XML indexing Ak indices

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XML indexing A(k) indices

• - Ragini
  Rahalkar
• 
  - Roshith Rajagopal

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Outline

• Introduction
• Motivation
• Labeled graph and index graph
• Bisimilarity and A(k) index
• Construction of A(k) index
• Query Evaluation
• Approximate index handling
• Implementation and testing
• Summary

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Introduction

• Structural summaries
• Evaluating Path Expressions
• A(K) index
• Indexing scheme for large graph data like XML
• Not all structure is interesting
• Paths longer than k
• Smaller and faster
• Schemaless data
• Competitive for arbitrary path expressions

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Prior Schemes

• 1-index Milo, Suciu 1999
• NFA rather than DFA (smaller)
• split graph nodes into equivalence classes based
  on incoming paths from the root
• Go for refinements (approximations)
• similarity
• bisimilarity

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Limitations of Prior Work

• Size
• Each and every path is indexed which is not
  necessary (does not exploit local similarity)
• 1-index size can be big too!
• Designed to answer queries involving arbitrarily
  complex paths, but...
• such paths may never show up in queries

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Labeled Graph

• G(Vg, Eg, root, SG, label, oid, value)
• Node path and label path
• Path expression
• Regular language

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Index Graph I(G)

• Extent of a node
• Regular expression execution with I(G)
• Safe extent mapping
• Containment of results of path expressions
• Precise index graph
• 1-index graph never bigger than data graph
• Can be computed in O( m log n )

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Notion of Bisimilarity

• Symmetric and binary relation
• For two nodes u and v , u b v if
• u and v have same labels
• If u is a parent of u, then there is a parent v
  of v such that u v and vice versa
• Objects 8 and 9 are bisimilar
• Objects 21 and 23 are not bisimilar

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The A(k) index

• Local similarity
• Using Equivalence class partition
• Grouping according to labels
• Notion of false paths
• Classification by labelbusiness and cultural
• Absolute precision and grouping similar data to
  allow index size affected by updates in the
  values of k

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K-bisimilarity

• Defined inductively as
• for any two nodes, u and v, u 0 v if u and v
  have same labels
• u k v iff
• u k-1 v and
• For every parent u of u and v of v
• u k-1 v

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1
1
A
A
1
1
A
A
B
B
C
2
3
2
3
C
C
2
3
2
3
B
B
C
4
5
D
4
5
4
5
4,5
D
D
D
D
D
D
6,7
E
E
E
E
E
E
7
6
6,7
7
6
A (0)
G
A (1)
A (2) 1-INDEX
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A(k) index properties

• If nodes u and v are k-bisimilar, then the set of
  labelpaths of length k into them is the same.
• The set of label-paths of length k into an
  A(k)-index node is the set of label-paths of
  length k into any node in its extent.
• The A(k)-index is precise for any simple path
  expression of length less than or equal to k.
• The A(k)-index is safe, i.e., its result on a
  path expression always contains the graph result
  for that query.
• The (k 1)-bisimulation is either equal to or is
  a refinement of the k-bisimulation.
• Let v x y be three nodes such that the shortest
  path to x from v or to y from v contains more
  than k edges. If an edge is added or deleted
  going from a node u to v, this update does not
  affect the k-bisimilarity relationship between x
  and y

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A(k) index construction

• Partitioning compute_k_bisim
• Notion of successor of a node
• Notion of stability
• Two sets of nodes A and B- Partition as
• A n SUCC(B)
• A SUCC(B)
• Computation of k1 bisimulation from k
  bisimulation
• Copy of k bisimulation divided into equivalence
  classes until they are stable with equivalence
  classes of k bisimulation
• Time O(km) Space- O(m) where m is no of edges

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Compute_k_bisim(G,k)

• Begin
• Q and X are each a list of node-sets
• Q partition VG by label
• X (a copy of) Q
• for i1 to k do
• foreach X1 in X do
• compute Succ(X1)
• for each Q1 in Q do
• replace Q1 by Q1 n Succ(X1) and Q1-
  Succ(X1)
• if there was no split then
• break
• X (a copy of) Q
• End

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Compute_A(k)_index(G,k)

• Begin
• Compute_k_bisim(G,k)
• foreach equiv. class in k-bisimulation do
• create an index node I
• extI data nodes in the equivalence Class
• foreach edge from u to v in G do
• Iu index node containing u
• Iv index node containing v
• if there is no edge from Iu to Iv then
• add an edge from Iu to Iv
• End

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Query Evaluation Schemes

• Index is queried using regular path expressions.
• Path expressions are of the form
• P Root.R
• Query Evaluation Techniques
• Forward Evaluation
• Backward Evaluation

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Query Evaluation Techniques

• Forward Evaluation Strategy
• Simulation of NFA on the graph
• Index graph traversed breadth first , making
  corresponding transitions
• Backward Evaluation Strategy
• Find nodes bearing final labels in R
• R evaluated in reverse manner from these nodes
• Intuition end of the expression more selective
  than the earlier paths, thus processing cheaper

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Approximate Index Graphs

• While evaluating R on Index graph, we add nodes
  in the ExtB rather than B to the result set.
• A(k) index is safe
• Result set for R is superset of the target set in
  the data graph.

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Approximate Index Graphs

• When node B is accepted along a path of length
  ltK in the A(k) Index Graph , a node in ExtB
  must be in the target set of R
• When index node accepted by a longer path, the
  data node initially added to a maybe set M
  instead of result set.
• Nodes in M are validated by reverse execution of
  the automation on the data graph beginning with
  each node in M

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Implementation Data Structures

• Data Graph Representation
• Element_HT
• Hashtable (NodeID, Element) Pairs
• Attribute_HT
• Hashtable
• (NodeID -1, IDREF Attribute) Pairs
• The Key of this hashtable is the NodeID of the
  element of this attribute.

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Implementation

• Index Tree
• EqClass_HT - Hashtable
• (EqClassID, Vector of NodeIDs in that EqClass)
• Generated from Compute_K_Bisim
• Link Table
• Linktable_HT - Hashtable
• (EqClassID, Vector of Child EqClassIDs)
• Generated from Compute_A(K)_index

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Sample ResultsSize of Index graph v/s K
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Summary

• Generalization of 1-index
• Value of k and tradeoff between the size of the
  index graph and accuracy
• Small values of k perform better than 1-Index
• Future scope
• Use in schema extraction and query optimization

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References

• Exploiting Local Similarity for Indexing Paths in
  Graph-Structured Data Raghav Kaushik, Ehud Gudes
  et all
• Index Structures for Path Expressions Milo,
  Suciu 1999

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• THANK YOU