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

2. Time series Analysis Many Applications: What’s needed:
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

3. 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

4. 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
This database people have overlooked great classical works in the searching text sequences is ignored
Kmp is not only appears to be more efficient but also doesn’t have dependency to the alphabet. It

5. 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
Fix this

6. 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
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)

7. 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

8. Optimal Pattern Search (OPS)
Search path for naive algorithm vs. optimized algorithm:

9. 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.
SELECT X.date AS start_date, X.price
U.date AS end_date, U.price
FROM quote
PARTITION BY name
ORDER 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

10. Beyond KMP: Star Patterns
Relaxed Double Bottom:
Only considering increases and decreases that are more than 2% *Z
(less than 2% change) *U *W *Y *R *V *T

11. Relaxed Double Bottom: Ninety fold improvement

12. Relaxed Double Bottom in June 1990

13. 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

14. 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
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)

15. General Predicates--Cont
p1(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

16. Matrices q and j: Input tested on pj is now tested against pk
pj succeeded:
pj failed:
Input that was tested against pj
Combing values of these lower triangular matrices ( j ³ k),
We derive the values of next(j) and shift (j)

17. Example

18. 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 < 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

19. Handling Star Patterns
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

20. Possible Transitions U  j,k+1 1 j,k+1
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:
j,k+1
j+1,k j+1,k+1
Elements j and k are not star predicates:
j,k j,k+1

21. Implication Graph