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23rd	  Interna3onal	  Symposium	  on	  Algorithms	  and	  Computa3on,	  ISAAC	  2012	                                     ...
Overview1.    Introduction2.    Disjoint line-segments        a)    Disconnected regions        b)    Differences with poi...
1. Introduction
Nearest Neighbor Voronoi DiagramThe nearest neighbor Voronoi diagram is thepartitioning of the plane into maximal regions,...
Higher Order Voronoi DiagramThe order-k Voronoi diagram is the partitioning of theplane into maximal regions, such that al...
Related Work¤  Higher order Voronoi diagram of points:   ¤  Structural complexity                [Lee 82, Edelsbrunner 8...
2. Disjoint Line-Segments     a) Disconnected regions
Disconnected RegionsA single order-k Voronoi region may disconnect t      faces                                        2-­...
Disconnected RegionsAn order-k Voronoi region may disconnectto         bounded faces.For          Order-­‐2	  Voronoi	  di...
Disconnected RegionsAn order-k Voronoi region may disconnectto     unbounded faces.For              k                     ...
Disconnected RegionsAn order-k Voronoi region may disconnectto     unbounded faces.For              F1                    ...
2. Disjoint Line-Segments     b) Differences with points
For Points:¤  Order-k Voronoi regions are connected convex polygons¤  The number of faces [Lee82]¤  The symmetry proper...
For Line-Segments:¤  A single order-k Voronoi region may disconnect to   faces¤  The number of faces [this paper]¤  NO ...
2. Disjoint Line-Segments     c) Structural complexity
Structural complexity¤  Let F be a face of region             in¤  The graph structure of             enclosed in F    i...
Structural complexity¤  Generalizing Lee’s approach we prove:¤  For            this formula already implies¤  For      ...
Bounding¤  We use well-known point-line duality transformation.¤  We transform every line-segment to a wedge [Aurenhamme...
Bounding¤  Consider an arrangement of dual wedges                                                             w5         ...
Bounding¤             maximum complexity of¤             maximum complexity of¤  The previous observation implies:
Bounding¤             maximum complexity of¤             maximum complexity of¤  The previous observation implies:¤  F...
Bounding¤             maximum complexity of¤             maximum complexity of¤  The previous observation implies:¤  F...
Structural complexity¤  The number of unbounded faces:¤  The total number of unbounded faces [this paper]:¤  We can bou...
Structural complexity¤  The number of faces in the order-k Voronoi diagram of n    disjoint line-segments:
3. Planar Straight-Line Graph
Planar straight-line graph¤  Challenge: Define order-k line-segment Voronoi diagram of a    planar straight-line graph co...
Definition¤  We extend the notion of the order-k Voronoi diargam.¤  A set H is called an order-k subset iff   ¤  type-1...
Planar straight-line graph                                                                         Order-1 Voronoi Diagram...
Planar straight-line graph                                                                               Order-2 Voronoi D...
Planar straight-line graph V (1, 5, 6, 7)                                 V (4, 5, 6)                                     ...
4. Intersecting Line-Segments
Intersecting line-segments¤  Number of faces:   ¤  Nearest neighbor Voronoi diagram of line-segments   ¤  Farthest Voro...
Intersecting line-segments¤  The number of faces in the order-k Voronoi diagram of n    intersecting line-segments
Summary¤  Lower bounds for disconnected regions¤  Structural complexity for disjoint line-segments:¤  Consistent defini...
References1.     P. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher o...
Thank you! h[p://zavermax.github.com	  
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On the higher order Voronoi diagram of line-segments (ISAAC2012)

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We analyze structural properties of the order-k Voronoi dia- gram of line segments, which surprisingly has not received any attention in the computational geometry literature. We show that order-k Voronoi regions of line segments may be disconnected; in fact a single order- k Voronoi region may consist of Ω(n) disjoint faces. Nevertheless, the structural complexity of the order-k Voronoi diagram of non-intersecting segments remains O(k(n − k)) similarly to points. For intersecting line segments the structural complexity remains O(k(n − k)) for k ≥ n/2.

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On the higher order Voronoi diagram of line-segments (ISAAC2012)

  1. 1. 23rd  Interna3onal  Symposium  on  Algorithms  and  Computa3on,  ISAAC  2012   Taipei,  Taiwan,  December  2012   On  higher  order  Voronoi  diagrams     of  line  segments   Maksym Zavershynskyi Evanthia Papadopoulou University  of  Lugano,  Switzerland   Supported  in  part  by  the  Swiss  Na3onal  Science  Founda3on  (SNF)  grant  200021-­‐127137.    Also  by  SNF  grant  20GG21-­‐134355  within  the  collabora3ve  research  project  EuroGIGA/VORONOI  of  the  European  Science  Founda3on.  
