An Abstract Framework for Generating Maximal Answers to Queries

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An Abstract Framework for Generating Maximal
Answers to Queries

• Sara Cohen, Yehoshua Sagiv

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Motivation
Queries and Databases
Answers and Semantics
Graph Properties
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The Problem

• In many different domains, we are given the
  option to query some source of information
• Usually, the user only gets results if the query
  can be completely answered (satisfied)
• In many domains, this is not appropriate, e.g.,
• The user is not familiar with the database
• The database does not contain complete
  information
• There is a mismatch between the ontology of the
  user and that of the database
• The query is a search that is not expected to
  be correct

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Search for papers by Smith that appeared in
ICDT 2004
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Sorry, no matching record found
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Search for buses from Haifa-Technion to Ben
Gurion Airport
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There is no direct bus line between the required
destinations
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Search for buses to Ben Gurion Airport
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Must choose From and To
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What Do Users Need?

• Users need a way to get interesting partial
  answers to their queries, especially if a
  complete answer does not exist
• These partial answers should contain maximal
  information
• Main Problems
• What should be the semantics of partial answers?
• How can all partial answers be efficiently
  computed?

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Previous Work

• Many solutions have been given for the main
  problems
• solutions differ, according to the problem domain
• Examples
• Full disjunctions Galindo-Legaria (94),
  Rajaraman, Ullman (96), Kanza, Sagiv (03)
• Queries with incomplete answers over
  semistructured data Kanza, Nutt, Sagiv (99)
• FleXPath Amer-Yahia, Lakshmanan, Pandit (04)
• Interconnections Cohen, Kanza, Sagiv (03)

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Our Contribution

• In the past, for each semantics considered, the
  query evaluation problem had to be studied anew.
  In this paper, we
• Present a general framework for defining
  semantics for partial answers
• Framework is general enough to cover most
  previously studied semantics
• Query evaluation problem can be solved once
  within this framework and reused for new
  semantics
• Results improve upon previous evaluation
  algorithms
• Presents relationship between this problem and
  that of the maximal P-subgraph problem

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Motivation
Queries and Databases
Answers and Semantics
Graph Properties
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Databases

• Databases are modeled as data graphs (V, E, r,
  lV, lE)
• r Can have a designated root
• lV Labels on the vertices
• lE Labels on the edges
• Note
• Nodes correspond to data items
• Even databases that do not have an inherent graph
  structure can be modeled as graphs, e.g.,
  relational databases

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XML as a Data Graph
University
Name
Dept
Dept
Technion
Name
Name
Faculty
Faculty
Computer Science
Biology
Professor
Lecturer
Teaches
Teaches
Teaches
Name
Name
Avi Levy
Chana Israeli
Bioinformatics
Databases
Molecular Biology
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Relational Database as a Data Graph
Sites
Climates
Country Climate Country Climate
Canada diverse
UK temporate
USA temporate
Country City Site Country City Site Country City Site
UK London Buckingham
USA NY Metropolitan
Accommodations
Country City Hotel Country City Hotel Country City Hotel
UK London Plaza
Canada Montreal Hitlon
Canada Toronto Ramada
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Relational Database as a Data Graph
Sites
Climates
Country Climate Country Climate
Canada diverse
UK temporate
USA temporate
Country City Site Country City Site Country City Site
UK London Buckingham
USA NY Metropolitan
Accommodations
Country City Hotel Country City Hotel Country City Hotel
UK London Plaza
Canada Montreal Hitlon
Canada Toronto Ramada
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Relational Database as a Data Graph
Sites
Country City Site Country City Site Country City Site
UK London Buckingham
USA NY Metropolitan
Accommodations
Country City Hotel Country City Hotel Country City Hotel
UK London Plaza
Canada Montreal Hitlon
Canada Toronto Ramada
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Relational Database as a Data Graph
Sites
Country City Site Country City Site Country City Site
UK London Buckingham
USA NY Metropolitan
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Relational Database as a Data Graph
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Queries

• Queries are modeled as query graphs (V, E, r,
  CV, CE, s)
• r Can have a designated root
• CV Vertex constraints on the vertices
  (basically, a boolean function on vertices)
• CE Edge constraints on the edges (basically, a
  boolean function on pairs of vertices)
• s A structural constraint, one of the letters C,
  R, N (defines the required structure of answers,
  i.e., connected, rooted or none)
• Note Nodes correspond to query variables

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XML Query as a Graph

• Returns faculty members from the Biology
  Department

University
Is Descendent
Dept and ContainsText(Biology)
Is Child
Structural Constraint Rooted
Faculty
Is GrandChild
Name
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Join Query as a Graph

• C A S

Structural Constraint Connected
Belongs to C
q1
C.Country A.Country
C.Country S.Country
q2
q3
Belongs to A
Belongs to S
A.Country S.Company and A.City S.City
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Motivation
Queries and Databases
Answers and Semantics
Graph Properties
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Assignment Graphs

