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 IIS and IIS

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
σCountry = KR(Addresses)  SID,Address(Local)£{KR}
σCountry  KR(Addresses)  Foreign
Schema evolution
Names
SID,
Name
Local
Address
Foreign
Address,
Country
Students  Name,Address,Country(Names ⋈ Addresses)
Names
SID,
Name
Addresses
Address,
Country
Students
Name,
Address,
Country
...
m12
m23 S1 S2 S3

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 ⋈1=3 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 ⋈1=3 S) µ 2,2(S)
Data integration, data exchange – GLAV
R ⋈ S µ T ⋈ U

6. Mapping composition S1 S2 S3 Names  Names
σCountry = KR(Addresses)  SID,Address(Local)£{KR}
σCountry  KR(Addresses)  Foreign
m23
m12  m23
Students  Name,Address, Country (Names ⋈
(SID,Address(Local)£{KR} [ Foreign))
Names
SID,
Name
Local
Address
Foreign
Address,
Country
m12
Students  Name,Address,Country(Names ⋈ Addresses)
Names
SID,
Name
Addresses
Address,
Country S2
Students
Name,
Address,
Country 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:
R(∙,∙,∙)
S(∙,∙)
T(∙,∙)
U(∙,∙,∙)
V(∙,∙) S1 S2 S3
m12
m23
R ⊆ S⋈T
S ⊆ (U),
T = V – W
W(∙,∙)
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(2=3(UU)) ⊆ U, 1,4(2=3(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  R ⊆ S ⋈(V – W), (V – W)  X ⊆ (U)
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
Doesn’t matter what else is in body
Can substitute everywhere

13. (2) Left compose (V) ⊆ R – U, R ⊆ S ⋈ T  (V) ⊆ (S ⋈ T) – U
“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 Output: 12K lines of C# code, command-line tool
# Test case 13: PODS05 example 2
SCHEMA
R(2), S(2), T(2)
CONSTRAINTS
R <= S,
P_{0,2} J_{0,1:1,2} (S S) <= R,
S <= T
ELIMINATE S;
Output:
P_{0,2} J_{0,1:1,2}(R R) <= R,
R <= 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?