Generating Efficient Plans for Queries Using Views

Chen LiStanford University

with Foto Afrati (National Technical University of Athens)

and Jeff Ullman (Stanford University)

SIGMOD, Santa Barbara, CA, May 23, 2001

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Answering queries using views

How to answer a query using only the results of views? [LMSS95]

Many applications:

– Data warehouses

– Data integration

– Query optimization

– …

Base relations

ViewsV1 V2 … Vn

QueryQ

R1 Rm…

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An example

View: V1(M, D, C) :- car(M, D), loc(D, C)

Query Q: Q(M, C) :- car(M, anderson), loc(anderson, C)

Rewriting P1: Q(M, C) :- V1(M, anderson, C)

car

BMW AlisonHonda Anderson

Make Dealer

… …Ford Varsity

loc

Anderson Palo altoVarsity Redwood City

Dealer City

Alison Mountain View… …

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Existing algorithms

Bucket algorithm [LRO96], Inverse-rule algorithm [DG97], MiniCon algorithm [PL00], …

However, instead of generating

P1: Q(M, C) :- V1(M, anderson, C)

they generate rewriting

P2: Q(M, C) :- V1(M, anderson, C1), V1(M1, anderson, C)

Why P2, not P1?– These algorithms take the Open-World Assumption (OWA): “P2 P1.”

– However, under the Closed-World Assumption (CWA): “P1 = P2.”

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Differences between OWA and CWA

W1(Make, Dealer) :- car(Make, Dealer)

W2(Make, Dealer) :- car(Make, Dealer)

All car tuplesW1 = W2 =

CWA

– W1 and W2 have all car tuples.

– E.g.: W1 and W2 are computed from the same car table in a database.

– W1 and W2 have some car tuples.

– E.g.: W1 and W2 are from two different web sites.

OWA

W1 W2

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Our problem: generating efficient plans using views under CWA

Base relations

Materialized views V1 V2 … Vn

Query Q

Existing algorithms work under both assumptions.

Our study– takes the CWA assumption.

– considers efficiency of rewritings.

Efficient plans?

R1 R2 Rm

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Challenge: in what space should we generate rewritings?

Rewritings: P1: Q(S, C) :- V1(M, a, C), V2(S, M, C)

P2: Q(S, C) :- V3(S), V1(M, a, C), V2(S, M, C)

P2 could be more efficient than P1!

car(Make, Dealer)

loc(Dealer, City)

part(Store, Make, City)

Q(S, C) :- car(M, a), loc(a, C), part(S, M, C)

V1(M, D, C) :- car(M, D), loc(D, C)

V2(S, M, C) :- part(S, M, C)

V3(S) :- car(M, a), loc(a, C), part(S, M, C)

a = ‘anderson’

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FocusViews

V1,V2,…,Vn

Query QStep 1: generate a rewriting P (logical plan)

Step 2: generate an efficient physical plan from P

We focus on the logical level (step 1).– Prune rewriting space to generate “good” rewritings.

– Different from the one-step approach: [CKPS95, ZCLPU00].

Both steps are cost-based.

Consider select-project-join queries, i.e., conjunctive queries.

Cost model CM

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Three cost models:

– CM1: number of subgoals in a physical plan

– CM2: sizes of views and intermediate relations

– CM3: CM2 + dropping attributes in intermediate relations

Experimental results

Conclusion and future directions

Rest of the talk

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CM1: number of subgoals in a physical plan

– Goal: generate rewritings with minimum number of subgoals

Motivations:

– Reduce the number of joins

– Reduce the number of view accesses

Example:

– P1: Q(S, C) :- V1(M, a, C), V2(S, M, C) more efficient

– P2: Q(S, C) :- V1(M1, a, C), V1(M, a, C1), V2(S, M, C)

A view can appear more than once in different “forms.”

Cost model CM1

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Results under CM1

Analyze the rewriting space:

– Find an interesting structure of the space;

– Show a procedure to reduce number of subgoals in a rewriting.

Develop an algorithm CoreCover:

– Input: a query Q, views V1, …, Vn

– Output: rewritings with minimum number of subgoals

Optimality: if there is a rewriting, then CoreCover guarantees to find a rewriting with minimum number of subgoals.

