1. Konstantinos Tsakalidis 1 Dynamic Data Structures: Orthogonal Range Queries and Update Efficiency Konstantinos Tsakalidis PhD Defense 23 September 2011

2. Konstantinos Tsakalidis 2 Κωνσταντίνος Τσακαλίδης 2000-2006 B. Eng. Computer Engineering and Informatics Dpt., University of Patras, Greece Sum. 2007 Intern Google Inc., Mountain View, California, USA 2007-2009 Ph. D. Student (Part A) MADALGO, Aarhus University, Denmark Sum. 2010 Visiting Prof. Ian Munro D. Cheriton School of Computer Science, University of Waterloo, Canada 2009-2011 Ph. D. Student (Part B)

3. Konstantinos Tsakalidis 3 Overview  Dynamic Planar Orthogonal 3-Sided Range Reporting Queries  [ISAAC ‘09] “Dynamic 3-Sided Planar Range Queries with Expected Doubly Logarithmic Time”  [ICDT ’10] “Efficient Processing of 3-Sided Range Queries with Probabilistic Guarantees”  Dynamic Planar Orthogonal Range Maxima Reporting Queries  [ICALP ’11] “Dynamic Planar Range Maxima Queries”  Multi-Versioned Indexed Databases  [SODA ‘12] “Fully Persistent B-Trees”

4. Konstantinos Tsakalidis 4 Databases and Geometry NameAgeSalaryDatePhone… Andreas305.5002/2010555-4321… Maria6.5004/1998555-3214… John253.0005/2011555-2143… Helen344.0001/2000555-1432… Jacob287.00011/1989555-1234… Planar (D=2) Euclidean Space 38 Query Operation Question about stored data Update Operation/Transaction Insert/Delete Tuple Change Value N points D dimensions 29 Salary Age … Date Name Phone

5. Konstantinos Tsakalidis 5 Models of Computation Pointer Machine Record O(1) fields word-RAM I/O Model [Aggarwal, Vitter ‘88] Space w bits/cell O(1) Time N M<N N B B words N/B M/B I/O Operation #Occupied Records #Arithmetic Operations +#Pointer Traversals Time #Occupied Cells #Arithmetic Operations +#cell READ/WRITEs #Occupied Blocks #I/O Operations specialized database Memory Disk

6. Konstantinos Tsakalidis 6 Overview  Dynamic Planar Orthogonal 3-Sided Range Reporting Queries  [ISAAC ‘09] “Dynamic 3-Sided Planar Range Queries with Expected Doubly Logarithmic Time”  [ICDT ’10] “Efficient Processing of 3-Sided Range Queries with Probabilistic Guarantees”  Dynamic Planar Orthogonal Range Maxima Reporting Queries  [ICALP ’11] “Dynamic Planar Range Maxima Queries”  Multi-Versioned Indexed Databases  [SODA ‘12] “Fully Persistent B-Trees”

7. Konstantinos Tsakalidis 7 Orthogonal Range Reporting Queries Salary Age 1000 Contour Query Report all points with: Salary > 1000 Dominance Query Report all points with: Salary > 1000 and Age > 35 35 2000 3-Sided Query Report all points with: 2000 > Salary > 1000 and Age > 35 Employees

8. Konstantinos Tsakalidis 8 I/O Model SpaceQuery I/OsUpdate I/Os External Priority Search Tree [Arge’99] amo. [ICDT ’10] Amortized Expected w.h.p. [ICDT ’10] Expected w.h.p. Amortized Expected w.h.p. [ISAAC‘09] Expected w.h.p.Amortized Expected [ISAAC ’09] Expected w.h.p. Expected amortized Worst-Case Efficient Dynamic 3-Sided Range Reporting word-RAM SpaceQuery TimeUpdate Time Fusion Tree [Willard’00] [Mortensen’06] I/O Model SpaceQuery I/OsUpdate I/Os External Priority Search Tree [Arge’99] amo. SpaceQuery TimeUpdate Time Priority Search Tree [McCreight’85] Pointer Machine word-RAM [ICDT ’10] Expected w.h.p. [ICDT ’10] Expected w.h.p. Expected w.h.p. X, Y: μ-random X: smooth Y: restricted X: smooth X, Y: μ-random X: smooth Y: restricted X: smooth Average-Case Efficient Dynamic 3-Sided Range Reporting

