1. Kinetic Data Structures: for computational geometry and for graph drawing Sue Whitesides Computer Science Department

2. introduction We’ve now seen geometry problems for things that move: reconfiguration problems for chains of links, localization and navigation problems problems for low degree of freedom objects (points, vacuum cleaner robot) And we’ve seen geometry used for expressing the binary relationships of a graph G = (V,E), and an FPT approach for a hard graph drawing problems This talk is a synthesis of these themes: data structures for things that move, including graph vertices (kinetic graph drawing) 2

3. 3 three parts: 1.Introduction to Kinetic Data Structures (KDS) 2.KDS approach to two computational geometry problems: k-nearest neighbors, and reverse k-nearest neighbors 3.KDS approach for a graph drawing problem: point set embeddibility for plane graphs on moving points

4. 1. Introduction to Kinetic Data Structures KDS 4

5. Introducing … the Kinetic Data Structure (KDS) Framework collection of algorithms and data structures designed to deal with moving points; concept introduced by Basch, Guibas, and Herschberger in the late 1990’s; now a familiar concept with an associated body of scholarly work some implementations: D. Russel’s PhD thesis, Stanford U., INRIA

6. motivation The technology is there to track moving objects and communicate the data and also to create moving points (virtual reality, games, simulations) … what computational problems arise from this? 6

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9. 9 Free Game Art: Traffic Jamwww.vickiwenderlich.com

10. 10 Idea of a KDS Framework (rough idea) Situation: set P of moving points (bodies) moving according to nice, known (or estimated) trajectories question we want to ask at (unknown) future times current answer to the question (“attribute” of P) What we want: (from our data storage “framework”): correct responses to our question at future times – i.e., “maintain the attribute over time” methods to do updates when the trajectory estimates change from time to time good performance (space, running time)

11. 11 Input: a set of moving points Output: an attribute of moving points over time Kinetic Data Structure (KDS) More about the KDS framework

12. 12 Standard KDS Performance Criteria

13. 2. KDS approach to two computational geometry problems on moving points: k-nearest neighbors, and reverse k nearest neighbors 13

14. The K-nearest Neighbors Problem and the Reverse k-nearest Neighbors Problem for Moving Points: a kinetic data structure approach Zahed Rahmati, Valerie King, and Sue Whitesides Computer Science Department, University of Victoria 14

15. Outline: problem statements (static case) potential applications our results some intuition/insight

16. The k-Nearest Neighbor Problem Given: a set P of n points in the plane (or more generally, in R d ) Find: for each point p i in P, the point p j ≠ p i closest to p i 16 The k-nearest neighbor problem asks for the k nearest neighbors of each point in P.

17. k = 1: all NNs (& Closest Pair) Solution represented as the directed nearest neighbor graph (regarded as an “attribute” of the point set) 17

18. The Reverse k-Nearest Neighbor Problem Given: a set P of n points in the plane (or R d ) and a query point q  P find: all the points p i of P for which q is among the k nearest neighbors of p i in P  { q } (for a given q, there may be no such points, or many such points) 18

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20. What if the points are moving? The solutions to the problems change from time to time … can we update the solutions without having to recompute from scratch whenever there is a change? 20

21. State-of-the-art pre-Rahmati et al. 21

22. 22 prior to work of Rahmati et al. at UVic:

23. 23 Some Example New Results

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28. 3. Stationary and Kinetic Point Set Embeddibility for Plane Graphs 28

29. Kinetic and Stationary Point-Set Embeddibility for Plane Graphs Zahed Rahmati, Sue Whitesides, and Valerie King Department of Computer Science University of Victoria 20 th International Symposium on Graph Drawing Redmond, Washington 19-21 Sept 2012 Presented by Rahnuma Islam Nishat

30. Point-Set Embedding (PSE) without mapping drawing of a plane graph on a point-set such that each vertex is mapped to a point and no two edges intersect. if each point moves with predictable trajectory, how can we maintain the PSE over time? the trajectory is an algebraic functions of constant maximum degree s: assume no three or more points are collinear in any positive interval of time. using Taylor series, any trajectory function can be approximated by a sum of terms. 30Kinetic & Stationary Point-Set Embeddability 9/21/2012

31. Related Work examples 31Kinetic & Stationary Point-Set Embeddability 9/21/2012 kinetic proximity graphs closest pair, Delaunay triangulation, Euclidean MST Guibas et al. [1991], Basch et al. [1997], Rahmati et al. [2011], Abam et al. [2012] Stationary PSE k-bend drawingKaufmann and Wiese[2001], Bose [2002], Di Giacomo et al. [2005], Cabello [2006], Biedl et al. [2012], Durocher et al. [2012] Kinetic graph drawing  no previous results  [this paper] introduces the kinetic data structure (KDS) framework for maintaining the PSE problem.

32. Kinetic Framework maintaining the PSE, such that: –a point is outside the triangle created by other three points. –no three or more points are collinear in any positive interval of time. we define: set of certificates –a set of certificates certifying the correctness of the embedding. 9/21/2012Kinetic & Stationary Point-Set Embeddability 32

33. Kinetic Framework 9/21/2012Kinetic & Stationary Point-Set Embeddability 33

34. 9/21/2012Kinetic & Stationary Point-Set Embeddability 34

35. 9/21/2012Kinetic & Stationary Point-Set Embeddability 35

36. 9/21/2012Kinetic & Stationary Point-Set Embeddability 36

37. [3-bend Drawing alg.] (1) order the points 9/21/2012Kinetic & Stationary Point-Set Embeddability 37

38. [3-bend Drawing alg.] (2) find the Ham. cycle 9/21/2012Kinetic & Stationary Point-Set Embeddability 38 - Giacomo et al. [2005] - Kaufmann and Wiese [2002]

39. [3-bend Drawing alg.] (2) find a Ham. Cycle 9/21/2012Kinetic & Stationary Point-Set Embeddability 39 - Giacomo et al. [2005] - Kaufmann and Wiese [2002]

40. [3-bend Drawing alg.] (3) add dummy points 9/21/2012Kinetic & Stationary Point-Set Embeddability 40

41. [3-bend Drawing alg.] (4) map HC to the points 9/21/2012Kinetic & Stationary Point-Set Embeddability 41

42. [3-bend Drawing alg.] (5) map the plane graph 9/21/2012Kinetic & Stationary Point-Set Embeddability 42

43. Dynamic & Kinetic Tournament Tree Maintenance of the max slope of the consecutive edges: 9/21/2012Kinetic & Stationary Point-Set Embeddability 43

44. Order Event 9/21/2012Kinetic & Stationary Point-Set Embeddability 44

45. Order Event 9/21/2012Kinetic & Stationary Point-Set Embeddability 45

46. Order Event 9/21/2012Kinetic & Stationary Point-Set Embeddability 46

47. Performance Analysis 47

48. Future Directions our approach also gives a compact KDS for 2-bend drawing; find a KDS for 2-bend drawing that satisfies all four performance criteria. find a KDS for straight-line drawings for some special graphs like outerplanar graphs and trees. propose meaningful kinetic GD problems

49. Thank you for your attention. Thank you for your wonderful hospitality and friendship. 49