Optimization of Sequence Queries in Database Systems

Optimization of Sequence Queries in Database Systems

Reza Sadri Carlo [email protected] [email protected]

[email protected]

Amir Zarkesh Jafar Adibi [email protected] [email protected]

Time series Analysis Many Applications:

Querying purchase patterns for marketing Stock market analysis Studying meteorological data

What’s needed: Expressive query language for finding

complex patterns in database sequences Efficient and scalable implementation:

Query Optimization

SQL-TS A query language for finding

complex patterns in sequences Minimal extension of SQL—only the

from clause affected A new Query optimization technique

based on extensions of the Knuth, Morris & Pratt (KMP) string-search algorithm

Text Search Optimization

Boyer and Moore Precomputed shift functions for each character and sub-

pattern Dependent to the alphabet size Works best for non-repeating patterns O(mn) worst case time

Knuth, Morris and Pratt (KMP) Independent of the alphabet size Most efficient in general: O(m+n) time

Karp and Rabin Prefix Hashing Dependent to the alphabet size O(mn) worst case time

Optimized string search:KMP

Consider text array text and pattern array p:

i 1 2 3 4 5 6 7 8 9 10 11 text[i] a b a b a b c a b c aj 1 2 3 4 5 6pattern[j] a b a b c a

After failing, use the information acquired so to: - backtrack to shift(j), rather than i+1, and - only check pattern values after next(j)But in SQL-TS we have general predicates & star patterns

shift and next Success for first j-1 elements of pattern. Failure

for jth element (when input is at i) Any shift less than shift(j) is guaranteed to lead

to failure, Match elements in the pattern starting at next(j)

Shifted Pattern

i – j + 1

1

1

i – j + shift(j) + 1 i - j + shift(j) + next(j)

shift(j) + 1 shift(j) + next(j)

i

j

next(j) j - shift(j)

Input

Pattern

shift(j)

Equality Predicates: KMP suffices

Find companies whose closing stock price inthree consecutive days was 10, 11, and 15.

SELECT X.name FROM quote CLUSTER BY name

SEQUENCE BY date AS (X, Y, Z)

WHERE X.price =10 AND Y.price=11

AND Z.price=15

But in SQL-TS we have general predicates

Optimal Pattern Search (OPS)

Search path for naive algorithm vs. optimized algorithm:

Beyond KMP: General PredicatesFor IBM stock prices, find all instances where the pattern of two successive

drops followed by two successive increases, and the drops take the price to a value between 40 and 50, and the first increase doesn't move the price beyond 52.

SELECT X.date AS start_date, X.price U.date AS end_date, U.price FROM quote CLUSTER BY name SEQUENCE BY date AS (X, Y, Z, T, U) WHERE X.name='IBM' AND Y.price < X.price AND Z.price < Y.price AND 40 < Z.price < 50 AND Z.price < T.price AND T.price < 52 AND T.price < U.price

*Z(less than 2% change)

*U(less than 2% change)

*W(less than 2% change)

*Y *R

*V*T

Beyond KMP: Star Patterns

Relaxed Double Bottom: Only considering increases and decreases

that are more than 2%

Relaxed Double Bottom: Ninety fold improvement

Relaxed Double Bottom in June 1990

Conclusion

Significant speedups—from 6 to 900 times faster

Queries, partial ordered domains, aggregates also treated in this approach

Many other optimization opportunities: e.g., parallel search for multiple patterns

shift and next Success for first j-1 elements of pattern. Failure

for jth element (when input is at i) Any shift less than shift(j) is guaranteed to lead

to failure, Match elements in the pattern starting at next(j)

Shifted Pattern

i – j + 1

1

1

i – j + shift(j) + 1 i - j + shift(j) + next(j)

shift(j) + 1 shift(j) + next(j)

i

j

next(j) j - shift(j)

Input

Pattern

shift(j)

General Predicates--Contp1(t) = (t.price < t.previous.price)

p2(t) = (t.price < t.previous.price) (40<t.price<50)

p3(t) = (t.price > t.previous.price) (t.price<52)

p4(t) = (t.price > t.previous.price)

And we need to find the implication between this pattern elements

Matrices and : Input tested on pj is now tested against pk

otherwiseU

ppif

Falsepandppif

kj

jkj

kj 0

1

otherwiseU

ppif

Truepandppif

kj

jkj

kj 0

1

Combing values of these lower triangular matrices ( j k),We derive the values of next(j) and shift (j)

pj succeeded:

pj failed:

Example

00

0

0

010

0

0

UUUU

UUUU

UUU

U

5,2,5021,20,10,95

:

654321

xxxxxx

pppppp

patternfollowingtheConsider

10100

10100

1010

1

10

1

U

UU

STAR PatternsSELECT X.NEXT.date, X.NEXT.price, S.previous.date, S.previous.priceFROM quote CLUSTER BY name, SEQUENCE BY date AS (*X, Y, *Z, *T, U, *V, S)WHERE X.name='IBM‘ AND X.price > X.previous.price AND 30 < Y.price AND Y.price < 40 AND Z.price < Z.previous.price AND T.price >

T.previous.price AND 35 < U.price AND U.price < 40 AND V.price < V.previous.price AND S.price < 30

Same input, Transitions on Original Pattern vs. Transitions on Pattern after the index set back j-k21

31 32

41 42 43

Example: Elements j and k are star predicates and jk is U: U j,k+1

j+1,k j+1,k+1

Handling Star Patterns

Possible Transitions Elements j and k are star predicates and jk is U: U j,k+1

j+1,k j+1,k+1

Elements j and k are star predicates and jk is 1:

1 j,k+1

j+1,k j+1,k+1

Elements j and k are not star predicates: j,k j,k+1

j+1,k j+1,k+1

Implication Graph

UUUU

UU

UUU

U

U

U

GP

00

010

1

01

0