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CoreCover: example

Query: Q(S, C) :- car(M, a), loc(a, C), part(S, M, C)

Evaluate views on D:

V1(M, D, C) :- car(M, D), loc(D, C) V1(m0, a, c0)

V2(S, M, C) :- part(S, M, C) V2(s0, m0, c0)

V3(S) :- car(M, a), loc(a, C), part(S, M, C) V3(s0)

Construct database D = { car(m0, a), loc(a, c0), part(s0, m0, c0) }

D

View tuples: V1(M, a, C), V2(S, M, C), V3(S)

Intuition: translate the problem to a set-covering problem.

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CoreCover: example (cont.)

Find minimal covers of query subgoals using view tuples

Q(S, C) :- V1(M, a, C) , V2(S, M, C)

Find query subgoals “covered” by each view tuple:

V1(M, a, C) car(M, a) V2(S, M, C) loc(a, C) V3(S) part(S, M, C)

View tuples: V1(M, a, C), V2(S, M, C), V3(S)

V1(M, D, C):- car(M, D), loc(D, C)

V2(S, M, C) :- part(S, M, C)

V3(S) :- car(M, a), loc(a, C), part(S, M, C)

Query: Q(S, C) :- car(M, a), loc(a, C), part(S, M, C)

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Algorithm: CoreCover

Q

Construct database D from QD

Find minimal covers of query subgoals using view tuples.rewritings

Evaluate views on D

“View tuples”

Viewtuples

T1

T2

…

Tk

Find query subgoals “answered” by each view tuple.

Query subgoals

G1

G2

G3

…

Gm

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Cost model CM2: considering sizes of views and intermediate

relationsMotivation: cost of V1 V2 is related to size(V1) and size(V2).

Cost = size(V1) + size(V2) + … + size(Vn)

+ size(IR1) + size(IR2) + … + size(IRn)

Physical plan:

Q( ) :- V1, V2, V3, …, Vn

IR1 IR2 IRn “IR”: intermediate relation

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Results under CM2

Observation: Adding more views may make a rewriting more efficient.

P1: Q(S, C) :- V1(M, a, C), V2(S, M, C)

P2: Q(S, C) :- V3(S), V1(M, a, C), V2(S, M, C)

If V3(S) is very selective, P2 can be more efficient than P1.

Larger search space: rewritings using view tuples produce an optimal physical plan under CM2.

– Modify CoreCover to find these rewritings.

– We discuss how to condense rewritings.

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Cost model CM3: dropping nonrelevant attributes

CM2: assumes all attributes are kept in IRs.

CM3: assumes attributes can be dropped in IRs to reduce sizes.

Bad news: didn’t find a space that guarantees to produce an optimal physical plan.

Good news: found a heuristic for optimizer to drop more attributes.

IRi

Y

Q( ) :- … Vi Vi+1 …

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Drop what attributes?

Drop Y if: (1) Y is not used in later joins, and

(2) Y is not in the answers.

Called the “supplementary-relation approach.” [BR87]

IRi

Y

Q( ) :- … Vi Vi+1 …

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Search space under CM3?

Q(A) :- r(A, A), t(A, B), s(B, B)

V1(A, B) :- r(A, A), s(B, B)

V2(A, B) :- t(A, B), s(B, B)

r(A,B)

s(C,D)

t(E,F)

Rewritings using view tuples may not produce optimal physical plans!

Rewriting using view tuples: P1: Q(A) :- V1(A, B), V2(A, B)

A more efficient rewriting: P2: Q(A) :- V1(A, C), V2(A, B)

Note: P1 and P2 both compute the answers to Q.

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Targeting rewritings to facilitate dropping of attributes

Goal: after the transformation, we may drop more attributes.

Main idea: given a sequence of subgoals, rename variables.

If Y Y’, the new rewriting is still equivalent to Q, then drop Y’ in IRi even if Y appears in later joins.

IRi

YY’

Q( ) :- … Vi Vi+1 …

P1: Q(A) :- V1(A, B), V2(A, B)

P2: Q(A) :- V1(A, C), V2(A, B)

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Experimental study

Purpose: – Test how fast CoreCover generates rewritings (cost model CM1).

– Analyze its efficiency and scalability.

