Approximate XML Joins

1. Approximate XML Joins Huang-Chun Yu Li Xu

2. Introduction • XML is widely used to integrate data from different sources. • Perform join operation for XML documents: • Two documents may convey approximately or exactly the same information but may be different on structure. • Even when two documents have the same DTD, the structures may be different due to optional elements or attributes.

3. paper paper publication conference title authors VLDB XML for the masses author author conference title authors type title author author Alice Bob VLDB XML for the masses author conference XML for the masses name name Alice VLDB Alice Rob Introduction • Example: We need approximately matching of XML documents!

4. Introduction • We also need a distance metric to quantify the differences between XML documents. • Tree edit distance is used in this paper for its generality and simplicity when quantifying the distance between trees. • Other distance metrics can be used as well.

5. Tree Edit Distance • Tree edit distance: the minimum number of tree edit operations (node insertion, deletion, label substitution) required to transform one tree to another. • Given two trees T1 and T2, there is a well known algorithm to compute the tree edit distance in O(|T1| |T2| h(T1) h(T2)).

6. Tree Edit Distance • Find a mappingM between T1 and T2 such that the editing cost is minimized. • The mapping consists pairs of integers (i, j) such that: • 1≤ i ≤ |T1| and 1≤ j ≤ |T2| • For any (i1, j1), (i2, j2) in M • i1 = i2 iff j1 = j2 • t1[i1] is to the left of t1[i2] iff t2[j1] is to the left of t2[j2] (sibling order preserving) • t1[i1] is an ancestor of t1[i2] iff t2[j1] is an ancestor of t2[j2] (ancestor order preserving)

7. Tree Edit Distance • Example: tree edit distance is 3 (delete B, insert H, relabel C to I) T2: A A T1: B H B C D H D E F G E I F G

8. Problem Definition • Given two XML data source S1 and S2, and a distance thresholdτ. • TDist(d1, d2): a function that assesses the tree edit distance between two documents d1 S1 and d2 S2. • Approximate join: return all pairs of documents (d1, d2) S1 S2 such that TDist(d1, d2) ≤ τ.

9. Challenges • Evaluation of TDist function between two documents is a very expensive operation. (worst case: O(n4), for trees of size O(n) ) • Traditional techniques in join algorithms (sort merge, hash join, etc) cannot be used.

10. Lower Bounds • Let T be an ordered labeled tree. Let pre(T) denote the preorder traversal of T and post(T) denote the postorder traversal of T. • Let T1, T2 be ordered labeled trees. max{ed(pre(T1), pre(T2)), ed(post(T1), post(T2)} ≤ TDist(T1, T2) • This can be computed in O(n2) time.

11. Upper Bounds • Additional constraint is imposed on the original TDist algorithm. The search space is reduced and a faster algorithm is proposed. • For any triple (t1[i1], t2[j1]), (t1[i2], t2[j2]), (t1[i3], t2[j3]) M, let lca( ) be the lowest common ancestor function. • t1[lca(t1[i1], t1[i2])] is a proper ancestor of t1[i3] iff t2[lca(t2[j1], t2[j2])] is a proper ancestor of t2[j3] • Two distinct subtrees of T1 will be mapped to two distinct subtrees of T2 . • It can be calculated in O(|T1||T2|) time.

12. T2: A A T1: B C D H D E F G E I F G Upper Bounds • Example: the upper bound is 5 (delete B, delete E, insert H, insert E, relabel C to I )

13. Upper Bounds Algorithm for Upper Bound:

14. Outline • Reference set • Choosing reference set • Approximate join algorithms

15. Outline • Reference set • Choosing reference set • Approximate join algorithms

16. Reference Set • S1, S2: two sets of XML documents • Reference set K S1∪S2 • a chosen set of XML documents • vi: a vector for document di S1∪S2 • dimensionality = |K| • vit = TDist(di, kt), kt K,1 ≤t≤|K|

17. Reference Set • | vit - vjt | ≤ TDist(di, dj) ≤vit + vjt , 1 ≤t ≤|K| • Essentially the above procedure “projects” documents di , dj onto the reference set K • τ: distance threshold • uij = min t,1 ≤ t ≤ |k|vit + vjt • uij ≤ τ: the pair is certainly within distance τ • lij = max t,1 ≤ t ≤ |k| |vit – vjt| • lij > τ: the pair can’t be within distance τ

