Implementing Mapping Composition

1
Implementing Mapping Composition

• Todd J. Green
• University of Pennsylania
• with Philip A. Bernstein (Microsoft Research),
• Sergey Melnik (Microsoft Research),
• Alan Nash (UC San Diego)
• VLDB 2006 Seoul, Korea
• Work partially supported by NSF grants
  IIS0513778 and IIS0415810

2
Schema mappings

• Mapping a correspondence between instances of
  different schemas

Names SID, Name
Students Name, Address
m
Addresses SID, Address
S1
S2
Students ? ?Name,Address (Names ? Addresses)
3
Applications of mappings

• Names ? Names?
• sCountry KR(Addresses) ? ?SID,Address(Local)KR
  
• sCountry ? KR(Addresses) ? Foreign

• Schema evolution

Students ? ?Name,Address,Country(Names ?
Addresses)
...
m12
m23
S3
S2
S1
4
Applications of mappings

• Data integration, data exchange

Sn
Addresses SID, Address, Country
Names SID, Name
...
m1
mn
Students ? ?Name,Address (Names ?
Addresses)
Names? ? Names Local ? ?SID,Address(?Country
KR(Addresses)) Foreign ? ?Country ? KR(Addresses)
S1
Sn-1
Foreign SID, Address, Country
Students Name, Address, Country
Names? SID, Name
Local SID, Address
...
5
Requirements for constraints

• First attribute in R is a key for R
• ?2,4(R ?13 R) µ ?2,2(R)
• View V equals R joined with S
• V µ R ? S, V R ? S
• Second attribute of R is a foreign key in S
• ?2(R) µ ?1(S)
• ?2,4(S ?13 S) µ ?2,2(S)
• Data integration, data exchange GLAV
• R ? S µ T ? U

6
Mapping composition
S1
S3
7
Composition is hard

• Hard part write composition in the same language
  as the input mappings. Depending on language
• Not always possible
• Not even decidable whether possible
• Strategy 1 use powerful (second-order) mapping
  language closed under composition FKPT04
• Not supported by DBMS today
• Expensive to check
• Source-target restriction
• Strategy 2 settle for partial solutions NBM05
• Containment mappings ? easier integration with
  DBMS
• The strategy we adopt in this work

8
Our contributions

• New algorithm for composition problem
• Incorporates view unfolding and left-composition
  (new technique)
• Makes best effort in failure cases
• Algebraic rather than logic-based mappings
• Use of monotonicity to handle more operators
• Modular and extensible factoring of algorithm
• First implementation of composition
• Experimental evaluation

9
Formal definition of composition

• Mapping set of pairs of instances of db schemas
• The composition m12 m23 is the mapping
• hA,Ci (9B)(hA,Bi 2 m12 and hB,Ci 2 m23)
• where A,B,C are instances of S1,S2,S3
• Composition problem find constraints in same
  language as input mappings giving the composition
  of the input mappings
• Example

S1 R, S2 S,T, S3 U,V,W R ? S?T, S ?
?(U), T V W
) R ? ?(U)?(V - W)
10
Best-effort composition problem

• Composition not always possible
• Best-effort composition problem compute set of
  constraints equivalent to input constraints, but
  with as many symbols from S2 eliminated as
  possible
• R ? U, R ? V,
• ?1,4(?23(U?U)) ? U, ?1,4(?23(V?V)) ? V,
• U ? T, V ? T
• Can eliminate U (cross out left column) or V
  (right column), but not both NBM05

11
Composition algorithm overview

• For each relation R in S2
• Try to eliminate R via (1) view unfolding
• Replace by pairs of ?, ?
• For each relation R in S2 not yet eliminated
• Try to eliminate R via (2) left compose
• Else, try to eliminate R via (3) right compose
• Output
• New constraints and list of relations
  successfully eliminated

12
(1) View unfolding

• Idea exploit equality constraints (if we have
  any)
• Standard technique substitute view definition
  for occurrences of view relation in mappings
• T V W, R ? S ?T, T ? X ? ?(U)
• ? R ? S ?(V W), (V W) ? X ? ?(U)
• Body must not mention view relation itself
• Doesnt matter what else is in body
• Can substitute everywhere

13
(2) Left compose

• View unfolding for containment constraints
• ?(V) ? R U, R ? S ? T
• ? ?(V) ? (S ? T) U
• Needs monotonicity of expressions in R.
• E1 ? E2(R), R ? E3 E1 ? E2(E3)
• if E2(R) is monotone in R (and R not in E3)
• Partial check for monotonicity
• Is S (T R) monotone in R?