• Assignment graphs are used to compactly represent
  assignments of query nodes to database nodes
• Basically, assignment graph for Q and D, written
  Q?D has
• Node (q,d) for each pair q? Q and d? D such that
  d satisfies the constraint on q
• Edge ((q,d), (q,d)) if there is an edge (q,q)
  in Q and (d,d) satisfies the constraint on
  (q,q)
• May also have a root (details omitted)

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(q3, s1)
(q2, a1)
(q1, c1)
(q2, a2)
(q1, c2)
(q1, c3)
(q2, a3)
(q3, s2)
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Partial Assignment

• A partial assignment is any subgraph of Q?D that
  does not contain two different nodes (q,d) and
  (q,d)
• otherwise, would map the node q to two different
  database nodes
• Can distinguish special types of partial
  assignments
• vertex complete
• edge complete
• structurally consistent

Every query node must appear in the partial
assignment
The partial assignment satisfies the querys
structural constraint
Every edge constraint between query variables in
the partial assignment holds
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Example
(q3, s1)
(q2, a1)
(q1, c1)
(q2, a2)
(q1, c2)
(q1, c3)
(q2, a3)
(q3, s2)
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Semantics

• All partial assignments for Q over D that satisfy
  the vertex and edge constraints are encoded in
  Q?D
• A semantics defines which subgraphs of the answer
  graph (i.e., which partial assignments) are in
  fact answers, e.g.,
• Sves allows all partial assignments that are
  vertex complete, edge complete and structurally
  consistent
• Ses allows all partial assignments that are edge
  complete and structurally consistent
• Ss allows all partial assignments that are
  structurally consistent
• Usually, we are only interested in maximal
  partial assignemnts

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Example Join
(q3, s1)
Using semantics Sves we get the natural join
(q2, a1)
(q1, c1)
(q2, a2)
(q1, c2)
(q1, c3)
(q2, a3)
(q3, s2)
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Example Join becomes a Full Disjunction
(q3, s1)
Using semantics Ses we get the full disjunction
(q2, a1)
(q1, c1)
(q2, a2)
(q1, c2)
(q1, c3)
(q2, a3)
(q3, s2)
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Other Examples

• Queries with incomplete answers over
  semistructured data Kanza, Nutt, Sagiv (PODS 99)
• Weak semantics modeled by Ses Or-semantics
  modeled by Ss
• FleXPath Amer-Yahia, Lakshmanan, Pandit (Sigmond
  04)
• Modeled by Ses
• Interconnections Cohen, Kanza, Sagiv (03)
• Complete interconnection can be modeled by Ses
  Reachable interconnection can be modeled by Ss

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Motivation
Queries and Databases
Answers and Semantics
Graph Properties
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Semantics are a type of Graph Property

• A graph property P is a set of graphs, e.g.,
• is a clique
• is a bipartite graph
• A semantics defines a set of graphs, for every Q,
  D (these graphs are subgraphs of Q?D)
• Therefore, semantics are a type of graph property

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Hereditary Graph Properties and their Variants

• There are several interesting types of graph
  properties that have been studied in graph theory
• A graph property P is hereditary if every induced
  subgraph of a graph in P, is also in P (e.g.,
  clique, is a forest)
• A graph property P is connected-hereditary if
  every connected induced subgraph of a graph in P,
  is also in P (e.g., is a tree)
• Can define rooted-hereditary similarly

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Semantics are usually Hereditary

• Most semantics for partial answers considered in
  the past are hereditary (in some sense), i.e.,
  subgraphs of a partial answer are also partial
  answers
• Many semantics require connectivity of results
  (e.g., full disjunctions)
• Some require answers to be rooted (e.g., FlexPath)

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Maximal P-Subgraph Problem

• Given a graph property P, and a graph G The
  maximal P-subgraph problem is Find all maximal
  induced subgraphs of G that have property P
• Therefore, the problem of finding all maximal
  answers for a query over a database, under a
  given semantics, is a special case of the maximal
  P-subgraph problem

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Efficient Query Evaluation

• There are efficient algorithms that find all
  maximal P-subgraphs for hereditary, connected
  hereditary and rooted hereditary properties
• Efficient in terms of the input and the output
  (i.e., incremental polynomial time)
• Use these algorithms to find maximal query
  answers, e.g., to find full disjunctions, weak
  answers, or-answers, etc.
• Improves upon previous results

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Conclusion

• Presented abstract framework
• Can model many different types of queries,
  databases and semantics in the framework
• Semantics in the framework are graph properties
• Solve the maximal P-subgraph problem once and
  reuse it to find maximal query answers

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Future Work

• It is convenient to define ranking functions and
  return answers in ranking order
• How/when can this be done in our framework?
• Note From the modeling it is immediately
  apparent that ranking cannot always be performed
  efficiently
• The problem of finding a maximal P-subgraph of
  size k is NP complete for hereditary and
  connected-hereditary graph properties
  (Yannakakis, STOC 78)

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Thank you!Questions?