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Boolean + Ranking: Querying a Database by K-Constrained Optimization

Boolean + Ranking: Querying a Database by K-Constrained Optimization. Zhen Zhang Seung-won Hwang Kevin C. Chang Min Wang Christian A. Lang Yuan-chi Chang Presented ACM SIGMOD Conference (SIGMOD 2006), Chicago, June 2006. Presented By : Pavan Kumar M.K. (1000618890)

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Boolean + Ranking: Querying a Database by K-Constrained Optimization

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  1. Boolean + Ranking: Querying a Database by K-Constrained Optimization Zhen Zhang Seung-won Hwang Kevin C. Chang Min Wang Christian A. Lang Yuan-chi Chang Presented ACM SIGMOD Conference (SIGMOD 2006), Chicago, June 2006 Presented By : Pavan Kumar M.K. (1000618890) Aditya Mangipudi (1000649172)

  2. Outline • Introduction • Motivation • A* Search Algorithm • A*-Driven State Space Construction • Optimization Driven Configuration • OPT* Search Algorithm • Experiments • Conclusion

  3. Motivation • The wide spread of databases for managing structured data, compounded with the expanded reach of the Internet, has brought forward interesting data retrieval and analysis scenarios to RDBMS • Only the Top-K results are of interest to the user.

  4. K-Constrained Optimization Query QUERY: Select the Top-5 2nd year students in CSE with highest GPA Boolean query: dept = CSE and year = 2 Qualifying constraint Find top answers + B: dept = CSE and year = 2 O:GPA Ranking query: Top 5 ranked by GPA Quantifying function

  5. K-Constrained Optimization Query • Query Q = (G, k) • G - Goal Function G = B . O • k – Retrieval Size

  6. What is the query evaluation mechanism? Boolean query Ranking query + How to answer?

  7. … … … D D B R Goal function G Current techniques lack of global search mechanism • If evaluated as separate operators • If search by an overall goal function G as a ranking function Boolean query B Ranking query R Boolean query B Ranking query R • Current techniques optimize only condition-by-condition

  8. Threshold Algorithm

  9. Assumptions • Threshold Algorithm essentially relies on a rigid assumption that G functions are Monotonic. • The monotonicity requires G to be decreasing if all its parameters are decreasing.

  10. Non-Monotonic Functions • Consider the example query as below to find houses in a certain price range with good price/sqrft ratio • The function G here in Non-Monotonic. Selecth.addressfrom House h, Whereh.price ≤ 200k ν h.price ≥ 400k Order byh.size/|h.price-300k|

  11. New Algorithm

  12. Need for encoding as a search problem • Existing algorithms build upon their problem-specific assumptions on the goal functions or index traversals. • For example, Threshold Algorithm assumes the monotonicity of G and the use of sorted accesses (interleaf navigation), based on which the search is implicitly hardwired. • In a Boolean Query like B = price > 100K, such a search is straightforward as the constraint expressions B explicitly suggests how to carry out a focused search, eg., visiting only the nodes with locality potentially satisfying B.

  13. Need for encoding as a search problem • In contrast, for a general k-constrained optimization query potentially involving arbitrary ranking combined with Boolean conditions and joining multiple relations, eg.. Q maximizing size/price ratio, it is no longer clear how to focus the search. • By encoding into a generic search with no assumptions on G, the search is generalized to support arbitrary G over potentially multiple indices and a combination of both hierarchical and interleaf traversals.

  14. A* Algorithm • A* is a well known search algorithm that finds the Shortest Path, given an initial and a designated goal state. • Widely used in the field of Artificial Intelligence. • Uses Best-First Search Traversal. • Uses heuristic information to carry out the search in a guided manner. • A* is guaranteed to find the correct answer (Correctness) by visiting the least number of states (Optimality) • Ex: GPS, Google Maps, A lot of puzzles, games etc.

