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

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)

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

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 ...

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

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

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

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

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)

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]

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

(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

(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?”

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

(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!

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

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

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

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)

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

Schema reconciliation (1) Random schema (30 relations of 2-10 attributes), random edits Point represents median time of reconciliation step of 500 runs

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

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.

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?