  2. 2. Overview1.  Introduction2.  Disjoint line-segments a)  Disconnected regions b)  Differences with points c)  Structural complexity3.  Planar straight-line graph4.  Intersecting line-segments
  3. 3. 1. Introduction
  4. 4. Nearest Neighbor Voronoi DiagramThe nearest neighbor Voronoi diagram is thepartitioning of the plane into maximal regions, suchthat all points within a region have the same closestsite.
  5. 5. Higher Order Voronoi DiagramThe order-k Voronoi diagram is the partitioning of theplane into maximal regions, such that all points withinan order-k region have the same k nearest sites. 2-­‐order  Voronoi     diagram  
  6. 6. Related Work¤  Higher order Voronoi diagram of points: ¤  Structural complexity [Lee 82, Edelsbrunner 87] ¤  Construction algorithms ¤  Iterative in time [Lee 82] ¤  Randomized incremental in expected time [Agarwal et al 98]¤  Farthest Voronoi diagram of line-segments [Aurenhammer et al 06]¤  Higher order Voronoi diagram of line-segments NOT STUDIED!
  7. 7. 2. Disjoint Line-Segments a) Disconnected regions
  8. 8. Disconnected RegionsA single order-k Voronoi region may disconnect t faces 2-­‐order  Voronoi     diagram  
  9. 9. Disconnected RegionsAn order-k Voronoi region may disconnectto bounded faces.For Order-­‐2  Voronoi  diargam  of  6  line-­‐segments  
  10. 10. Disconnected RegionsAn order-k Voronoi region may disconnectto unbounded faces.For k n-k Order-­‐4  Voronoi  diargam  of  7  line-­‐segments   *  Generalizing  [Aurenhammer  et  al  06].  
  11. 11. Disconnected RegionsAn order-k Voronoi region may disconnectto unbounded faces.For F1 Line-­‐segments  can  be  untangled!   F2 F3 Order-­‐4  Voronoi  diargam  of  7  line-­‐segments   F4 *  Generalizing  [Aurenhammer  et  al  06].  
  12. 12. 2. Disjoint Line-Segments b) Differences with points
  13. 13. For Points:¤  Order-k Voronoi regions are connected convex polygons¤  The number of faces [Lee82]¤  The symmetry property for the number of unbounded faces:¤  The k-set theory [Edelsbrunner 87, Alon et al 86] implies bounds:¤  The structural complexity is
  14. 14. For Line-Segments:¤  A single order-k Voronoi region may disconnect to faces¤  The number of faces [this paper]¤  NO symmetry property for unbounded faces!¤  NO k-set theory available for unbounded faces!¤  The structural complexity is
  15. 15. 2. Disjoint Line-Segments c) Structural complexity
  16. 16. Structural complexity¤  Let F be a face of region in¤  The graph structure of enclosed in F is a connected tree that consists of at least one edge Vk (S) Vk−1 (S)
  17. 17. Structural complexity¤  Generalizing Lee’s approach we prove:¤  For this formula already implies¤  For we need to bound
  18. 18. Bounding¤  We use well-known point-line duality transformation.¤  We transform every line-segment to a wedge [Aurenhammer et al 06]
  19. 19. Bounding¤  Consider an arrangement of dual wedges w5 w4 w3 w2 p q w1¤  An unbounded edge in order-k Voronoi diagram corresponds to a vertex of *  for  direc3ons  from  π  to  2π#
  20. 20. Bounding¤  maximum complexity of¤  maximum complexity of¤  The previous observation implies:
  21. 21. Bounding¤  maximum complexity of¤  maximum complexity of¤  The previous observation implies:¤  Formula for complexity of of Jordan curves [Sharir, Agarwal 95]¤  Complexity of lower envelope of wedges [Edelsbrunner et al 82]
  22. 22. Bounding¤  maximum complexity of¤  maximum complexity of¤  The previous observation implies:¤  Formula for complexity of of Jordan curves [Sharir, Agarwal 95]¤  Complexity of lower envelope of wedges [Edelsbrunner et al 82]
  23. 23. Structural complexity¤  The number of unbounded faces:¤  The total number of unbounded faces [this paper]:¤  We can bound:
  24. 24. Structural complexity¤  The number of faces in the order-k Voronoi diagram of n disjoint line-segments:
  25. 25. 3. Planar Straight-Line Graph
