Constructing Search Space for Materialized View Selection

DOLAP'04 - Washington DC 1

Constructing Search Space for Materialized View Selection

Dimiti Theodoratos Wugang Xu

New Jersey Institute of Technology


Problem (1)

• Many problems in Databases require the selection of views to materialize.

• A general form of these problems is the following: – Given a set of queries, select a set of views to

materialize such that a cost function is optimized and a number of constraints is satisfied.


Problem (2)

• Examples of view selection problems in DWing.

– Given a set of queries to be satisfied by the DW, select a set of views to materialize such that the combination of the query evaluation and view maintenance cost is minimized and the size of the materialized views does not exceed the space allocated for materialization.

– Find the best global evaluation plan for multiple incremental maintenance expressions for materialized views.


Problem (3)

• Solving view selection problems requires the identification of common sub-expressions between queries.

• Usually, this is done by identifying equivalent (or subsumed) view nodes in query evaluation plans of two queries in a bottom-up way.

• However, for this approach to be successful, all the alternative query evaluation plans of the queries need to be considered – an unfeasible task.


Example - Query Evaluation Plans and common subexpressions

R S T U

A=A

A=B

B=B

R S T U

A=A

A=B

B=B

Q1Q2



R S T U

A=A

A=B

B=B

Q1

R S U T

A=A

A=B

B=B

Q2

V V



R S T U

A=A

A=B

B=B

Q1

R S U T

A=A

A=B

B=B

Q2

V

V



R S T U

A=A

A=B

B=B

Q1

R S T U

A=A

A=B

B=B

Q2

W

W



R S T U

A=A

A=B

B=B

Q1

R S T U

A=A

A=B

B=B

Q2


Our approach

Q1 Q2

V (CCD)

Optimal evaluationPlan of Q1

Optimal evaluationPlan of Q2

Evaluation plans of V

(search space of viewsTo materialize)

Rewriting of Q1 using V

Rewriting of Q2 using V


Goals

• Formalize the concept of ‘closeness’ of a common subexpression to two queries.

• Design algorithms for computing common sub-expressions that are as close to the queries as possible (these common subexpressions are called Closest Common Derivators).

• We address these problems starting with SPJ queries that involve self-joins.


Example

Q1Select R1.A, R2.B, R3.C From U, R as R1, R as R2, R as R3, S as S1Where U.A=R1.A and R1.B<=R2.B and R2.C<=R3.B

and R3.C=S.C and R2.B<3 and R3.A>=4 and R3.A<=7 and S1.D>=3

Q2Select R4.C, R5.A, S3.CFrom S as s2, R as R4, R as R5, S as S3, TWhere S2.C<=R4.C and R4.C=R5.B and R5.C<=S3.C

and S3.D=T.D and R4.B=3 and R5.A>=5 and R5.A<=9 and S3.D>=3


Query Graph Representation

U

R1[R]:A R2[R]:B R3[R]:C

S1[S]

S2[S]

R4[R]:C R5[R]:A S3[S]:C

T

U.A=R1.AR1.B≤R2.B R2.C≤R3.B

R3.C=S1.C

R2.B<3 R3.A≥4^R3.A≤7

S1.D≥3

R4.B=3 R5.A≥5^R5.A≤9 S3.D≥3

S2.C≤R4.C

R4.C=R5.B R5.C≤S3.CS3.D=T.D

Q1

Q2


Query rewritings• A rewriting Q’ of a query Q using view V is a query that

references V and possibly base relations such that replacing V by its definition results in a query equivalent to Q. Notation: Q |-- V. If there is a rewriting of Q that references only V (no base relations), we call it complete rewriting. Notation: Q ||-- V. Otherwise, we call it a partial rewriting.

• A rewriting Q’ of query Q using a view V is called simple rewriting if view V has a single occurrence in Q’.

• A rewriting Q’ of a query Q using a view V is minimal if for every relation R that has n, n>0, occurrences in Q, R has k, 0 ≤ k≤ n, occurrences in V and n- k occurrences in Q’. Notation: Q |--m V.


Common Derivator (CD) of two queries

• Let Q1 and Q2 be two queries and R1, R2 be two sets of relation occurrences from Q1 and Q2, respectively, that have the same number of relation occurrences of each relation. A common derivator (CD) of Q1 and Q2 over the respective sets R1 and R2 is a view V such that there is a minimal rewriting of Q1 (resp. Q2) using V that involves V and only those relation occurrences of Q1 (resp. Q2) that do not appear in R1 (resp. R2.)


Example - Common Derivator

U

R1[R]:A R2[R]:BR3[R]:C

S1[S]

S2[S]

R4[R]:C R5[R]:A S3[S]:C

T


R3.C=S1.C

R2.B<3 R3.A≥4^R3.A≤7

S1.D≥3

R4.B=3 R5.A≥5^R5.A≤9 S3.D≥3

S2.C≤R4.C


Q1

Q2

R6[R] R7[R]

R6.C≤R7.B

f1

f2

f1

f2

V1

R1={R2, R3}R2={R4, R5}

CD V1 over R1 and R2


Example - Common Derivator

U

R1[R]:A R2[R]:B R3[R]:C

S1[S]

S2[S]

R4[R]:C R5[R]:A S3[S]:C

T


R3.C=S1.C

R2.B<3 R3.A≥4^R3.A≤7

S1.D≥3

R4.B=3R5.A≥5^R5.A≤9 S3.D≥3

S2.C≤R4.C


Q1

Q2

R6[R]

