1. Computational Geometry for DAC: Partitioning Algorithms Joseph S. B. Mitchell and Girishkumar Sabhnani Stony Brook University

2. Optimal Subdivisions in CG Triangulations: Triangulations: Min-weight (total edge length) triangulation: NP-hard, in general, but O(n 3 ) for simple polygons, by dynamic programming)Min-weight (total edge length) triangulation: NP-hard, in general, but O(n 3 ) for simple polygons, by dynamic programming) Other objectives: Max the min angle (Delaunay), min the max angle, etc.Other objectives: Max the min angle (Delaunay), min the max angle, etc. 2

3. Optimal Subdivisions in CG Convex partitions: Min # convex pieces Convex partitions: Min # convex pieces Exact methods: DPExact methods: DP Approximation: Hertel-Mehlhorn algorithmApproximation: Hertel-Mehlhorn algorithm 3

4. Optimal Subdivisions in CG Load-balancing: Load-balancing: Rectangular partition of n x n matrix A, into k rectangles in order to minimize the max weight of a rectangle is NP-hard – [Muthukrishnan et al ’98]Rectangular partition of n x n matrix A, into k rectangles in order to minimize the max weight of a rectangle is NP-hard – [Muthukrishnan et al ’98] The element values used in the reduction are constants  4The element values used in the reduction are constants  4 4

5. 5 2D problem Hardness of special case Form small bundles of straight trajectories associated with each element of matrix, laid out in rectangular grid Form small bundles of straight trajectories associated with each element of matrix, laid out in rectangular grid Sectorization in the form of rectangles will solve the NP-hard problem Sectorization in the form of rectangles will solve the NP-hard problem

6. Related Problem: Districting 6 An example of "cracking" style Gerrymandering; where the urban (and mostly liberal) concentration of Columbus, Ohio is split into thirds and then each segment outweighted by attachment to largely conservative suburbs. Source: Wikipedia

7. A gerrymandered Congressional District, the 11th CD of CA (now occupied by Democrat Jerry McNerney), drawn to favor Republican Richard Pombo. While the Danville area is a traditional Republican stronghold, Morgan Hill is not, and that largely Democratic district was added to obtain the proper population numbers for the 11th after Livermore was assigned to the 10th at the behest of the incumbent Democrat (Ellen Tauscher), since it contains the Lawrence Livermore National Laboratory (located near the "580" shield) and she sits in the House Energy Committee. The 10th CD is immediately north of the 11th in Contra Costa and Solano Counties. See the California 11th congressional district election, 2006 for an unexpected result that overcame this gerrymander.Jerry McNerneyRichard Pombo LivermoreEllen TauscherLawrence Livermore National LaboratoryCalifornia 11th congressional district election, 2006 Image:The Gerry-Mander.png Gerrymandering

8. Fundamental Mathematical Tool: Ham Sandwich Theorem 8 “The volumes of any n solids of dimension n can always be simultaneously bisected by an (n – 1) dimensional hyperplane” [Steinhaus 1938] Corollary of Borsuk-Ulam theorem (1933): “any continuous function from an n -sphere into R n maps some pair of antipodal points to the same point” Ack: Armbruster, Carlsson, Ye

9. Discrete Set Partition [Bespamyatnikh, Kirkpatrick, Snoeyink 2000] and [Ito, Uehara, Yokoyama 1998] address a similar problem: “Given gn red points and gm blue points in the plane in general position, find a subdivision of the plane into g disjoint convex polygons, each of which contains n red points and m blue points.” Ack: Armbruster, Carlsson, Ye

10. Finding Equitable Convex Partitions of Points and Applications Benjamin Armbruster, John Gunnar Carlsson, Yinyu Ye n points are scattered inside a convex polygon P (in 2D) with m vertices. Does there exist a partition of P into n sub-regions satisfying the following: Each sub-region is convex Each sub-region contains one point All sub-regions have equal area

11. Related Problem: Voronoi Diagram In the Voronoi Diagram, we satisfy the first two properties (each sub-region is convex and contains one point), but the sub-regions have different areas. Ack: Armbruster, Carlsson, Ye

12. Result Not only such an equitable partition always exists, but also we can find it exactly in running time O(Nn log N), where N = m + n. Ack: Armbruster, Carlsson, Ye