14
Normalization for left compose

• Need one constraint of form R ? E1
• Use identities to normalize, e.g.
• R ? E1 and R ? E2 iff R ? E1 ? E2
• E1 ? E2 ? E3 iff E1 ? E3 and E2 ? E3
• ?(E1) ? E2 iff E1 ? E2 ? Dr
• More identities in paper
• After left compose, try to eliminate D

15
(3) Right compose

• Dual to left compose, from NBM05
• Example
• S ?T ? R, R U ???(V)
• ? (S ?T) U ? ?(V)
• Monotonicity check needed here too
• Normalization may introduce Skolem functions
• E1 ? ?(E2) iff f(E1) ? E2
• Must eliminate Skolem functions after composition
• Lots of effort coding this step!

16
User-defined operators

• User specifies
• Monotonicity of operator in its arguments
• If E1 monotone in R and E2 antimonotone in R or
  independent of R, then E1 E2 monotone in R
• if E1 monotone in R or independent of R and E2
  antimonotone in R, then E1 E2 monotone in R
• Identities for normalization
• E1 E2 ? E3 iff E1 ? E2 ? E3
• User-defined operators and standard relational
  operators treated uniformly

17
Implementation

• 12K lines of C code, command-line tool
• Test case 13 PODS05 example 2
• SCHEMA
• R(2), S(2), T(2)
• CONSTRAINTS
• R lt S,
• P_0,2 J_0,11,2 (S S) lt R,
• S lt T
• ELIMINATE
• S
• Output
• P_0,2 J_0,11,2(R R) lt R,
• R lt T

18
Experimental evaluation

• First attempt at a composition benchmark
• Schema editing and schema reconciliation
  scenarios
• Add a column to R to produce S ?(R) S
• Measure
• of symbols eliminated
• Running time
• As a function of
• Editing primitives allowed, length of edit
  sequence, presence/absence of keys, starting
  schema size,
• Synthetic data

19
Summary of results

• Algorithm often effective in eliminating most or
  even all relation symbols from S2
• Running time in subsecond range even for large
  problems containing hundreds of constraints
• Certain schema editing primitives problematic
• Key constraints did not reduce effectiveness,
  although did increase running time (and output
  size)

20
Schema editing

• Random starting schema (30 relations of 2-10
  attributes)
• 100 random edits
• 100 different runs, sorted by execution time

21
Schema reconciliation (1)

• Random schema (30 relations of 2-10 attributes),
  random edits
• Point represents median time of reconciliation
  step of 500 runs

22
Schema reconciliation (2)

• Random schema (variable relations of 2-10
  attributes)
• 100 random edits
• 100 different runs, sorted by execution time

23
Related work

• MH03 J. Madhavan, A. Y. Halevy. Composing
  mappings among data sources. VLDB, 2003.
• FKPT04 R. Fagin, Ph. G. Kolaitis, L. Popa, W.C.
  Tan. Composing schema mappings second-order
  dependencies to the rescue. PODS, 2004.
• NBM05 A. Nash, P. A. Bernstein, S. Melnik.
  Composition of mappings given by embedded
  dependencies. PODS, 2005.

24
Conclusion and future work

• We motivated and described the mapping
  composition problem
• We presented an implementation of a practical new
  algorithm for the composition problem
• We also presented an experimental evaluation
• To do theoretical analysis of impact of
  user-defined operators
• To do output constraints from algorithm can be a
  mess! How to clean up?