Incremental Maintenance of XML Structural Indexes

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
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experiments
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intro
algorithm
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exp
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algorithm
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proof
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A(k)-index
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1-index
about
proof
about
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uses
4
Label Path Expressions
paper
/paper/section/algorithm
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section
section
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title
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experiments
exp
intro
algorithm
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exp
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title
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title
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algorithm
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proof
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A(k)-index
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1-index
about
proof
about
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uses
5
Structural Indexes

• Why do we need them?
• 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
  listsKaushik et al. 04
• Focus on maintenance
• Has a major effect on index efficiency
• Remains an overlooked issue

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

• Constructed by using bisimilarity
• Definition based on stability
• Partition data nodes into index nodes
• dnode (v) and inode (Iv)
• Iu is vs index parent if u is vs parent
• An inode is stable if all of its dnodes have the
  same index parents
• In a 1-index, all inodes are stable

Iu
u
Iv
v
8
1-Index Example
paper
paper
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1
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section
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title
section
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2,4,8,13
section
8
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3
exp
exp
title
algorithm
exp
algorithm
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title
15,16
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3,5,9,14
6,10
6
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algorithm
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proof
about
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proof
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proof
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about
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about
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proof
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/paper/section/algorithm
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data graph
1-index
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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
2,4,8,13
2,4
8,13
exp
title
algorithm
15,16
3,5,9,14
6,10
proof
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17,18
inodes
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- 1 X 100
about
proof
inodes in the minimum 1-index
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uses
Ideal quality 0
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Previous Results

• Construction
• 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)
R
R
R
A
B
A
B
A
B
C1
C2
C3
C1, C2
C3
C3
C1
C2
D1
D2
D3
D3
D1, D2
D3
D1, D2
Data Graph
1-Index
Split 1
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Edge Insertion An Example (2)
R
R
R
A
B
A
B
A
B
C3
C1
C2
C2, C3
C1
C2, C3
C1
D3
D1
D2
D3
D1
D2
D2, D3
D1
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
B2
B1
B2
B1
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
paper
1
13
section
section
2
title
14
title
3
8
section
4
section
experiments
exp
intro
algorithm
15
16
exp
5
title
6
9
title
10
algorithm
7
proof
17
A(k)-index
11
18
1-index
about
proof
about
12
uses
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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
C3
C1
C2
C2,C3
C1
C2,C3
C1
C1,C2,C3 C4,C5,C6
C6
C4
C5
C4
C5,C6
C4,C5,C6
Data graph A(2) (1-index)
A(1) A(0)
Maintenance of A(i)-index requires the
information in A(i-1)-index
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A(k)-index Refinement Tree
R
R
R
R
A
B
A
B
A
B
A
B
C3
C1
C2
C2,C3
C1
C2,C3
C1
C1,C2,C3 C4,C5,C6
C6
C4
C5
C4
C5,C6
C4,C5,C6
Data graph A(2) (1-index)
A(1) A(0)
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A(k)-index Refinement Tree
R
R
R
R
A
B
A
B
A
B
A
B
C3
C1
C2
C
C
C
C
C
C6
C4
C5
C
C
C
Data graph A(2)
A(1) A(0)

• Reduce storage cost
• Reduce maintenance cost

0.5 13 additional storage
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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

a minimal
the minimum
21
Outline
paper
1
13
section
section
2
title
14
title
3
8
section
4
section
experiments
exp
intro
algorithm
15
16
exp
5
title
6
9
title
10
algorithm
7
proof
17
A(k)-index
11
18
1-index
about
proof
about
12
uses
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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)

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Experiment Results 1-index
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Experiment Results A(k)-index
running times
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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