1. By A. Aboulnaga, A. R. Alameldeen and J. F. Naughton Vldb’01
Estimating the Selectivity of XML Path Expressions for Internet Scale Applications By
A. Aboulnaga, A. R. Alameldeen and J. F. Naughton
Vldb’01
Presented by
Kan Kin Fai

2. Outline of Presentation
Introduction
Path Trees
Markov Tables
Experimental Findings

3. Motivation
XML enables Internet scale applications that query data from many sources
Niagara, Xyleme, …
Queries over XML data use path expressions

4. An XML Document <readings> <play>
<title>Pygmalion</title>
<author>Bernard Shaw</author>
</play>
<novel>
<title>David Copperfield</title>
<author>Charles Dickens</author>
</novel>
</readings>

5. Querying XML Data
FOR $n_auth IN document("*")//novel/author
$p_auth IN document("*")//play/author
WHERE $n_auth/text() = $p_auth/text()
RETURN $n_auth
Optimizing this query requires estimating the selectivity of the path expressions
This requires information about the structure of the XML data

6. Goal of this Paper
Build database statistics that capture the structure of XML data
Ensure that the statistics fit in a small amount of memory
For efficient query optimization
Important for Internet scale applications
Use the statistics to estimate the selectivity of simple XML path expressions

7. Simple Path Expression
A sequence of tags that represents a navigation through the tree structure of the XML data starting anywhere in the tree
//t1/t2/…/tn
Assumes an unordered model of XML
Does not consider navigations based on IDREF attributes or on predicates on the attributes values

8. Path Trees A tree representing the structure of an XML document
Every node represents a path starting from the root of the XML document
The root node represents the root element

9. Path Trees
A node has a child node for every distinct element directly nested in any of the elements reachable by the path it represents
Every node is labeled with the tag name of the elements reachable by the path it represents and with the number of such elements (i.e. frequency of the node)

10. Path Trees <A> <B> </B> <B>
<D> </D>
</B>
<C>
<E> </E>
</C>
</A> A 1 B 2 C 1 D 1 D 1 E 3

11. Summarizing Path Trees
Path trees contain all the information needed for selectivity estimation
Problem: May not fit in available memory
Small available memory
Internet scale
Remove low frequency nodes
Removed nodes replaced with *-nodes
Tag name: * meaning "any tag"
Frequency: Average frequency of replaced nodes
Sibling-*, Level-*, Global-*, No-*

12. Sibling-* Summarization
Repeatedly choose the path tree node with the lowest frequency and mark it for deletion
Check its siblings to see if any of them is either a *-node or a regular node that has been marked for deletion
If yes, coalesce the node with such sibling node into one *-node.
Coalescing the children of coalesced nodes if they have the same tag name.

13. Sibling-* Summarization
During summarization, all path tree nodes store the number of nodes in the original unsummarized path tree that they represent and the total frequency of these nodes.
When the path tree becomes small enough, traverse the tree and compute for every *-node the average frequency of the multiple deleted nodes that it represents.