  15. Goal Function For a tuple t with m attribute values, Goal Function G(t) maps the tuple to a positive numeric score. R(t) if B(t) is true 0if B(t) isfalse G(t) = B(t)*R(t) = (ie, lowest score)

  16. Query Model Selecth.addressfrom House h, Whereh.price ≤ 200k ν h.price ≥ 400k Order byh.size/|h.price-300k|

  17. Landscape of Score Function - G

  18. OPT* Framework • To realize k-constrained optimization over databases, this paper develops the OPT* framework. • Objective: To Optimize G with the help of indices as access methods over tuples in D. • Discrete State Search: From the view of using indices, we are to search the maximizing tuples on the index nodes as “discrete states”. • Continuous Function Optimization: From the view of maximizing goal functions, we are to optimize G.

  19. G OPT* D D Evaluate query as its nature suggests! Function optimization of G Optimize G over D Discrete state search over D

  20. B+ Tree Structure Indices Value Space

  21. Some definitions first.. • States : States in a search graph represent “localities” of values at different granularity– from coarse to fine, and eventually reach tuples in the database. • Region State • Tuple State • Transitions : While states of space give “locations” in the map, transitions further capture possible paths followed to reach our destination of query answers. Example : for two states u and v, there is a transition (u, v) if v ∈ Next(u)

  22. b1 0-250 250-600 b2 b3 0-100 100-250 250-350 350-600 b6 b7 ……… 2 5 1 a1 0-3000 3000-4500 a2 a3 0-1500 1500-3000 3000-4000 4000-6000 a6 a7 ……… 5 1 We view compound index as discrete space Price (k) 600 1 350 2 5 250 4 3 100 6 size 1500 3000 4000 4500

  23. We view compound index as discrete space Price (k) Mij = (ai, bj) b1 0-250 250-600 600 b2 b3 M11 1 350 0-100 100-250 250-350 350-600 b6 2 M32 M23 M33 b7 5 M22 ……… 250 2 5 1 4 3 … 100 … M76 M55 M75 M56 M66 M77 6 size 1500 3000 4000 4500 4 2 5 1 a1 M67 0-3000 3000-4500 a2 a3 0-1500 1500-3000 3000-4000 4000-6000 a6 a7 ……… 5 1

  24. M22 M32 M23 M33 We view compound index as discrete space conceptually, combined space Price (k) Mij =(ai, bj) b1 0-250 250-600 600 b2 b3 M11 1 350 0-100 100-250 250-350 350-600 b6 2 b7 5 ……… 250 2 5 1 4 3 100 … M55 M75 M56 M66 M77 M67 M76 6 size 1500 3000 4000 4500 4 2 5 1 a1 0-3000 3000-4500 a2 a3 0-1500 1500-3000 3000-4000 4000-6000 a6 a7 ……… 5 1

  25. Challenge 1: What is the search mechanism?

  26. Encoding the problem into shortest path is challenging • > A* Gives Shortest Path to testable goal. • > The goal is to find optimal tuple states with maximal G-Score.

  27. Transformation needed…. • How to encode a tuple to a path? • Adding a virtual target t* only reachable through tuples • How to encode maximal tuple with minimal path? • Quality of path depends solely on the tuple it passes by • For tuple state t D(t, t*) = - G(t) • For two states r, u D(r, u) = 0 M11 0 0 M22 M32 M23 M33 0 0 … M66 M67 M76 M75 M56 M77 M55 0 0 4 2 5 1 - G(1) - G(4) t*

  28. Challenge 2: How to guide the search?

  29. Functional Optimization perspective… • Function optimization measures quality of states • Function optimization aspects: • Defines Proper Heuristics • Identifies a set of initial states to start search.

