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Caching Dynamic Skyline Queries

Caching Dynamic Skyline Queries. D. Sacharidis 1 , P. Bouros 1 , T. Sellis 1,2 1 National Technical University of Athens 2 Institute for Management of Information Systems – R.C. Athena. Outline. Introduction Skyline (SL) and dynamic skyline queries (DSL) Related work

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Caching Dynamic Skyline Queries

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  1. Caching Dynamic Skyline Queries D. Sacharidis1, P. Bouros1, T. Sellis1,2 1National Technical University of Athens 2Institute for Management of Information Systems – R.C. Athena SSDBM'08

  2. Outline • Introduction • Skyline (SL) and dynamic skyline queries (DSL) • Related work • Evaluating dynamic skyline queries • Computing orthant skylines (OSL) • Computing dynamic skyline via caching • LRU, LFU, LPP cache replacement policies • Experimental evaluation • Conclusions and Future work SSDBM'08

  3. Skyline queries (SL) • Given a dataset of d-dimensional points • SL contains points not dominated by others • x dominates y iff x as good as y in all dimensions and strictly better in at least one SSDBM'08

  4. Skyline queries (SL) • Given a dataset of d-dimensional points • SL contains points not dominated by others • x dominates y iff x as good as y in all dimensions and strictly better in at least one Price Distance from sea • Example • Dataset of hotels • Prefer cheap hotels close to the sea SSDBM'08

  5. Skyline queries (SL) • Given a dataset of d-dimensional points • SL contains points not dominated by others • x dominates y iff x as good as y in all dimensions and strictly better in at least one Skyline points Price Distance from sea • Example • Dataset of hotels • Prefer cheap hotels close to the sea SSDBM'08

  6. Skyline queries (SL) • Given a dataset of d-dimensional points • SL contains points not dominated by others • x dominates y iff x as good as y in all dimensions and strictly better in at least one Skyline points Price p1 Distance from sea • Example • Dataset of hotels • Prefer cheap hotels close to the sea SSDBM'08

  7. Skyline queries (SL) • Given a dataset of d-dimensional points • SL contains points not dominated by others • x dominates y iff x as good as y in all dimensions and strictly better in at least one Skyline points Price p1 p2 Distance from sea • Example • Dataset of hotels • Prefer cheap hotels close to the sea SSDBM'08

  8. Dynamic skyline queries (DSL) • Extension of skyline queries • Given a query point q • DSL contains points not dynamically dominated by others w.r.t q • x dynamically dominates y iff x as preferable as y w.r.t. q in all dimensions and strictly more preferable w.r.t. q in at least one • Can be treated as static SL • Transform points w.r.t. q SSDBM'08

  9. Dynamic skyline queries (DSL) • Extension of skyline queries • Given a query point q • DSL contains points not dynamically dominated by others w.r.t q • x dynamically dominates y iff x as preferable as y w.r.t. q in all dimensions and strictly more preferable w.r.t. q in at least one • Can be treated as static SL • Transform points w.r.t. q Query point q Price Distance from sea • Example • User defines “ideal” hotel q SSDBM'08

  10. Dynamic skyline queries (DSL) • Extension of skyline queries • Given a query point q • DSL contains points not dynamically dominated by others w.r.t q • x dynamically dominates y iff x as preferable as y w.r.t. q in all dimensions and strictly more preferable w.r.t. q in at least one • Can be treated as static SL • Transform points w.r.t. q Dynamic Skyline points Price q Distance from sea • Example • User defines “ideal” hotel q SSDBM'08

  11. Dynamic skyline queries (DSL) • Extension of skyline queries • Given a query point q • DSL contains points not dynamically dominated by others w.r.t q • x dynamically dominates y iff x as preferable as y w.r.t. q in all dimensions and strictly more preferable w.r.t. q in at least one • Can be treated as static SL • Transform points w.r.t. q p4 p5 Dynamic Skyline points Price q Distance from sea • Example • User defines “ideal” hotel q SSDBM'08

  12. Intuition (1) • Traditional SL algorithms need to run anew for each DSL query • Our idea • Exploit results from past queries to reduce processing cost for future DSL queries • Cache past queries • Decide which queries in cache are useful SSDBM'08

