Akindex : Exploiting Local Similarity to Index Paths in Graph Data

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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)

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Outline

• Problem statement
• Prior work and limitations
• Background
• A(k)-index
• Query Evaluation
• Preliminary experiments
• Update
• Conclusions

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Data Model

• Rooted, node-labeled graph with unique root root
  has unique label
• Nodes - objects
• Arcs - object-subobject relationship
• In 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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Example index graph
0
0
2
1
2
1
3,4
4
3
5,6
6
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Data graph Index graph
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Index Graph (contd)

• 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 (contd)

• 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)
• similarity
• bisimilarity

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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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Local Similarity
ROOT
metro
cultural
neighborhoods
business
museum
museum
hotel
nhd.
nhd.
nearby
attr.
attr.
cult.
cult.
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Main Contributions

• New family of approximate index structures
• Applicable to
• Approximate Schema
• Statistics
• Query evaluation using approximate indexes
• Preliminary performance study
• Update algorithms

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Approximate Indexes

• Motivation
• Smaller
• More efficient query processing
• Limited update cost - maintain local information
• Approximate 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 statement
• Prior work and limitations
• Background
• A(k)-index
• Query Evaluation
• Preliminary experiments
• Update
• Conclusions

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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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Graph Bisimulation (contd)
and vice-versa!
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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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Bisimulation on example
ROOT
metro
cultural
neighborhoods
business
museum
museum
hotel
nhd.
nhd.
nearby
attr.
attr.
cult.
cult.
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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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Example for k-bisimulation
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A(2) for example
ROOT
metro
cultural
neighborhoods
business
museum
museum
hotel
nhd.
nhd.
nearby
attr.
attr.
cult.
cult.
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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
• If k-bisim k1-bisim then k-bisim bisim
• Size 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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Example Query Evaluation
Automaton Graph
Index Graph
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Approximate Indexes

• Caveat False positives possible
• Approach 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 verification not required
• Second 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 (contd)

• 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 statement
• Prior work and limitations
• Background
• A(k)-index
• Query Evaluation
• Preliminary experiments
• Update
• Conclusions

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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 graph
• Performance metrics are abstract
• Cost total number of nodes visited (graph
  index)

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Bisimulation Sizes
IMDB Nodes 190,000 ODP Nodes 143,000
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Query Evaluation Plans
1. Forward eval 2. Backward eval (assume a label
index)
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Short Queries - IMDB
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Long Queries - IMDB
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Queries beginning with _
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Queries containing _
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Approximate Answers
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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 vs parents

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A(k)-index Update (contd)

• A(k)-index
• only nodes to be considered are those at distance
  
• Maintain tree of splits
• Work 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) update
• R - 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 relation
• B - any refinement of B
• B(G) - index graph built using B
• B(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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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
• Storage
• Application of update algorithms to triggers