMOD02, ICDE03, ACM TODS06 March
Presenter
Jim
9/20/2018
Dept. of Computer Science

2. Dept. of Computer Science
Outline
Motivation
Proposed approach
Result
Conclusion
9/20/2018
Dept. of Computer Science

3. Global Query Interface
Motivation
Data integration
A popular application on Internet
Query the different data sources
eg. an integration of search engine
Global Query Interface
Result
Query
DBLP
CiteSeer
GoogleScholar
9/20/2018
Dept. of Computer Science

4. Global Query Interface
Motivation
Data integration
Identify same data object in different data sources
eg. query papers from the global query interface
Global Query Interface
Need to identify the same paper from the query result!
Result
Query
DBLP
CiteSeer
GoogleScholar
9/20/2018
Dept. of Computer Science

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Motivation
XML Data integration
Identify same data object in different schema
General case
Find out pair of object from two object set
DBLP
CiteSeer
9/20/2018
Dept. of Computer Science

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Problem
Approximately matching XML documents
Given
Two set of XML documents, each set conforms to a schema
A distance metric Dist between XML documents and a distance threshold t
Result
All pair of documents (Ti,Tj), such Dist(Ti,Tj) <= t
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Dept. of Computer Science

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Distance metric
String edit distance
Minimum step of edit operation (insert, delete, change) to change S1 to S2
eg1. ed(“abcde”,”bCdef”)=3,
edit operation: del ‘a’, ‘c’ -> ‘C’, ins ‘f’
eg2. ed(“hilton hotel”,”hilson hotel”)=1
edit operation: ‘t’ -> ‘s’
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Dept. of Computer Science

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Distance metric
String edit distance
Time complexity
O(|S1|*|S2|) worst case complexity
Dynamic programming style
Recursion step
Ci,j is the cost of match Si and Tj (edit distance)
Ci,j = Ci-1,j-1 if S[i]=T[j],
1+min(Ci-1, j, Ci, j-1, Ci-1 ,j-1), else.
9/20/2018
Dept. of Computer Science

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Tree edit distance
Tree edit distance
Generalization of string edit distance
Minimum cost sequence of tree edit operation to transform T1 to T2
Operation: insertion, deletion and re-label of nodes
Method
Dynamic programming style
Compute a mapping between nodes of T1 & T2 with minimum cost
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Dept. of Computer Science

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Tree edit distance
Tree edit distance – example
Mappings (‘e’ means empty)
(A,A), (B,e), (D,D), (E,E), (e,H), (C,I), (F,F), (G,G)
Do NOT care position of node
Cost (edit operations)
delete ‘B’, insert ‘H’,
re-label ‘C’->‘I’
TreeEditDist(T1,T2)=3
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Dept. of Computer Science

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Tree edit distance
Tree edit distance – basic constraints
For any pairs (i1,j1), (i2,j2)
i1=i2 iff j1=j2 one-to-one
i1 is to the left of i2 iff
j1 is to the left of j2
sibling order preserving
i1 is an ancestor of i2 iff
j1 is an ancestor of j2
ancestor order preserving
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Dept. of Computer Science

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Tree edit distance
Tree edit distance – algorithms
Dynamic programming style
Recursion step
v and w ib the rightmost nodes of F1 and F2
|F1|*|F2| sub-problem for computing dist(F1,F2)
Overall Worst case complexity
O(|T1|2*|T2|2)
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Dept. of Computer Science

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Tree edit distance
Modelling XML
ordered, labelled trees
Distance between XML documents
Tree edit distance
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Dept. of Computer Science

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Tree edit distance
Problem
High time complexity of TreeEditDist()
Worst case O(|T1|2*|T2|2)
Improved: O(|T1|*|T2|*depth(T1)*depth(T2))
Not practical for large XML documents
How to efficiently perform approximate join using TreeEditDist()?
9/20/2018
Dept. of Computer Science

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Proposed approaches
Without indexing
Bounding TreeEditDist()
Lower & upper bound as filter
Sampling based bounding
Sampling the dataset and pre-compute the distance as filter
Add indexing
R-tree indexing
9/20/2018
Dept. of Computer Science

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Lower bound
Idea
Transform a tree to a string
Compute string edit distance as lower bound
Method
Use pre-order & post-order traversal of a tree
Rational
Operation on the string -> operation on the tree, but NOT vice versa
dist(T1,T2)>=dist(S1,S2)
9/20/2018
Dept. of Computer Science