  13. Intuition (2) Price Distance from sea SSDBM'08

  14. Intuition (2) • 2 past DSL queries • qa, qb • Each query partitions space in 4quadrants qa Price qb Distance from sea SSDBM'08

  15. Intuition (3) p4 • A new query q arrives • Consider DSL for qa • p1 is contained DSL(qa) • p1 dominates p2, p3, p4 • p1 lies in upper right quadrant w.r.t. qa • qa lies in upper right quadrant w.r.t. q • p1 dominates also p2, p3,p4 w.r.t. to q • Exclude p2, p3, p4 from dominance test for DSL(q) p2 p1 p3 qa q Price qb Distance from sea • Shaded area denotes points dominated by p1 SSDBM'08

  16. Contribution in brief • Caching past DSL queries cannot reduce processing cost for future ones • We need more information about dominance relationships • Introduce orthantskylines (OSL) and examine their relationship with DSL • ExtendBitmap algorithm to compute OSL in parallel with DSL • Cache OSL to enhance DSL queries evaluation • Present 3 cache replacement policies • LRU, LFU, LPP • Experimental evaluation of caching mechanism SSDBM'08

  17. Related work • Non-indexed methods • Block-Nested Loops (BnL) • Bitmap • Multidimensional Divide and Conquer (DnC) • Sort First Scan (SFS) • Index-based methods • B-tree • sort points according to the lowest valued coordinate • R-tree • Nearest neighbor based (NN) • Branch and bound (BBS) SSDBM'08

  18. Related work • Non-indexed methods • Block-Nested Loops (BnL) • Bitmap • Multidimensional Divide and Conquer (DnC) • Sort First Scan (SFS) • Index-based methods • B-tree • sort points according to the lowest valued coordinate • R-tree • Nearest neighbor based (NN) • Branch and bound (BBS) SSDBM'08

  19. Bitmap • BnL variant • Suitable for domains with low cardinality and discrete • In brief • Computes a bitmap representation of the points in the dataset • Examines each point separately (dominance test) • Checks whether it is contained in the skyline or not • Exploits fast bitwise operations OR/AND SSDBM'08

  20. Bitmap – Dominance test • For each point p • Define A = A1 & A2 & … & Ad • Denotes the points as good as p in all dimensions • Define B = B1 | B2 | … | Bd • Denotes the points strictly better than p in at least one dimension • Dominance test: • If C = A & B has all bits set to 0 then p is in SL SSDBM'08

  21. Orthant skyline (OSL) • OSL provides more information about dominance relationships than DSL • Useful for pruning • Given a dataset of d-dimensional points and a query point q • Space partitioned in 2d orthants • o-th orthant skyline (OSL) of q contains points of the o-th orthantnot dynamically dominated by others inside orthant o w.r.t q SSDBM'08

  22. Orthant skyline (OSL) Quadrant 1 Quadrant 0 • OSL provides more information about dominance relationships than DSL • Useful for pruning • Given a dataset of d-dimensional points and a query point q • Space partitioned in 2d orthants • o-th orthant skyline (OSL) of q contains points of the o-th orthantnot dynamically dominated by others inside orthant o w.r.t q Query point q Price Distance from sea Quadrant 2 Quadrant 3 SSDBM'08

  23. Orthant skyline (OSL) Quadrant 1 Quadrant 0 • OSL provides more information about dominance relationships than DSL • Useful for pruning • Given a dataset of d-dimensional points and a query point q • Space partitioned in 2d orthants • o-th orthant skyline (OSL) of q contains points of the o-th orthantnot dynamically dominated by others inside orthant o w.r.t q Query point q Price Distance from sea Quadrant 2 Quadrant 3 SSDBM'08

  24. Orthant skyline (OSL) Quadrant 1 Quadrant 0 • OSL provides more information about dominance relationships than DSL • Useful for pruning • Given a dataset of d-dimensional points and a query point q • Space partitioned in 2d orthants • o-th orthant skyline (OSL) of q contains points of the o-th orthantnot dynamically dominated by others inside orthant o w.r.t q Query point q Quadrant 2 skyline points Price Distance from sea Quadrant 2 Quadrant 3 SSDBM'08

