Optimization of Sequence Queries in Database Systems Reza Sadri Carlo Zaniolo Amir Zarkesh Jafar Adibi

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 text[i] a b a b a b c a b c a j pattern[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 i – j + shift(j) + 1i - j + shift(j) + next(j) shift(j) + 1shift(j) + next(j) i j next(j)j - shift(j) Input Pattern shift(j)

Equality Predicates: KMP suffices Find companies whose closing stock price in three 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 Predicates For 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. S ELECT 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 i – j + shift(j) + 1i - j + shift(j) + next(j) shift(j) + 1shift(j) + next(j) i j next(j)j - shift(j) Input Pattern shift(j)

General Predicates--Cont p 1 (t) = (t.price < t.previous.price) p 2 (t) = (t.price < t.previous.price)  (40<t.price<50) p 3 (t) = (t.price > t.previous.price)  (t.price<52) p 4 (t) = (t.price > t.previous.price) And we need to find the implication between this pattern elements

Matrices  and  : Input tested on p j is now tested against p k Combing values of these lower triangular matrices ( j  k), We derive the values of next(j) and shift (j) p j succeeded: p j failed:

Example

STAR Patterns SELECT X.NEXT.date, X.NEXT.price, S.previous.date, S.previous.price FROM 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 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-k  21   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