9. Konstantinos Tsakalidis 9  Unknown non-changing μ-Random probabilistic distribution  (f,g)-Smooth distribution  Not exceed a specific bound, no matter how small subinterval  Includes regular, uniform distributions  Any distribution is (f,Θ(n))-smooth  Restricted class of distributions  Few elements occur very often  Many elements occur rarely  Zipfian, Power Law Distributions Probabilistic Distributions Smooth Restricted

10. Konstantinos Tsakalidis 10 Priority Search Tree [McCreight’75] Move Up Maximum Y Space: O(n) Update: Update: O(log n) Pointer Machine

11. Konstantinos Tsakalidis 11 Query by X-Coordinate: logn + t PathSubtreesInX( s) Pointer Machine O(logn)

12. Konstantinos Tsakalidis 12 Query by Y-Coordinate: logn + t u ulul urur [Alstrup, Brodal, Rauhe ‘00] 1D Range Maximum Queries (Children) u Find next point to be reported in O(1) time O(1) time Pointer Machine word-RAM

13. Konstantinos Tsakalidis 13 [ISAAC ‘09] Update:O(log log n) exp. amo. Query: O(log log n+t) exp. w.h.p. Space: O(n) Weight i =Θ(2 2 i )  O(loglogn) expected w.h.p. [Mehlhorn, Tsakalidis ’93, Kaporis et al. ’06] [Andersson, Thorup ‘07] RMQ O(1) expected amortized word-RAM

14. Konstantinos Tsakalidis 14 I/O Model SpaceQuery I/OsUpdate I/Os [ISAAC‘09] Expected w.h.p.Amortized Expected Average-Case Efficient Dynamic 3-Sided Range Reporting SpaceQuery TimeUpdate Time [ISAAC ’09] Expected w.h.p. Expected amortized word-RAM X: smooth

15. Konstantinos Tsakalidis 15 Overview  Dynamic Planar Orthogonal 3-Sided Range Reporting Queries  [ISAAC ‘09] “Dynamic 3-Sided Planar Range Queries with Expected Doubly Logarithmic Time”  [ICDT ’10] “Efficient Processing of 3-Sided Range Queries with Probabilistic Guarantees”  Dynamic Planar Orthogonal Range Maxima Reporting Queries  [ICALP ’11] “Dynamic Planar Range Maxima Queries”  Multi-Versioned Indexed Databases  [SODA ‘12] “Fully Persistent B-Trees”

16. Konstantinos Tsakalidis 16 Orthogonal Range MAXIMA Reporting Queries OR “Generalized Planar SKYLINE Operator” Dominance Maxima Queries Report all maximal points among points with x in [x l,+∞) and y in [y b,+∞) Contour Maxima Queries Report all maximal points among points with x in (-∞, x l ] 3-Sided Maxima Queries Report all maximal points among points with x in [x l, x r ] and y in [y b,+∞) Salary Age Employees 4-Sided Maxima Queries Report all maximal points among points with x in [x l, x r ] and y in [y b,y t ] Interesting Points Oldest and Best Payed Maximal Point Dominates: Is “Above” Is NOT Dominated xlxl ybyb xlxl ybyb xrxr ybyb xlxl xlxl xrxr ybyb ytyt

17. Konstantinos Tsakalidis 17 Worst-Case Efficient Dynamic Range MAXIMA Reporting Pointer MachineInsertDelete Overmars, van Leeuwen ‘81 logn + t-log 2 n Frederickson, Rodger ‘90logn + tlog 2 n+t logn(1+t) lognlog 2 n Janardan ‘91logn + t lognlog 2 n Kapoor ‘00logn + t amo.-logn [ICALP ’11]logn + t logn word-RAMInsertDelete [ICALP ’11]

18. Konstantinos Tsakalidis 18 Tournament Tree Copy Up Maximum Y Y-Winning Paths Pointer Machine

19. Konstantinos Tsakalidis 19 Tournament Tree Right(u)MAX( ) u Pointer Machine Find next point to be reported in O(1) time

20. Konstantinos Tsakalidis 20 3-Sided Range Maxima Queries Query Time: log n + t MAX( ) Pointer Machine Subtrees(Paths) O(logn)