Experiment setup:

– A query generator (in Java). Input parameters:• Number of base relations

• Number of attributes in a relation

• Number of views (1-1000), queries (5)

• Number of subgoals in a view and a query

• Shape of queries and views (star, chain, …)

– Implemented in Java on a dual-processor Sun Ultra 2 workstation, running SUNOS 5.6, 256MB memory

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Star queries and views Each query has 8 subgoals, and each view has 1, 2, or 3 subgoals.

No attribute projection in the head of the queries/views.

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Chain queries and views Each query has 8 subgoals, and each view has 1, 2, or 3 subgoals.

1 variable is projected in the head of the queries/views.

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Conclusion

Generating efficient plans using views under CWA:– Cost model CM1: number of subgoals in a plan

• Analysis of the rewriting space• A search space for rewritings

• CoreCover: finding rewritings with minimum number of subgoals

– Cost model CM2: sizes of views and IRs• A search space for rewritings• Condense rewritings

– Cost model CM3: dropping irrelevant attributes in IRs• A heuristic to help optimizer drop attributes

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Future work

More complicated queries and views:– Arithmetic comparisons ( <=, >=, …)

– Aggregations

Different assumptions:– Open-world assumption

– Maximally-contained rewritings

Constraints: – Functional dependencies

– Foreign-key constraints

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Thank you!

Questions?

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Differences between CoreCover and MiniCon

CoreCover takes CWA, and MiniCon takes the OWA.

MiniCon tries to minimize the number of query subgoals, but it has no guarantee.

Technical differences:

– CoreCover is more “aggressive” than MiniCon about finding query subgoals answered by a view tuple.

– Finding set covers of query subgoals: CoreCover allows overlapping, and MiniCon does not allow it.

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Difference from earlier studiesViews

V1,V2,…,Vn

Query QStep 1: generate a rewriting P (logical plan)

Step 2: generate an efficient physical plan from P

One-step approach: [CKPS95, ZCLPU00].

We focus on the logical level (step 1).– Prune rewriting space to generate “good” rewritings.

– Cost-based.

Cost model CM

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Rewriting space

All rewritingsMinimalrewritings

Locallyminimalrewritings

Containmentminimalrewritings

Globallyminimalrewritings

P

P’

Rewriting P P’:

Remove its redundant subgoals [Chandra & Merlin 77]:

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P’ P’’: Remove its subgoals while retaining its equivalence to Q:

P3: Q(S, C) :- V3(S), V1(M, a, C), V2(S, M, C)

V3(S) can still be removed.

Rewriting space (cont.)

All rewritingsMinimalrewritings

Locallyminimalrewritings

Containmentminimalrewritings

Globallyminimalrewritings

PP’

P’’

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Rewriting space (cont.)

All rewritingsMinimalrewritings

Locallyminimalrewritings

Containmentminimalrewritings

Globallyminimalrewritings

PP’

P’’ P*: transform P’’ using the mapping from the expansion of P’’ to the query:

P1: Q(S,C) :- v1(M1,a,C),v1(M,a,C1),v2(S,M,C)

P2: Q(S,C) :- v1(M,a,C), v2(S,M,C)

P’’

P*

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Concise representation of rewritings

Problem: as the number of views increases, the number of rewritings could be large!

Solution:

– Group views into equivalence classes

– Group view tuples into equivalence classes based on their covered query subgoals.

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Advantages

Advantages:

– Number of equivalence classes bounded by the number of query subgoals.

– The optimizer finds efficient physical plans by considering the “representative rewritings,” then decides how to make them more efficient by adding more view tuples.

– The optimizer can replace a view tuple in a rewriting by another view tuple in the same equivalence class to have another rewriting.

Equivalence classesViews

V1

V2

…

Vn

{V1, V3}

{V4,V10,V15}

{V2, V9}

…

Equivalence classesView tuples

T1

T2

…

Tn

{T2, T5}

{T1,T6,T9}

{T3}

…

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Main results of experiments

CoreCover has good efficiency and scalability.

By grouping views and view tuples into equivalence classes, we can reduce the number of views and view tuples used by CoreCover.

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Star queries and views:Number of equivalence classes

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Star queries and views: Number of view tuples