18. Outline • Reference set • Choosing reference set • Approximate join algorithms

19. Choosing Reference Set • S = S1∪S2 • S is well separated, if • S can be divided into k clusters s.t. • Documents within a cluster have small distance (say less than τ/2) • Documents in different clusters have large distance (say larger than 3τ/2)

20. Choosing Reference Set • S is well separated • choose a single point from each of the k ( the size of the reference set ) largest clusters to be in the reference set • If k is not known • fi : the fraction of points in the first i clusters • Choose k ≥ i ≥ 2 , such that

21. Choosing Reference Set • choose d C1 in the reference set K • (di C1, djC1) should be in the output • TDist(di, dj) ≤TDist(di, d) + TDist(dj, d) ≤τ/2 + τ/2 = τ • C1 containing n1 documents • Saving n1 *(n1 - 1)/2 evaluations of TDist() • (di C1 , dj C2) should not be in the output • TDist(di, dj) ≥ |TDist(di, d) - TDist(dj, d)| > 3τ/2 - τ/2 = τ • Saving n1 *(|S| - n1) evaluations of TDist()

22. Algorithm • do{ 1.1 randomly pick a point d from the data set S 1.2 put all the points within τ/2 distance with d in one cluster } until (all documents in S belong to some cluster ) 2. choose the k largest clusters 3. pick a random point from each cluster to be in the reference set K

23. Outline • Reference set • Choosing reference set • Approximate join algorithms

24. Bounds Algorithm • Naïve algorithm • Nested loop join + TDist algorithm • Bounds algorithm for each di S1 { for each djS2 { if (UBDist(di ,dj) ≤ τ ) output (di , dj); if (LBDist(di , dj) ≤ τ ) if (TDist(di , dj ) ≤ τ ) output(di , dj); } }

25. Pruning with a Reference Set • for each pair (di S1, djS2 ) • uij = min t,1 ≤ t ≤ |k|vit + vjt • lij = max t,1 ≤ t ≤ |k| |vit – vjt| • uij ≤ τ: the pair belongs to the output • lij > τ: the pair can be pruned away • lij ≤ τ < uij : apply TDist(di, dj) to identify the distance between di and dj • refer to this algorithm as RS (ReferenceSets) • Drawback • need to perform (| S1| + |S2|) * |K| invocations of TDist() to compute vectors

26. Applying Both Optimizations • if RS algorithm indicates that TDist() should be invoked between a pair • can be possibly avoid by applying the computational cheaper LBDist() and UBDist() • refer to this algorithm as RSB (RSBounds)

27. RSC Algorithm • potentially more evaluation of TDist() • because of the construction of vectors • two vectors for document di S1∪S2 • vector vli :vlit = LBDist(di, kt), kt K,1 ≤t≤|K| • vector vui : vuit = UBDist(di, kt), kt K,1 ≤t≤|K| • | vlit - vujt | ≤ TDist(di, dj) ≤vuit + vujt , 1 ≤t ≤|K| • uij = min t,1 ≤ t ≤ |k|vuit + vujt • lij = max t,1 ≤ t ≤ |k| |vlit – vujt| • Refer to this algorithm as RSCombined(RSC) • Drawback: double the size of vectors

28. Performance Evaluation • Run time vs. number of nodes

29. Performance Evaluation • Run time vs. distance threshold (XMark) • Run time vs. distance threshold (DBLP)

30. Performance Evaluation • Run time vs. distance threshold (XMark) • Number of TDist calculation vs. distance threshold (XMark)

31. Conclusion & Future Work • The algorithms are not scalable for huge data sets. • The performance of these algorithms has a strong correlation with the data itself. • The performance of the reference set depends on the clustering algorithm chosen. • Try to incorporate other distance matrices into the algorithms. • Try to explore the various indexingschemes which can be used in the algorithms.

32. References • S. Guda, H. V. Jagadish, N. Koudas, D. Srivastava, and T. Yu, Approximate XML Joins, Proceedings of ACM SIGMID, 2002. • K. Zhang and D. Shasha, Tree Pattern Matching, Oxford University Press, 1997. • S. Guha, R. Rastogi, and K. Shim, CURE: An Efficient Clustering Algorithm for Large Databases, Proceedings of ACM SIGMOD, 1998. • T. Zhang, R. Ramakrishnan, and M. Livny, BIRCH: An Efficient Data Clustering Method for Very Large Databases, Proceedings of ACM SIGMOD, 1996.