  25. OSL and DSL relationship Quadrant 1 Quadrant 0 Price q Distance from sea Quadrant 2 Quadrant 3 SSDBM'08

  26. OSL and DSL relationship Quadrant 1 Quadrant 0 Price q Distance from sea Quadrant 2 Quadrant 3 SSDBM'08

  27. OSL and DSL relationship Quadrant 1 Quadrant 0 • Map points from quadrants 1,2,3 to points inside quadrant 0 Price q Distance from sea Quadrant 2 Quadrant 3 SSDBM'08

  28. OSL and DSL relationship Quadrant 1 Quadrant 0 • Map points from quadrants 1,2,3 to points inside quadrant 0 • Compute DSL w.r.t. q Price q Distance from sea Quadrant 2 Quadrant 3 SSDBM'08

  29. OSL and DSL relationship Quadrant 1 Quadrant 0 • Map points from quadrants 1,2,3 to points inside quadrant 0 • Compute DSL w.r.t. q • Union of all OSLs is superset of DSL w.r.t. to q Price q Distance from sea Quadrant 2 Quadrant 3 SSDBM'08

  30. OSL and DSL relationship Quadrant 1 Quadrant 0 • Map points from quadrants 1,2,3 to points inside quadrant 0 • Compute DSL w.r.t. q • Union of all OSLs is superset of DSL w.r.t. to q p1 p2 Price q Distance from sea Quadrant 2 Quadrant 3 SSDBM'08

  31. OSL and DSL relationship Quadrant 1 Quadrant 0 • Map points from quadrants 1,2,3 to points inside quadrant 0 • Compute DSL w.r.t. q • Union of all OSLs is superset of DSL w.r.t. to q p2 Price q p3 Distance from sea Quadrant 2 Quadrant 3 SSDBM'08

  32. Computing orthant skylines • Algorithm DBM • Extends Bitmap to compute DSL and OSLs at the same time • Method: • Compute bitmap representation • Transform each point coordinates w.r.t. to query q • Dominance test, point p, orthant o • p not in OSLo and not in DSL • p not in DSL, but in OSLo • p in DSL and in OSLo SSDBM'08

  33. Dynamic skylines Via Caching • Cache OSLs instead of DSLs • Query cache contains (query point qj, OSLs) • OSLs encode by bitmaps • Algorithm cDBM • OSL contains information about dominance test inside orthant • Discard points inside orthants from dominance tests • Method: • Compute bitmap representation • For each point p consider its position (orthant) w.r.t. to cache queries qj • If p in the same orthant o w.r.t qj as qj w.r.t. q and p not in OSLo(qj) then exclude it from OSLo(q), DSL(q) SSDBM'08

  34. Cache Replacement Policies • General idea • Limited cache space • Identify least useful query point in cache • Replace it with new one SSDBM'08

  35. Usage-based policies • Only a few queries in cache are useful • Log cache query usage • Given a new query q • Consider as input the query point cache Q • Only query points in OSL of Q w.r.t. q are useful • Update cache - remove: • Least Recently Used (LRU) query point • Least Frequently Used (LFU) query point SSDBM'08

  36. Usage-based policies • Only a few queries in cache are useful • Log cache query usage • Given a new query q • Consider as input the query point cache Q • Only query points in OSL of Q w.r.t. q are useful • Update cache - remove: • Least Recently Used (LRU) query point • Least Frequently Used (LFU) query point qd qb qc qa Price q Distance from sea SSDBM'08

  37. Usage-based policies Redundant queries • Only a few queries in cache are useful • Log cache query usage • Given a new query q • Consider as input the query point cache Q • Only query points in OSL of Q w.r.t. q are useful • Update cache - remove: • Least Recently Used (LRU) query point • Least Frequently Used (LFU) query point qd qb qc qa Price q Distance from sea SSDBM'08

  38. Usage-based policies Redundant queries • Only a few queries in cache are useful • Log cache query usage • Given a new query q • Consider as input the query points in cache Q • Only query points inOSL of Q w.r.t. q are useful • Update cache - remove: • Least Recently Used (LRU) query point • Least Frequently Used (LFU) query point qd qb qc qa Price q Distance from sea SSDBM'08