14. Sibling-* Summarization1 B 13 C 9 D 7 E 5 F 15 G 10 H 6 I 2 J 4 K 11 K 12

15. Sibling-* Summarization1 B 13 C 9 D 7 E 5 F 15 G 10 H 6 I 2 J 4 K 11 K 12

16. Sibling-* Summarization1 B 13 C 9 D 7 E 5 F 15 G 10 H 6 I J 4 K 11 K 12 2

17. Sibling-* Summarization1 B 13 C 9 D 7 E 5 F 15 G 10 H 6 I J K 11 K 12 2 4

18. Sibling-* Summarization1 B 13 C 9 D 7 E 5 F 15 G 10 H 6 *
f=6
n=2 K 11 K 12

19. Sibling-* Summarization1 B 13 C 9 D 7 E F 15 G 10 H 6 5 *
f=6
n=2 K 11 K 12

20. Sibling-* Summarization1 B 13 C 9 D 7 E F 15 G 10 H 5 6 *
f=6
n=2 K 11 K 12

21. Sibling-* Summarization1 B 13 C 9 D E F 15 G 10 H 7 5 6 *
f=6
n=2 K 11 K 12

22. Sibling-* Summarization1 B 13 C 9 *
f=12
n=2 F 15 G 10 H 6 *
f=6
n=2 K 11 K 12

23. Sibling-* Summarization1 B 13 C 9 *
f=12
n=2 F 15 G 10 H 6 *
f=6
n=2 K 11 K 12

24. Sibling-* Summarization1 B 13 C 9 *
f=12
n=2 F 15 G H 10 6 *
f=6
n=2 K 11 K 12

25. Sibling-* Summarization1 B 13 C 9 *
f=12
n=2 F 15 *
f=16
n=2 *
f=6
n=2 K 11 K 12

26. Sibling-* Summarization1 B 13 C 9 *
f=12
n=2 F 15 *
f=16
n=2 *
f=6
n=2 K
f=23
n=2

27. Sibling-* Summarization1 B 13 C 9 * F 15 * 6 8 * K
f=23
n=2 3

28. Original Path Tree A 1 B 13 C 9 D 7 E 5 F 15 G 10 H 6 I 2 J 4 K 11 K12

29. Sibling-* Summarization1 B 13 C 9 * F 15 * 6 8 * K
f=23
n=2 3
Try to preserve the exact position of the deleted nodes in the original path tree
May need to delete 2n nodes to reduce the size of the tree by n nodes

30. Selectivity Estimation
Try to match the tags in the path expressions with tags in the path tree to find all path tree nodes to which the path expression leads
The estimated selectivity is the total frequency of all these nodes.

31. Selectivity Estimation
When we can’t match a tag in the path expression to a path tree node with a regular tag, try to match it to a *-node that can take its place.
E.g. //A/B/C would match all of //A/*/C, //A/*/* and //*/B/*
Allow matches with any number of *-nodes as long as they include at least one node with a regular tag name

32. Level-* Summarization
Has a *-node for every level of the path tree representing all deleted nodes at this level
All nodes deleted at any given level of the path tree are coalesced into the *-node for this level
Preserves only the level in the path tree of the deleted nodes, not their exact position as in sibling-*
Need to delete n+h nodes to reduce the size of the path tree by n nodes, where h is the number of levels in the tree

33. Level-* Summarization1 B 13 C 9 D 7 E 5 F 15 G 10 H 6 I 2 J 4 K 11 K 12

34. Level-* Summarization1 B 13 C 9 D E F 15 G 10 H 7 5 6 I J K 11 K 12 2 4

35. Level-* Summarization1 B 13 C 9 * 6 F 15 G 10 * 3 K 11 K 12

36. Global-* Summarization
A single *-node represents all low-frequency nodes deleted from anywhere in the path tree
Preserves less information about the deleted nodes than sibling-* or level-*
Needs to delete only n+1 nodes to reduce the size of the path tree by n nodes

37. Global-* Summarization1 B 13 C 9 D 7 E 5 F 15 G 10 H 6 I 2 J 4 K 11 K 12

38. Global-* Summarization1 B 13 C 9 D E F 15 G 10 H 7 5 6 I J K 11 K 12 2 4

39. Global-* Summarization3 * B 13 C 9 D F 15 G 10 H 7 6 K 11 K 12

40. No-* Summarization
Low-frequency nodes are simply deleted and not replaced with *-nodes
Deletes exactly n nodes to reduce the size of a path tree by n nodes

41. No-* Summarization A 1 B 13 C 9 D 7 E 5 F 15 G 10 H 6 I 2 J 4 K 11 K12

42. No-* Summarization A 1 B 13 C 9 D E F 15 G 10 H 7 5 6 I J K 11 K 12 24

43. No-* Summarization B 13 C 9 D E F 15 G 10 H 7 5 6 K 11 K 12

44. Markov Tables
A table of all distinct paths of length up to m and their frequencies
For paths of length greater than m, combine paths from the Markov table
Example:
Uses "short memory" or "Markov" property
f(B/C/D)
f(B/C)
f(A/B/C/D) = f(A/B/C)

45. Markov Tables Path Freq A 1 AC 6 B 11 AD 4 C 15 BC 9 D 19 BD 7 AB CD 8

46. Summarizing Markov Tables
Exact selectivities for paths of length up to m
Approximate selectivities for paths longer than m
Problem: May not fit in available memory
Remove low frequency paths
Discard removed paths of length > 2
Replace removed paths of length 1 or 2 with *-paths
Suffix-*, Global-*, No-*

47. Suffix-* Summarization
Two special *-paths
*: all deleted paths of length 1
*/*: all deleted paths of length 2
Adds low-frequency path of length 1 to *
Keeps a set of deleted paths of length 2, SD
Deletes low-frequency path of length 2
Looks for suffix-* path (e.g. A/*) with the same starting tag in the Markov table
Looks for path with the same starting tag in SD

48. Suffix-* Summarization
Deletes low-frequency path of length 2
Adds deleted suffix-* path to */*
At the end, add deleted paths in SD to */* and compute the average frequencies of all *-paths.
Selectivity Estimation
Use the frequencies of suffix-* paths and *-paths if any of the required paths is not found
Return 0 if only *-paths are used for estimation