21. Konstantinos Tsakalidis 21 Update Operation Pointer Machine Previous Update: O(log 2 n)

22. Konstantinos Tsakalidis 22 U URUR ULUL Update Operation Pointer Machine MAX(Right(u R )) MAX(Right(u)) MAX(Right(u L )) [Sundar ‘89] Priority Queue with Attrition O(1) time

23. Konstantinos Tsakalidis 23 Reconstruct Rollback Update Operation Pointer Machine Partially Perstistent Priority Queue with Attrition O(1) time, space overhead per update step [Brodal ‘96] worst case [Driscol et al. ‘89] amortized Space:O(n) Update:O(logn)

24. Konstantinos Tsakalidis 24 [ICALP ‘11] [ICALP ’11]SpaceInsertDelete Pointer Machinenlogn+tlogn word-RAMn Pointer Machinenlognlog 2 n+tlog 2 n [ICALP ’11]SpaceInsertDelete

25. Konstantinos Tsakalidis 25 Rectangular Visibility Queries 4x4x (+∞,+∞) (+∞,-∞) (-∞,+∞) (-∞,-∞) Proximity Queries/Similarity Search 4-Sided Range Maxima Queries

26. Konstantinos Tsakalidis 26 Worst-Case Efficient 4-Sided Range MAXIMA Reporting and Rectangular Visibility Queries Pointer MachineSpaceInsertDelete Overmars, Wood ‘88nlognlog 2 n+tlog 2 nlog 3 n Overmars, Wood ‘88nlognlog 2 n +t logn log 2 n [ICALP ’11]nlognlog 2 n+tlog 2 n

27. Konstantinos Tsakalidis 27 Overview  Dynamic Planar Orthogonal 3-Sided Range Reporting Queries  [ISAAC ‘09] “Dynamic 3-Sided Planar Range Queries with Expected Doubly Logarithmic Time”  [ICDT ’10] “Efficient Processing of 3-Sided Range Queries with Probabilistic Guarantees”  Dynamic Planar Orthogonal Range Maxima Reporting Queries  [ICALP ’11] “Dynamic Planar Range Maxima Queries”  Multi-Versioned Indexed Databases  [SODA ‘12] “Fully Persistent B-Trees”

28. Konstantinos Tsakalidis 28 B-Trees [Bayer,McCreight ‘72] NameAgeSalary… Andreas305.500… Maria386.500… John253.000… Helen344.000… Jacob287.000… Indexed Database Space: O(N/B) blocks Update:O(log B N) I/Os Access: O(log B N) I/Os Multi-Versioned Databases Btrfs Data Platform

29. Konstantinos Tsakalidis 29 Fully Persistent B-Trees I/O ModelSpaceQuery I/OsUpdate I/Os Amortized Lanka, Mays ‘91n/B(log B n + t/B)log B mlog B n log B m [SODA ’12]n/Blog B n + t/Blog B n + log 2 B n elements in one version m update operations = #versions B block size

30. Konstantinos Tsakalidis 30 [SODA ‘12] Incremental B-Trees  Lazy Updates  O(log B N) READs  O(1) WRITEs that make O(1) changes to a block Result Space O(N/B) Query O(log B N+t/B) I/Os Update O(log B N + log 2 B) I/Os I/O-Efficient Full Persistence  Interface of Primitive Operations  READ  WRITE  Input is a pointer-based Structure  Node occupies O(1) blocks  Node has indegree O(1)  O(1) I/O-Overhead per access to a block  O(log 2 B) I/O-Overhead per change to a block  [Driscol et al.’89] Node-Splitting Method  ACCESS  NEW_NODE  NEW_VERSION

31. Konstantinos Tsakalidis 31 Mange Tak Konstantinos Tsakalidis Ph.D. Student tsakalid@madalgo.au.dk Tsakalidis K., et al. [ISAAC ‘09] “Dynamic 3-Sided Planar Range Queries with Expected Doubly Logarithmic Time” [ICDT ’10] “Efficient Processing of 3-Sided Range Queries with Probabilistic Guarantees” [ICALP ’11] “Dynamic Planar Range Maxima Queries” [SODA ‘12] “Fully Persistent B-Trees”