  39. Usage-based policies Redundant queries • Only a few queries in cache are useful • Log cache query usage • Given a new query q • Consider as input the query points in cache Q • Only query points in OSL of Q w.r.t. q are useful • Update cache - remove: • Least Recently Used (LRU) query point • Least Frequently Used (LFU) query point qd qb qc qa Price q Distance from sea SSDBM'08

  40. Pruning power-based policy • Usage-based policies do not indicate usefulness • Useful cached query • Great pruning power • Probability that a query can prune points of dataset from DSL computation • Depends on • Points dominated by query in an orthant j • Points contained in the antisymetric orthantof j • Update cache – remove • Query point with less pruning power (LPP) qa Price q Distance from sea SSDBM'08

  41. Pruning power-based policy • Usage-based policies do not indicate usefulness • Useful cached query • Great pruning power • Probability that a query can prune points of dataset from DSL computation • Depends on • Points dominated by query in an orthant j • Points contained in the antisymetric orthantof j • Update cache – remove • Query point with less pruning power (LPP) 5:24 2:24 qa 3:24 74:24 Price q Distance from sea SSDBM'08

  42. Pruning power-based policy • Usage-based policies do not indicate usefulness • Useful cached query • Great pruning power • Probability that a query can prune points of dataset from DSL computation • Depends on • Points dominated by query in an orthant j • Points contained in the antisymetric orthant of j • Update cache – remove • Query point with less pruning power (LPP) 5:74 2:34 qa 3:44 74:884 Price q Distance from sea SSDBM'08

  43. Pruning power-based policy • Usage-based policies do not indicate usefulness • Useful cached query • Great pruning power • Probability that a query can prune points of dataset from DSL computation • Depends on • Points dominated by query in an orthant j • Points contained in the antisymetric orthant of j • Update cache – remove • Query point with less pruning power (LPP) 5:74 2:3176 qa 3:44 74:884 Price q Distance from sea SSDBM'08

  44. Pruning power-based policy • Usage-based policies do not indicate usefulness • Useful cached query • Great pruning power • Probability that a query can prune points of dataset from DSL computation • Depends on • Points dominated by query in an orthant j • Points contained in the antisymetric orthant of j • Update cache – remove • Query point with less pruning power (LPP) 5:720 2:3176 qa 3:421 74:88222 Price q Distance from sea SSDBM'08

  45. Pruning power-based policy • Usage-based policies do not indicate usefulness • Useful cached query • Great pruning power • Probability that a query can prune points of dataset from DSL computation • Depends on • Points dominated by query in an orthant j • Points contained in the antisymetric orthant of j • Update cache – remove • Query point with less pruning power (LPP) 5:720 2:3176 qa 3:421 74:88222 Price q Distance from sea SSDBM'08

  46. Experimental Evaluation • Synthetic datasets • Distribution types • Independent, correlated, anti-correlated • Number of points N • 10k, 20k, 50k, 100k, • Dimensionality • d = {2,3,4,5,6} • Domain size for dimension • |D| = {10,20,50} • Compare • Bitmap (NO-CACHE) • cDBM with LFU,LRU,LPP cache replacement policies • Query cache • |Q| = {10,20,30,40,50} past query points • Cache size is |Q|*N bits uncompressed SSDBM'08

  47. Varying query cache size Independent Anti-correlated • Dataset: N = 50k points, with d = 4 dimensions of |D| = 20 domain size • LFU,LRU cache queries not representative for future ones • LPP caches queries with great pruning power SSDBM'08

  48. Effect of distribution parameters Correlated vary N Correlated vary d • Relative improvement in running time over NO-CACHE • Vary number of points N • d = 4 dimensions of |D| = 20 domain size • Vary number of dimensions d • N = 50k, |D| = 20 SSDBM'08

  49. Conclusions and Future work • Conclusions • Introduced orthant skylines(OSLs) and discussed its relationship with DSL • Extended Bitmap to compute OSLs and DSL at the same time (DBM algorithm) • Proposed caching mechanism of OSLs to reducecost for future DSL queries • LRU, LFU, LPPcache replacement policies • Experimentally verified the efficiency of caching mechanism • Future work • Apply caching mechanism to index-based methods • Further increase pruning power of cached queries SSDBM'08

  50. Questions ? SSDBM'08

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