49. Suffix-* Summarization
Path
Freq A 1 AC 6 B 11 AD 4 C 15 BC 9 D 19 BD 7 AB CD 8

50. Suffix-* Summarization
Path
Freq A 1 AC 6 B 11 AD 4 C 15 BC 9 D 19 BD 7 AB CD 8 * **

51. Suffix-* Summarization
Path
Freq A 1 AC 6 B 11 AD 4 C 15 BC 9 D 19 BD 7 AB CD 8 * **

52. Suffix-* Summarization
Path
Freq AC 6 B 11 AD 4 C 15 BC 9 D 19 BD 7 AB CD 8 *
f=1,n=1 **

53. Suffix-* Summarization
Path
Freq AC 6 B 11 AD 4 C 15 BC 9 D 19 BD 7 AB CD 8 *
f=1,n=1 **
SD= { }
Set of deleted paths of length 2

54. Suffix-* Summarization
Path
Freq AC 6 B 11 C 15 BC 9 D 19 BD 7 AB CD 8 *
f=1,n=1 **
SD= { (AD,4) }

55. Suffix-* Summarization
Path
Freq AC 6 B 11 C 15 BC 9 D 19 BD 7 AB CD 8 *
f=1,n=1 **
SD= { (AD,4) }

56. Suffix-* Summarization
Path
Freq AC 6 B 11 C 15 BC 9 D 19 BD 7 AB CD 8 *
f=1,n=1 **
SD= { (AD,4) }

57. Suffix-* Summarization
Path
Freq A*
f=10,n=2 B 11 C 15 BC 9 D 19 BD 7 AB CD 8 *
f=1,n=1 **
SD= { }

58. Suffix-* Summarization
Path
Freq A*
f=10,n=2 B 11 C 15 BC 9 D 19 BD 7 AB CD 8 *
f=1,n=1 **
SD= { }

59. Suffix-* Summarization
Path
Freq A*
f=10,n=2 B 11 C 15 BC 9 D 19 AB CD 8 *
f=1,n=1 **
SD= { (BD,7) }

60. Suffix-* Summarization
Path
Freq A*
f=10,n=2 B 11 C 15 BC 9 D 19 AB CD 8 *
f=1,n=1 **
SD= { (BD,7) }

61. Suffix-* Summarization
Path
Freq A*
f=10,n=2 B 11 C 15 BC 9 D 19 AB *
f=1,n=1 **
SD= { (BD,7), (CD,8) }

62. Suffix-* Summarization
Path
Freq A*
f=10,n=2 B 11 C 15 BC 9 D 19 AB *
f=1,n=1 **
SD= { (BD,7), (CD,8) }

63. Suffix-* Summarization
Path
Freq A*
f=10,n=2 B 11 C 15 BC 9 D 19 AB *
f=1,n=1 **
SD= { (BD,7), (CD,8) }

64. Suffix-* Summarization
Path
Freq A*
f=10,n=2 B 11 C 15 B*
f=16,n=2 D 19 AB *
f=1,n=1 **
SD= { (CD,8) }

65. Suffix-* Summarization
Path
Freq A*
f=10,n=2 B 11 C 15 B*
f=16,n=2 D 19 AB *
f=1,n=1 **
SD= { (CD,8) }

66. Suffix-* Summarization
Path
Freq B 11 C 15 B*
f=16,n=2 D 19 AB *
f=1,n=1 **
f=10,n=2
We gave A* a second chance but it lost it.
Done.
SD= { (CD,8) }

67. Suffix-* Summarization
Path
Freq B 11 C 15 B* 8 D 19 AB * 1 ** 6
Selectivity estimation.
SD= { }

68. Global-*, No-* Summarization
Two *-paths, * and **
Adds low-frequency path of length 1 or 2 to the appropriate *-path immediately
Deletes fewer paths than suffix-* to summarize the Markov table
No-*
No *-paths

69. Experimental Findings
Path trees
Query paths in data: Global-*
Query paths not in data: No-*
Markov tables
m = 2 is best (practical values: 2 and 3)
Query paths in data: Suffix-*

70. Explanation
Methods using *-nodes/*-paths aggressively assume nodes/paths that cannot be found did exist in the original path tree/Markov table.
No -* conservatively assumes that nodes/paths that cannot be found did not exist in the original path tree/Markov table.

71. Experimental Findings
When the data has many common sub-structures, Markov tables give more accurate estimation.
When the data does not have many common sub-structures, path trees give more accurate estimation.

72. Explanation
DBLP
Repeated sub-structures effectively captured by Markov tables
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