a(k)-index : exploiting local similarity to index paths in graph data raghav kaushik (uw) pradeep...
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A(k)-index :Exploiting Local Similarity to Index Paths in Graph Data
Raghav Kaushik (UW)Pradeep Shenoy (UWash)Philip Bohannon (Bell Labs)Ehud Gudes (BGU)
![Page 2: A(k)-index : Exploiting Local Similarity to Index Paths in Graph Data Raghav Kaushik (UW) Pradeep Shenoy (UWash) Philip Bohannon (Bell Labs) Ehud Gudes](https://reader035.vdocuments.site/reader035/viewer/2022062519/5697c0071a28abf838cc60cc/html5/thumbnails/2.jpg)
Outline
Problem statementPrior work and limitationsBackgroundA(k)-indexQuery EvaluationPreliminary experimentsUpdateConclusions
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Data Model
Rooted, node-labeled graph with unique root; root has unique label
Nodes - objectsArcs - object-subobject relationshipIn XML context
Index tag structure No distinction between elements and attributes No distinction between tree and idref arcs Order ignored
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Problem Statement
Practical indexing schemes for large graph data (like XML data) (100K - 1M nodes) Size ~10% of database size Efficient construction and update Tunable to a workload
Queries of the form R x, where R is a regular path expression
Schemaless data
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Flavor of Approach
Different from traditional value indices
Structural summaries for indexing paths
Both data and index are rooted graphs
Example: Dataguide
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Index Graph
Structural summary
Associate a set of data nodes with each index node, called its extent
Preserve data paths in index graph
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Data graph Index graph
Example index graph
5,6
3,4
21
00
1 2
3 4
5 6
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Index Graph (cont’d)
Can be constructed from any partition
Node for every equivalence class C
Edge between C and C’ if exists an
edge v v’ with v in C and v’ in C’
Preserves data paths, no false drops
Our structures are all index graphs
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Prior Schemes
Dataguide [Goldman, Widom 1997] Deterministic automaton corresponding
to data graph
Each set of data nodes that can be distinguished by a path query is summarized by a single node in the index
Can be exponential in size!
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Prior Schemes (cont’d)
1-index [Milo, Suciu 1999] NFA rather than DFA (smaller) split graph nodes into equivalence classes
based on incoming paths from the root Computing best split is PSPACE complete Go for refinements (approximations)
similaritybisimilarity
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Limitations of Prior Work
Size Dataguide sizes subject to exponential blow-
up 1-index size can be big too!
Update No known update algorithm for 1-index
Designed to answer queries involving arbitrarily complex paths, but... such paths may never show up in queries
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ROOT
metro
cultural business neighborhoods
museum museum hotel
nearby
nhd.nhd.
attr.attr.
cult.cult.
Local Similarity
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Main Contributions
New family of approximate index structures
Applicable to Approximate Schema Statistics
Query evaluation using approximate indexes
Preliminary performance studyUpdate algorithms
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Approximate Indexes
Motivation: Smaller More efficient query processing Limited update cost - maintain local
informationApproximate dataguide [Goldman, et.al]
path merging, object matching, etc no formal basis (but different goal) no study of effect on query processing
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Outline
Problem statementPrior work and limitationsBackgroundA(k)-indexQuery EvaluationPreliminary experimentsUpdateConclusions
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Graph Bisimulation
A bisimulation is a symmetric relation R between nodes
If A1 R A2 then A1 and A2 have the same labels and ...
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B1
A1 A2R
A1 A2
B1
R
B2R
and vice-versa!
Graph Bisimulation (cont’d)
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Bisimilarity
Two nodes a and b are bisimilar if they are related in some bisimulation
1-index is index graph constructed from bisimulation partition
Simulation partition: similar
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ROOT
metro
cultural business neighborhoods
museum museum hotel
nearby
nhd.nhd.
attr.attr.
cult.cult.
Bisimulation on example
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k-bisimulation
Nodes A1 and A2 are 0-bisimilar iff same label
A1 and A2 are k-bisimilar iff k-1 bisimilar and
if (B1, A1), exists (B2, A2): B1 and B2 are k-1 bisimilar, and vice versa
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Data graph
0
1 2
3 4
65
Example for k-bisimulation
0
1 2
3,4
5,6
0-bisimulation
0
1 2
3 4
5,6
1-bisimulation
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ROOT
metro
cultural business neighborhoods
museum museum hotel
nearby
nhd.nhd.
attr.attr.
cult.cult.
A(2) for example
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Properties
If a and b are bisimilar set of incoming paths into them is same
If a and b are k-similar or k-bisimilar set of incoming paths of length <= k are
sameIf k-bisim = k+1-bisim then k-bisim =
bisimSize: certainly smaller than bisimulation
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Query Evaluation
Only queries studied are regular path queries of the form R x
Query Evaluation Approach: Create automaton for regexp query Run automaton on the index graph Result is union of extents belonging to
index nodes accepted by automaton
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0
1 2
3,4
5,6
Automaton Graph Index Graph
Example Query Evaluation
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Approximate Indexes
Caveat: False positives possibleApproach: verify each node on data
graph by running reverse automaton Prohibitive cost?