14. USA Example Ack: Armbruster, Carlsson, Ye

15. 15 Airspace Sectorization Problem Joint work with A. Basu and G. Sabhnani

16. 16 Airspace Sectorization Problem k – no. of sectors k – no. of sectors b i – Workload(WL) of sector i b i – Workload(WL) of sector i Given the air-traffic pattern, decompose the domain of airspace into k sectors, “optimally” Given the air-traffic pattern, decompose the domain of airspace into k sectors, “optimally” Min-Max WLMin-Max WL Min-Sum WLMin-Sum WL OR given a max Workload B, minimize k OR given a max Workload B, minimize k We model as precise optimal geometric partitioning problem, for which we give provable results and heuristics to approximate

17. 17 1D Problem x t Greedy Algorithm

18. 18 1D problem Greedy = Optimal Greedy Optimal

19. 19 1D problem Running Time O(nlogn) Median ‘x’ Coordinate of the Vertices in Arrangement R. Cole, J. Salowe, W. Steiger, and E. Szemeredi SIAM J. Comput. ‘89 O(nlogn) Median ‘x’ Coordinate of the Vertices in Arrangement R. Cole, J. Salowe, W. Steiger, and E. Szemeredi SIAM J. Comput. ‘89 O(nlogn) : Compute the Max-WL after the interval is decided O(nlogn) : Compute the Max-WL after the interval is decided O(log(n 2 )) : Binary Search O(n 2 ) critical points O(log(n 2 )) : Binary Search O(n 2 ) critical points Total running time O(nlogn * log(n 2 )) = O(nlog 2 (n)) Total running time O(nlogn * log(n 2 )) = O(nlog 2 (n)) * k

20. 20 1D problem Time Avg WL Avg number of planes in a sector at any time Avg number of planes in a sector at any time Easier to sectorize in 1D O((k+n)logn) for given budget B O((k+n)logn) for given budget B Because the WL is a Because the WL is a piecewise linear and continuous function with breaks at end points (O(n))

21. 21 CG Concept: Binary Space Partition (BSP) BSP: Special case of a Convex Partition, P, of a domain D Defining property: Recursively obtained by cutting a face of P into two subfaces by a line/plane/hyperplane, cutting “all the way through” at each step D

22. 22 Example: ZFW Divided into 18 Sectors

23. BSP Trees in CG Used for “painter’s algorithm” Used for “painter’s algorithm” Applet Applet Applet 23

24. 24 Optimizing a Binary Space Partition (BSP) Dynamic Programming: Optimize over all BSP’s, exploiting the fact that BSP’s are recursively defined. Result: Provably optimal (among BSP partitions) method to partition an airspace into sectors, in a top-down fashion, for any specification/definition of “work load” in a sector (minimize the maximum workload in a sector – most balanced sectorization)

25. Optimal Load-Balancing BSP Partition Dynamic Program Dynamic Program Input: n points in a rectangle, R Input: n points in a rectangle, R Objective: BSP-Partition R into m rectangles, each with exactly k points, while maximizing the minimum aspect ratio (“niceness”) Objective: BSP-Partition R into m rectangles, each with exactly k points, while maximizing the minimum aspect ratio (“niceness”) 25 k=2 Subproblem: rectangle (x 1,x 2,y 1,y 2 )

26. 26 Optimal Smoothing Problem Given: Integer-programming-based solution (hex cells) (from Arash) Goal: Compute “optimal” smoothed boundaries (combine MILP and GeoSect techniques) Method: Use CG concepts of optimal paths, link distance, and optimal workload partitioning. Among all possible polygonal chains of k “links” (edges), that join two degree-3 vertices in the hex-cell map, find the optimal path according to an objective function based on min-max workload on each side of the path. Optimal 2-link partition path between two adjacent sectors of hex-cells

27. Polyline Simplification Mathematica site Mathematica site Mathematica site Mathematica site 27

28. 28 Dominant Flows: Trajectory Clustering Clustering of trajectory data Clustering of trajectory data Example: Example: Duality Custers

29. Patterns in Trajectories n trajectories, each with t time steps  n polygonal lines with t vertices n trajectories, each with t time steps  n polygonal lines with t vertices Already looked at most visited location Already looked at most visited location Computational Geometry and Spatial Data Mining, M. van Kreveld