R7[R]R6.C≤

R7.Bf1

f2

f1

f2

R7.A≥3^R7.A≤9

R7.C≤S4.CR1 = {R2, R3, S1}R2 = {R4, R5, S3}

f1

f2

S4[S]

CD V2 over R1 and R2


Closeness relationship between CDs

• Let Q1, Q2 be two queries, V= X(C(R)) is a CD of Q1 and Q2 over R1 and R2, V’= X(C’(R’)) be a CD of Q1 and Q2 over R1’ and R2’, and R1 R1’ and R2 R2’. CD V’ is closer to Q1 and Q2 than CD V if the following conditions are satisfied(a) V’ |-- V

(b) if C’(R’) ||-- C(R) then V ||─V’


Example – Closeness relationship

R6[R] R7[R]R6.C≤R7.B

f1

f2

f1

f2

R2

R4

R3

R5

R6[R]

R7[R]R6.C≤

R7.Bf1

f2

f1

f2

R2

R4

R3

R5

R7.A≥3^R7.A≤9

R7.C≤S4.C

V1 V2

R1={R2, R3}R2={R4, R5}

R1={R2, R3, S1}R2={R4, R5, S3}

f1

f2

S1

S3

S4[S]

• V2 is closer to Q1 and Q2 than V1



R6[R]

R7[R]R6.C≤

R7.Bf1

f2

f1

f2

R2

R4

R3

R5

R7.A≥3^R7.A≤9

R7.C≤S4.C

V2

R1={R2, R3, S1}R2={R4, R5, S3}

f1

f2

S1

S3

S4[S]

R6[R]

R7[R]R6.C≤

R7.Bf1

f2

f1

f2

R2

R4

R3

R5

R7.A≥3^R7.A≤9

R7.C≤S4.C

V3

R1={R2, R3, S1}R2={R4, R5, S3}

f1

f2

S1

S3

S4[S]R

6.B≤3

S4.D≥3




R6[R]

R7[R]R6.C≤R7.Bf1

f2

f1

f2

R2

R4

R3

R5

R7.A≥3^R7.A≤9

R7.C≤S4.C

V3

R1={R2, R3, S1}R2={R4, R5, S3}

f1

f2

S1

S3

S4[S]R

6.B≤3

S4.D≥3

R6[R]:A,B,C

R7[R]:A,B,C

R6.C≤R7.Bf1

f2

f1

f2

R2

R4

R3

R5

R7.A≥3^R7.A≤9

R7.C≤S4.C

V4

R1={R2, R3, S1}R2={R4, R5, S3}

f1

f2

S1

S3

S4[S]:D

R6.

B≤3

S4.D≥3



Closest Common Derivator (CCD)

• Let Q1 and Q2 be two queries. A Closest Common Derivator (CCD) of Q1 and Q2 over R1 and R2 is a CD V of Q1 and Q2 over R1 and R2 such that there exists no CD of Q1 and Q2 that is closer to Q1 and Q2 than V.


Example

R6[R]:A,B,C

R7[R]:A,B,C

R6.C≤R7.Bf1

f2

f1

f2

R2

R4

R3

R5

R7.A≥3^R7.A≤9

R7.C≤S4.CCCD1

R1={R2, R3, S1}R2={R4, R5, S3}

f1

f2

S1

S3

S4[S]:D

R6.

B≤3

S4.D

≥3

S5[S]:C,D R8[R]:A,B,CR6.C≤R7.B

f1

f2

f1

f2

S1

S2

R1

R4R1={R2, R3}R2={R4, R5}

CCD2


How to compute a CCD

• Query graph in Full Form

• Condition merging

• Candidate CCDs

• Comparison of Candidate CCDs over the same occurrence set


Full Form Condition and Query

• A condition C is in full form if:

1. For every atomic condition Ai such that C |= Ai, there is an atomic condition Aj in C such that Aj |= Ai (|= denotes logical implication)

2. Condition C does not include strongly redundant atomic conditions.

• A query X(C(R) is in full form if its condition C is in full from.


Example—Query graph Full Form

R S TR.A=S.A S.B=T.B

S T US.B=T.B T.C=U.C

B>

5

B>

7

S TS.B=T.B

Q1

Q2

‘CCD’


Example—Query graph full formR S T

R.A=S.A S.B=T.B

S T US.B=T.B T.C=U.C

B>

5

B>

7

S TS.B=T.B

Q1

Q2

CCD

B>5

B>7

B>

5


Condition Merging

• Two conditions C1 and C2 are mergeable if there is a non-valid condition C such that C1|=C and C2|=C and there exists no condition C', C'≡C, such that C1|=C', C2 |= C' and C’|= C. Condition C is called a merge of C1 and C2.

• We show how the merge of two conditions can be computed.


CCD Computation

• We introduce the concept of a candidate CCD: a graph representation of a CCD resulting by ‘merging’ common subparts of two query graphs.

• We show that a CCD of two queries is a candidate CCD.

• We express the CCD closeness relationship on candidate CCDs.


CCD Computation (2)

In order to compute all the CCDs of two queries:

• We compute all the candidate CCDs of two query graphs in full form.

• We discard a candidate CCD V if there is another CCD V’ that is closer to the queries than V.


Future work

• Extend the concept of a CCD so that it applies to a more general class of queries.

• Use the concept of a CCD to identify common sub-expressions within one query

• Use the concept of a CCD to design algorithms for different materialized view selection problems.


Thanks

Documents

Constructing Search Space for Materialized View Selection