  30. Structure of Procedure OPT • Input : G(x1,……,xm) and domain of values dom = xiε [xi1,xi2] • Output : <O,U> = OPT(G,dom) where O={gives local optima} U={Upper Bound Score} OPTPOINT gives O Component of OPT OPTMAX gives U Component of OPT • Approaches • Analytical Method • Seach based (Ex:Hill • Climbing) • Template Based

  31. States and Transitions High Medium Low Figure illustrates different states have different promises. Search should favor the choice of M77 over M67 because its more promising.

  32. 1. Define admissible heuristics: Measure tightest upper bound • To guarantee completeness • A* requires admissible heuristics, i.e., estimate optimistically • To ensure admissible heuristics • Function optimization gives tightest upper bound • Analytical approaches • Numeric analysis package H(region) = OPTMAX(G, region) i.e., maximal value of G in the region

  33. Consider Example… • h(M67) gives U=0 • However if we follow the link from M67 to M77, we can reach Tuple 1 with score 15. 600 M67 M77 1 350 2 5 250 4 3 100 6 1500 3000 4000 4500

  34. M22 M32 M23 M33 2. Configure descending space: disconnect uphills • To guarantee optimality • A* requires descending heuristics • To ensure descending heuristics • Remove uphill links M11 … M55 M75 M56 M66 M77 M67 M76 4 2 5 1

  35. M22 M32 M23 M33 Find right start point: Start from local optima • To guarantee correctness • Every tuple state must be reachable from start states • Taking only downhills requires start with high points • To ensure reachability • Initial states should contain all local optima M11 … M55 M75 M56 M66 M77 M67 M76 4 2 1 5

  36. Putting together: Executing A* on the configured space top-down M11 M22 M32 M23 M33 … M67 M76 M57 M55 M75 M56 M66 M77 4 2 5 1 • Search is implemented as priority queue driven traversal

  37. Need of States and Transitions • Example . Given a set of states constructed from the set of index graph I, the search, in principle, should follow those transitions to look for the tuple states maximizing the goal function.. The search may follow the path • M11 → M33 → M77 → 1  Top-down search • M57 → M77 → 1  Bottom-Up Search

  38. M22 M32 M23 M33 OPT* Search Algorithm M11 M55 M75 M56 M66 M77 M67 M76 4 2 1 5

  39. Optimality of OPT* • OPT* may result in different costs if started at different initial states. • Top down-> More hops | Bottom up->Less hops • Preference goes to Bottom Up but what if Goal functions G=1/(X-Y)2+1, any value satisfying X=Y maximizes the function.

  40. Experiments • Comparison vs. • Boolean then ranking • Ranking then boolean • Metrics: node accessed = Nl + Nt • Settings: • Benchmark queries over real dataset • Controlled queries over synthetic dataset

  41. BR_clustered BR_unclustered OPT* Benchmark queries • Datasets: • 19,706 real estate listing crawled online • Queries • Q1: size * bedrms/| price-450k| : [40k<=price<=50k] • Q2: size * ebedrms / |price-350k| : [price<400k^size>4000] • Q3: size/price : [bedrms=3 ν bedrms=4] Q1 Q2 Q3

  42. Controlled queries • Datasets • Three randomly generated datasets of 100k points • Uniform, gaussian, logvariatenormal • Queries • Linear average queries: (eg, 0.4*a + 0.6*b) • Nearest neighbor queries: (eg, (x-3)^2 + (y-4)^2) • Join queries: (0.4*R.a + 0.6*S.b: R.c=R.d)

  43. Conclusion • Problem • Study K-constrained optimization queries as boolean + ranking • Abstraction • Encode K-constrained optimization into shortest path problem • Framework • Develop OPT* to process K-constrained optimization

  44. References • Boolean + Ranking: Querying a Database by K-Constrained Optimization. Z. Zhang, S. Hwang, K. C.-C. Chang, M. Wang, C. Lang, and Y. Chang. In Proceedings of the 2006 ACM SIGMOD Conference (SIGMOD 2006), pages 359-370, Chicago, June 2006 • www.wikipedia.org

  45. Thank you! Questions?

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