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Lower bound
Example
ed(s1,s2)=3, ed(T1,T2)=3
For the case of ‘<‘, i.e., dist(s1,s2)<dist(T1,T2), suppose all the nodes has the same label
pre-order(T1): ABDECFG, pre-order(T2): ADHEIFG
9/20/2018
Dept. of Computer Science

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Lower bound
Lower bound theorem
Max(StrEditDist(pre(T1),pre(T2)), StrEditDist(post(T1),post(T2)))
<= TreeEditDist(T1,T2)
Complexity
O(|T1|*|T2|) using string edit distance
Note: O(|T1|2*|T2|2) for tree edit distance
Cheaper than TreeEditDist()
9/20/2018
Dept. of Computer Science

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Upper bound
Basic constraints of TreeEditDist()
one-to-one
sibling order preserving
ancestor order preserving
9/20/2018
Dept. of Computer Science

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Upper bound
Add one more constraint
Intuition
Two distinct subtree of T1 will be mapped to two distinct subtree of T2
For any pairs (i1,j1), (i2,j2), (i3,j3)
lca(i1,i2) is a proper ancestor of i3 iff lca(j1,j2) is a proper ancestor of j3
lca order preserving
9/20/2018
Dept. of Computer Science

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Upper bound
LCA constraint – example
(E,E) is not a mapping any more, as D & E are in the same subtree of A in T1, but in two distinct subtree of A in T2
TreeEditDist_UB(T1,T2)=5
delete ‘B’, insert ‘H’
re-label ‘C’->’I’
delete ‘E’, insert ‘E’
9/20/2018
Dept. of Computer Science

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Upper bound
Computing upper bound
Reduce the number of sub-problem
Reduce time complexity to O(|T1|*|T2|)
9/20/2018
Dept. of Computer Science

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Summary of bounding
Bounding method
Lower bound
Transform tree to string using traversal
Upper bound
Add LCA constraint to the tree edit distance
Filtering with bound
Given (T1,T2) and threshold t
if LBDist(T1,T2)>t, then prune (T1,T2)
if UBDist(T1,T2)<=t, then admit (T1,T2)
9/20/2018
Dept. of Computer Science

24. Sampling based bounding
TreeEditDist(), or simply Dist(), is a metric
Triangle inequality
|Dist(di,dr)-Dist(dr,dj)|<=Dist(di,dj)<= Dist(di,dr)+Dist(dr,dj)
Reference set
A set of K sample document
Sampling method later dr di dj
Reference Set
dr1
dr2 …
drk
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Dept. of Computer Science

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Reference set
For each XML document di
Pre-compute the Dist() between di and reference set
Distance vector vi (|vi|=K, vi1,vi2,…vik) dr di dj
Reference Set
dr1
dr2 …
drk
9/20/2018
Dept. of Computer Science

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Reference set
Reference set based bound
Given XML document pair (di,dj)
LB_RS=max(|vil-vjl|), 1<=l<=k
UB_RS=min(vil+vjl), 1<=l<=k
Use the bound
if LB_RS>t, then prune (di,dj)
if UB_RS<=t, then admit (di,dj) dr di dj
Reference Set
dr1
dr2 …
drk
9/20/2018
Dept. of Computer Science

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Reference set
Choose the reference set (or simply, RS)
Assumption
XML data set is well separated
Goal
Make the RS decisive with small size
Optimal RS
One XML file from one cluster
Reference Set
LB_RS=max(|vil-vjl|), 1<=l<=k
UB_RS=min(vil+vjl), 1<=l<=k
9/20/2018
Dept. of Computer Science

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Reference set
Choose the RS
Randomly sampling the all XML data set
Clustering the sampled XML data
Based on TreeEditDist()
Use CURE or BIRCH to clustering
One sample from one cluster
Reference Set
9/20/2018
Dept. of Computer Science

29. Dept. of Computer Science
Approximate join
Problem statement again
Given
Two set of XML documents
A distance metric Dist between XML documents
A distance threshold t
Result
All pair of documents (Ti,Tj), such Dist(Ti,Tj) <= t
Approximate join algorithm
Using bounding & reference set
No indexing
9/20/2018
Dept. of Computer Science

