1. Incremental Maintenance of XML Structural Indexes
Ke Yi1, Hao He1, Ioana Stanoi2 and Jun Yang1
1Department of Computer Science, Duke University
2IBM T. J. Watson Research Center

2. Motivation XML is gaining tremendously in popularity in recent years
Used to represent many kinds of data
Major DB vendors are rushing to incorporate solutions for native XML repositories and retrieval
IBM DB2, Oracle , Microsoft SQL Server
Tamino, Natix, X-Hive, …

3. Overview paper 1 13 section section 2 title 14 title 3 8 section 4
“experiments”
exp
“intro”
algorithm 15 16 5
title 6
exp
algorithm 9
title 10
proof 7 17
“A(k)-index”
“1-index” 11 18
about
proof
about 12
uses

4. Label Path Expressions
paper
/paper/section/algorithm 1 13
section
section 2
title 14
title 3 8
section 4
section
“experiments”
exp
“intro”
algorithm 15 16 5
title 6
exp
algorithm 9
title 10
proof 7 17
“A(k)-index”
“1-index” 11 18
about
proof
about 12
uses

5. Structural Indexes Why do we need them? Structural indexes
Speedup the evaluation of path expressions
Provides a structural summary of the data graph
Structural indexes
DataGuide [Goldman & Widom 97]
1-index [Milo & Suciu 99]
A(k)-index [Kaushik et al. 02], D(k)-index [Qun et al. 03], M(k)-index [He & Yang 04]
Integration of structural indexes and inverted lists [Kaushik et al. 04]
Focus on maintenance
Has a major effect on index efficiency
Remains an overlooked issue

6. Outline “1-index” paper 1 13 section section 2 title 14 title 3 8
“experiments”
exp
“intro”
algorithm 15 16 5
title 6
exp
algorithm 9
title 10
proof 7 17
“A(k)-index”
“1-index” 11 18
about
proof
about 12
uses

7. 1-Index: Definition Constructed by using bisimilarity
Definition based on stability
Partition data nodes into index nodes
dnode (v) and inode (I[v])
I[u] is v’s index parent if u is v’s parent
An inode is stable if all of its dnodes have the same index parents
In a 1-index, all inodes are stable
I[u] u
I[v] v

8. 1-Index: Example /paper/section/algorithm paper paper 1 1 13 section
title 14
section 2
2,4,8,13
section 8 4
section 3 15
exp
exp
title
exp
algorithm
title
algorithm 16 10
15,16
3,5,9,14
6,10 6 9
algorithm
title 5
title 18
about
proof
proof 17 11
17,18
proof 7 7
uses 11
about
about
proof 12 12
/paper/section/algorithm
uses
data graph
1-index

9. 1-Index: Quality
paper
Assigning dnodes that are bisimilar into different inodes
does not affect correctness,
but does affect efficiency
The quality of an index 1
section
2,4
2,4,8,13
8,13
exp
title
algorithm
15,16
3,5,9,14
6,10
proof 11
17,18
# inodes 7
− 1 X 100%
about
# inodes in
the minimum 1-index
proof 12
uses
Ideal: quality = 0%

10. Previous Results Construction Edge changes Subgraph addition
The PT algorithm [Paige & Tarjan 87], in time O(m log n)
m – # edges, n - # nodes
Edge changes
The propagate algorithm [Kaushik et al. 02]
Quality of the 1-index after update
No guarantee on the quality of the resulted index
3 ~ 5% after 500 edge insertions in experiments
Subgraph addition
Index-reconstruction

11. Edge Insertion: An Example (1)B A B A B C1 C2 C3
C1, C2 C3 C1 C2 C3 D1 D2 D3
D1, D2 D3
D1, D2 D3
Data Graph
1-Index
Split 1

12. Edge Insertion: An Example (2)B A B A B C1 C2 C3 C1
C2, C3 C1
C2, C3 D1 D2 D3 D1 D2 D3 D1
D2, D3
Split 2
Merge 1
Merge 2
Indeed the minimum 1-index
for the data graph after update
Not a coincidence!

13. Minimum & Minimal Indexes
Minimum: with the smallest number of inodes
Minimal: no two inodes can be merged R R R A1 A2 A1 A2
A1,A2 B1 B2
B1,B2 B1 B2
Data graph Minimum 1-index Minimal 1-index

14. Quality Guarantee
Theorem: The split/merge algorithm always maintains a minimal 1-index
Lemma: For acyclic data graphs, there is a unique minimal 1-index
The minimum 1-index is always maintained
For cyclic data graphs, there could be more than one minimal 1-index
One of them is maintained

15. Outline “A(k)-index” paper 1 13 section section 2 title 14 title 3 8
“experiments”
exp
“intro”
algorithm 15 16 5
title 6
exp
algorithm 9
title 10
proof 7 17
“A(k)-index”
“1-index” 11 18
about
proof
about 12
uses

16. A(k)-Index: Definition
k-bisimilarity
Definition based on stability
A(0)-index: partition by label …
A(k)-Index
An inode in A(k)-index is stable if all of its dnodes have the same index parents in A(k-1)-index
Only interested in paths of length ≤k
Shown to be much smaller and more efficient than 1-index [Kaushik et al. 02]
But, no efficient maintenance algorithms are known!

17. A(k)-index: Example R R R R A B A B A B A B C1 C2 C3 C1 C2,C3 C1 C2,C3
Data graph A(2) (=1-index) A(1) A(0)
Maintenance of A(i)-index requires the information in A(i-1)-index

18. A(k)-index: Refinement TreeB A B A B A B C1 C2 C3 C1
C2,C3 C1
C2,C3
C1,C2,C3
C4,C5,C6 C4 C5 C6 C4
C5,C6
C4,C5,C6
Data graph A(2) (=1-index) A(1) A(0)

19. A(k)-index: Refinement TreeB A B A B A B C1 C2 C3 C C C C C C4 C5 C6 C C C
Data graph A(2) A(1) A(0)
Reduce storage cost
Reduce maintenance cost
0.5% ~ 13% additional storage

20. Quality Guarantee
Theorem: The split/merge algorithm always maintains A(k)-index
Lemma: There is a unique minimal A(k)-index for any data graph, acyclic or cyclic
the minimum
a minimal
1-index
A(k)-index
Acyclic
minimum
Cyclic
minimal

21. Outline “experiments” paper 1 13 section section 2 title 14 title 3 8
“intro”
algorithm 15 16 5
title 6
exp
algorithm 9
title 10
proof 7 17
“A(k)-index”
“1-index” 11 18
about
proof
about 12
uses

22. Experiments on Edge Changes
Datasets
Real-life: IMDB (272,000 nodes)
Benchmark: XMark (198,000 nodes)
Setup
First delete a portion of existing ID-REF links
Then do random mixed insertions/deletions
Compare with
1-index: propagate (+ reconstruction)
A(k)-index: recompute affected portion (+ reconstruction)

23. Experiment Results: 1-index

24. Experiment Results: A(k)-index
speedup 2
1.35 3
6.15 4
16.6 5
15.3
running times

25. Conclusions
The first solutions for the maintenance (edge & subgraph additions/deletions) of 1-index and A(k)-index that are both effective and efficient
Effective: quality guarantee on the resulted index
Efficient: the algorithms themselves are fast
Thank you!

26. Graphical Illustration
size
valid 1-index
merge
split
index
the index can only grow in size due to splitting, if merging is not enforced