Then why use approx. indices? In fact, frequently more efficient than
data graph or precise index
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Improving Validation
First cut: Keep track of accepting-path-length for accepted nodes with path length <= k,
verification not requiredSecond step: Share traversals among
verification calls mark node-state pairs on a successful
verification path as accept similar marking for failed path
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Improving Validation (cont’d)
Third Step: Avoid needless
verification
Example: For _*.R queries, no need to
verify all the way up to the root
Generalize the above!
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Outline
Problem statementPrior work and limitationsBackgroundA(k)-indexQuery EvaluationPreliminary experimentsUpdateConclusions
![Page 30: A(k)-index : Exploiting Local Similarity to Index Paths in Graph Data Raghav Kaushik (UW) Pradeep Shenoy (UWash) Philip Bohannon (Bell Labs) Ehud Gudes](https://reader035.vdocuments.site/reader035/viewer/2022062519/5697c0071a28abf838cc60cc/html5/thumbnails/30.jpg)
Preliminary Experiments
Data used: Internet Move Database (http://www.imdb.com) 250,000 movies & TV shows 460,000 actors, etc XML version = ~1GB
We used subsets of this database ranging from 200 - 2000 movies
Whole database --> future work!
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Preliminary Experiments
Second source: Open Directory Project (http://www.dmoz.org) Entire source available in RDF format
Subsets: (entire subtree under a topic, say shopping)
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Storage Model
Results independent of any particular storage model
In-memory rooted graphPerformance metrics are abstract
Cost = total number of nodes visited (graph + index)
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IMDB#Nodes:190,000
ODP#Nodes:143,000 0
10
20
30
40
0 2 4 6 8 10 12 14 16
K parameter of Index
Pe
rce
nt
of
Da
ta G
rap
h S
ize
A(k)-Index, IMDBData
A(k)-Index, ODPData
Bisimulation Sizes
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Query Evaluation Plans
012345678
IMDB Short IMDB Long ODP Short
Workload
No
de
Vis
its
(L
og
Sc
ale
)
1-index(fwd)
1-index(back)
G (fwd)
G(back)
1. Forward eval
2. Backward eval (assume a label index)
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0
0.5
1
1.5
2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
K parameter of Index
Fra
cti
on
of
1-I
nd
ex
Co
st
Validation Cost
Index Cost
Short Queries - IMDB
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0
0.5
1
1.5
2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
K parameter of Index
Fra
cti
on
of
1-I
nd
ex
Co
st Validation Cost
Index Cost
Long Queries - IMDB
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0
0.5
1
1.5
2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
K parameter of Index
Fra
cti
on
of
1-I
nd
ex
Co
st
validcost
indexcost
Queries beginning with _*
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0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
K parameter of Index
Fra
cti
on
of
1-I
nd
ex
Co
st
validcost
indexcost
Queries containing _*
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Approximate Answers
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15K parameter of Index
Fra
cti
on
of
Co
rre
ct
Re
su
lt S
ize
False Positives
ToValidate
Guaranteed Results
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A(k)-index Update
Edge added from u to v
A(0)-index -> no change except possible addition of edge
A(1)-index -> index node containing v may change determined by set of labels in v’s
parents
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A(k)-index Update (contd)
A(k)-index only nodes to be considered are those at
distance < k from vMaintain tree of splitsWork iteratively:
find new A(1) position of v find new A(2) positions of v and its children …
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Updating the 1-index
One way is generalization of A(k) updateR - any binary relation on the nodes that
is reflexive transitively closed.
A refinement of R is any subset that is reflexive transitively closed
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Refinement
B - bisimulation relationB’ - any refinement of BB(G) - index graph built using BB’(G) - index graph built using B’
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Theorem
Theorem: B(B’(G)) = B(G)Intuition:
Similar nodes behave similarly So, fuse them together!
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Lazy Update
Basic Idea: G G’ , and meanwhile B(G) B(G’) Instead, “relax” the graph B(G) to B’(G’)
How? A “stable” partitioning of G is either
B(G) or its refinement. Propagate graph update on B(G) by
splitting nodes until stable.
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0
1
2
3
4
5
6
0 100 200 300 400 500
Number of edges added
Pe
rce
nt
inc
rea
se
in
ind
ex
siz
e Propagated Index
Accurate Index
Lazy Update Performance
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Conclusions
Novel approximate index structures and validation techniques
Experiments demonstrate k-bisimulation index is Efficiently constructed Effective for query answering
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Future Work
Handle more query types Branching queries Queries with selection
Annotating A(k) with statistics for query optimization
StorageApplication of update algorithms to
triggers