  26. 26. Planar straight-line graph¤  Challenge: Define order-k line-segment Voronoi diagram of a planar straight-line graph consistently ¤  Avoid artificial splitting of equidistant regions for abutting segments that cause degeneracies ¤  Do not alter the definition for disjoint line-segments (using 3 sites per line-segment) s1 s1 b(s1, s2) b(s1, s2) v b(s1, s2) v b(s1, s2) b s2 s2 (a) (b)
  27. 27. Definition¤  We extend the notion of the order-k Voronoi diargam.¤  A set H is called an order-k subset iff ¤  type-1: I(p) ¤  type-2: , where and , p is the set of line-segments incident to . x Representative¤  Order-k Voronoi region:
  28. 28. Planar straight-line graph Order-1 Voronoi Diagram of Planar Straight-Line Graph V (6, 5) V (3, 4) V (5) V (4) V (6) 5 4 6V (1, 6, 7) V (7, 8) 7 V (4, 5) 3 V (7) 8 V (3) 1 V (1) V (8) 2 V (1, 2) V (2, 3, 8) V (2)
  29. 29. Planar straight-line graph Order-2 Voronoi Diagram of Planar Straight-Line Graph V (6, 5) V (4, 5) V (3, 4) V (5, 7, 8) 6 5 4V (1, 6, 7) V (7, 5) V (6, 7) 7 V (3, 4, 5) V (1, 7) V (7, 8) 3 1 8 V (3, 8) V (8, 4, 5) V (2, 7) V (2, 8) 2 V (2, 3, 8) V (1, 2)
  30. 30. Planar straight-line graph V (1, 5, 6, 7) V (4, 5, 6) Order-3 Voronoi Diagram of Planar Straight-Line Graph V (5, 6, 7) V (3, 4, 5) V (4, 5, 7) V (4, 5, 7, 8) 5 4 6 V (6, 7, 8)V (1, 6, 7) V (5, 7, 8) V (4, 5, 8) 7 V (3, 4, 8) V (1, 7, 8) V (3, 7, 8)V (1, 2, 6, 7) V (3, 4, 5, 8) V (2, 7, 8) 3 8 1 V (2, 3, 8) V (1, 2, 7) 2 V (1, 2, 8)
  31. 31. 4. Intersecting Line-Segments
  32. 32. Intersecting line-segments¤  Number of faces: ¤  Nearest neighbor Voronoi diagram of line-segments ¤  Farthest Voronoi diagram of line-segments where - # of intersections Intuitively, intersections influence small orders and the influence grows weaker as k increases.
  33. 33. Intersecting line-segments¤  The number of faces in the order-k Voronoi diagram of n intersecting line-segments
  34. 34. Summary¤  Lower bounds for disconnected regions¤  Structural complexity for disjoint line-segments:¤  Consistent definition for a planar straight-line graph.¤  Structural complexity for intersecting line-segments:
  35. 35. References1.  P. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27(3): 654-667 (1998)2.  N. Alon and E. Gyori. The number of small semispaces of a finite set of points in the plane. J. Comb. Theory, Ser. A 41(1): 154-157 (1986)3.  F. Aurenhammer, R. Drysdale, and H. Krasser. Farthest line segment Voronoi diagrams. Inf. Process. Lett. 100 (6): 220-225 (2006)4.  F. Aurenhammer and R. Klein Voronoi Diagrams in ”Handbook of computational geometry.” J.-R. Sack and J. Urrutia, North-Holland Publishing Co., 20005.  J.-D. Boissonnat, O. Devillers and M. Teillaud A Semidynamic Construction of Higher-Order Voronoi Diagrams and Its Randomized Analysis. Algorithmica 9(4): 329-356 (1993)6.  H. Edelsbrunner. Algorithms in combinatorial geometry. EATCS monographs on theoretical computer science. Springer, 1987., Chapter 13.47.  H. Edelsbrunner, H. A. Maurer, F. P. Preparata, A. L. Rosenberg, E. Welzl and D. Wood. Stabbing Line Segments. BIT 22(3): 274-281 (1982)8.  M. I. Karavelas. A robust and efficient implementation for the segment Voronoi diagram. In Proc. 1st Int. Symp. on Voronoi Diagrams in Science and Engineering, Tokyo: 51-62 (2004)9.  D. T. Lee. On k-Nearest Neighbor Voronoi Diagrams in the Plane. IEEE Trans. Computers 31(6): 478-487 (1982)10.  D. T. Lee, R. L. S. Drysdale. Generalization of Voronoi Diagrams in the Plane. SIAM J. Comput. 10(1): 73-87 (1981)11.  E. Papadopoulou Net-Aware Critical Area Extraction for Opens in VLSI Circuits Via Higher-Order Voronoi Diagrams. IEEE Trans. on CAD of Integrated Circuits and Systems 30(5): 704-717 (2011)12.  M. I. Shamos and D. Hoey. Closest-point problems. In Proc. 16th IEEE Symp. on Foundations of Comput. Sci.: 151-162 (1975)13.  M. Sharir and P. Agarwal. Davenport-Schinzel Sequences and their Geometric Applications. Cambridge University Press, 1995., Chapter 5.414.  C.-K. Yap. An O(n log n) Algorithm for the Voronoi Diagram of a Set of Simple Curve Segments. Discrete & Computational Geometry 2: 365-393 (1987)
  36. 36. Thank you! h[p://zavermax.github.com  

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