30. Patterns in Trajectories Flock: near positions of (sub)trajectories for some subset of the entities during some time Flock: near positions of (sub)trajectories for some subset of the entities during some time Convergence: same destination region for some subset of the entities Convergence: same destination region for some subset of the entities Encounter: same destination region with same arrival time for some subset of the entities Encounter: same destination region with same arrival time for some subset of the entities Similarity of trajectories Similarity of trajectories Same direction of movement, leadership,...... Same direction of movement, leadership,...... flockconvergence Computational Geometry and Spatial Data Mining, M. van Kreveld

31. Patterns in Trajectories Flocking, convergence, encounter patterns Flocking, convergence, encounter patterns Laube, van Kreveld, Imfeld (SDH 2004)Laube, van Kreveld, Imfeld (SDH 2004) Gudmundsson, van Kreveld, Speckmann (ACM GIS 2004)Gudmundsson, van Kreveld, Speckmann (ACM GIS 2004) Benkert, Gudmundsson, Huebner, Wolle (ESA 2006)Benkert, Gudmundsson, Huebner, Wolle (ESA 2006)...... Similarity of trajectories Similarity of trajectories Vlachos, Kollios, Gunopulos (ICDE 2002)Vlachos, Kollios, Gunopulos (ICDE 2002) Shim, Chang (WAIM 2003)Shim, Chang (WAIM 2003)...... Lifelines, motion mining, modeling motion Lifelines, motion mining, modeling motion Mountain, Raper (GeoComputation 2001)Mountain, Raper (GeoComputation 2001) Kollios, Scaroff, Betke (DM&KD 2001)Kollios, Scaroff, Betke (DM&KD 2001) Frank (GISDATA 8, 2001)Frank (GISDATA 8, 2001)...... Computational Geometry and Spatial Data Mining, M. van Kreveld

32. Patterns in Trajectories Flock: near positions of (sub)trajectories for some subset of the entities during some time Flock: near positions of (sub)trajectories for some subset of the entities during some time clustering-type patternclustering-type pattern different definitions are useddifferent definitions are used Given: radius r, subset size m, and duration T, a flock is a subset of size  m that is inside a (moving) circle of radius r for a duration  T Given: radius r, subset size m, and duration T, a flock is a subset of size  m that is inside a (moving) circle of radius r for a duration  T Computational Geometry and Spatial Data Mining, M. van Kreveld

34. Patterns in Trajectories Longest flock: given a radius r and subset size m, determine the longest time interval for which m entities were within each other’s proximity (circle radius r) Longest flock: given a radius r and subset size m, determine the longest time interval for which m entities were within each other’s proximity (circle radius r) Time = 016543278 longest flock in [ 1.8, 6.4 ] m = 3 Computational Geometry and Spatial Data Mining, M. van Kreveld

35. 35 Clustering: Dominant Routes Voronoi-based partitioning

36. 36 Optimal Design of Tubes Flexible airspace design Flexible airspace design Dynamic Airspace Configuration Dynamic Airspace Configuration Network of “tubes”, similar to highwaysNetwork of “tubes”, similar to highways Dynamically designed/optimized Dynamically designed/optimized High volume, multiple lanes High volume, multiple lanes Equipage requirements (navigation, communication) Equipage requirements (navigation, communication)

37. Design of Tubes: Parameters Cross section: width (# lanes), height (# levels) Cross section: width (# lanes), height (# levels) Flight (equipage) characteristics per lane/levelFlight (equipage) characteristics per lane/level Merge/split points? Merge/split points? # and separation# and separation On/off ramps, and separation standards: upper/lower bounds on off, on, b On/off ramps, and separation standards: upper/lower bounds on off, on, b 37 off on b

38. 38 Input: Airports and Demands Optimization formulation: V. Polishchuk

39. 39 Hubs e.g., clusters of airports

40. 40 Tube Network We formulate: Hub placement optimization problem

41. 41 Hub Placement Problem “Steiner” nodes

42. 42 Second-Order Cone Programming (SOCP) Linear Constraints Cone Constraints

43. 43 SOCP in Computational Geometry

44. 44 Hubs Placement: SOCP Formulation

45. 45 Hubs Placement: SOCP Formulation

46. 46 Additional Constraints: Bonus!

47. Limitations of SOCP Model Imposing lower bounds on distances (hub- hub, hub-airport, etc) makes it NP-hard Imposing lower bounds on distances (hub- hub, hub-airport, etc) makes it NP-hard Euclidean distance (vs. spherical, wind- optimized routes) Euclidean distance (vs. spherical, wind- optimized routes) Network topology must be given: Still need to search over different connectivity graphs. Network topology must be given: Still need to search over different connectivity graphs. 47