30. Dept. of Computer Science
Approximate join
Baseline algorithm
Nested loop join with TreeEditDist()
Using lower & upper bound ONLY
Nested loop join with bound
if LBDist(T1,T2)>t, then prune (T1,T2)
if UBDist(T1,T2)<=t, then admit (T1,T2)
Using reference set ONLY
if LB_RS(T1,T2)>t, then prune (T1,T2)
if UB_RS (T1,T2)<=t, then admit (T1,T2)
9/20/2018
Dept. of Computer Science

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Approximate join
Apply both bounding and sampling (1)
In sequence
Filter using reference set
Filter using lower & upper bound
Apply TreeEditDist()
9/20/2018
Dept. of Computer Science

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Approximate join
Apply both bounding and sampling (2)
In combination
Compute lower & upper bound between each XML data and RS, but Not compute TreeEditDist()
For each XML data, there are two vectors vl &vu
Change the bounding function
Given XML document pair (di,dj)
LB_RSC=max(|vium-vjlm|), 1<=m<=k
UB_RSC=min(vium+vjum), 1<=m<=k
9/20/2018
Dept. of Computer Science

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R-tree indexing
Use R-tree index to make improvement
Construct R-tree index for both dataset
Leaf node consists of document distance vector
DIR node consists of MBRs
MBR of the leaf
R-tree leaf node K
max(viu) K K K
min(vil) vu vu …… vu vl vl …… vl D1 D2 Dm
9/20/2018
Dept. of Computer Science

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R-tree indexing
Use index to make further improvement
Range query (given a Di and MBR node)
if exist j (1<=j<=d), max(vju)+vj<=t, admit Di with all leaf nodes in MBR
if exist j (1<=j<=d), min(vjl)-vj>t or vj-min(vju)>t, prune Di with all leaf nodes
ROOT ……
max(viu)
max(viu)
max(viu)
min(vil)
min(vil)
min(vil) …… vu …… vu vu …… vu vu …… vu
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Dept. of Computer Science vl …… vl vl …… vl vl …… vl

35. Dept. of Computer Science
R-tree indexing
Use index to make further improvement
R-tree join (given two set of documents)
Based on the range query
Slightly modify the bounding function from (point,MBR) to (MBR,MBR)
ROOT ……
max(viu)
max(viu)
max(viu)
min(vil)
min(vil)
min(vil) …… vu …… vu vu …… vu vu …… vu
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Dept. of Computer Science vl …… vl vl …… vl vl …… vl

36. Dept. of Computer Science
Summary
Baseline approximate join algorithm
Nested loop join with TreeEditDist() (Naïve)
Proposed algorithm
Lower & upper bound (B)
Reference set (RS)
Bound + reference set
Sequence (RSB)
Combination (RSC)
R-tree indexing support
9/20/2018
Dept. of Computer Science

37. Dept. of Computer Science
Experiment result
Data set
Data Set A: IBM XML data generator
Data Set B: arrange A into 8 clusters
Data Set C: DBLP data
Test query
Approximate self-join on each data set
9/20/2018
Dept. of Computer Science

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Experimental result
Tightness of lower & upper bound
x-axis: ratio of bound/TreeEditDist()
y-axis: number to pairs
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Dept. of Computer Science

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Experimental result
Processing time (on data set A)
y-axis: ratio to naïve (logarithmic scale)
x-axis: reference set size
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Dept. of Computer Science

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Experimental result
Processing time (on data set B)
Data set B has 8 clusters
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Dept. of Computer Science

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Experimental result
Component of computing time (on DS B)
Optimal: |RS|=8
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Dept. of Computer Science

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Experimental result
Processing time on RS A (no cluster)
x-axis: distance threshold
Larger threshold
Upper bound
Smaller threshold
Lower bound
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Dept. of Computer Science

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Experimental result
Using R-tree index
x-axis: distance threshold
y-axis: vector computation ratio to RSB
9/20/2018
Dept. of Computer Science

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Conclusion
Motivation
Data integration in the context of XML
Approximate join on XML document
Approaches
Lower & upper bound
Use reference set to filtering
R-tree indexing
9/20/2018
Dept. of Computer Science

45. Dept. of Computer Science
Q & A
Happy New Year!
9/20/2018
Dept. of Computer Science