Document Open Access Logo

Segment Proximity Graphs and Nearest Neighbor Queries Amid Disjoint Segments

Authors Pankaj K. Agarwal , Haim Kaplan , Matthew J. Katz , Micha Sharir

PDF
Thumbnail PDF

File

LIPIcs.ESA.2024.7.pdf
  • Filesize: 1.13 MB
  • 20 pages

Document Identifiers

Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Haim Kaplan
  • School of Computer Science, Tel Aviv University, Israel
Matthew J. Katz
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer Sheva, Israel
Micha Sharir
  • School of Computer Science, Tel Aviv University, Israel

Funding

  • Agarwal, Pankaj K.: Partially supported by NSF grants IIS-18-14493, CCF-20-07556, and CCF-22-23870.
  • Kaplan, Haim: Partially supported by Israel Science Foundation Grant 1156/23 and the Blavatnik research Foundation.
  • Katz, Matthew J.: Partially supported by Israel Science Foundation Grant 495/23.
  • Sharir, Micha: Partially supported by Israel Science Foundation Grant 495/23.

Cite AsGet BibTex

Pankaj K. Agarwal, Haim Kaplan, Matthew J. Katz, and Micha Sharir. Segment Proximity Graphs and Nearest Neighbor Queries Amid Disjoint Segments. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.7

Abstract

In this paper we study a few proximity problems related to a set of pairwise-disjoint segments in {ℝ}². Let S be a set of n pairwise-disjoint segments in {ℝ}², and let r > 0 be a parameter. We define the segment proximity graph of S to be G_r(S) := (S,E), where E = {(e₁,e₂) ∣ dist(e₁,e₂) ≤ r} and dist (e₁,e₂) = min_{(p,q) ∈ e₁× e₂} ‖p-q‖ is the Euclidean distance between e₁ and e₂. We define the weight of an edge (e₁,e₂) ∈ E to be dist(e₁,e₂). We first present a simple grid-based O(nlog² n)-time algorithm for computing a BFS tree of G_r(S). We apply it to obtain an O^*(n^{6/5}) + O(nlog²nlogΔ)-time algorithm for the so-called reverse shortest path problem, in which we want to find the smallest value r^* for which G_{r^*}(S) contains a path of some specified length between two designated start and target segments (where the O^*(⋅) notation hides polylogarithmic factors). Here Δ = max_{e ≠ e' ∈ S} dist(e,e')/min_{e ≠ e' ∈ S} dist(e,e') is the spread of S. Next, we present a dynamic data structure that can maintain a set S of pairwise-disjoint segments in the plane under insertions/deletions, so that, for a query segment e from an unknown set Q of pairwise-disjoint segments, such that e does not intersect any segment in (the current version of) S, the segment of S closest to e can be computed in O(log⁵ n) amortized time. The amortized update time is also O(log⁵ n). We note that if the segments in S∪Q are allowed to intersect then the known lower bounds on halfplane range searching suggest that a sequence of n updates and queries may take at least close to Ω(n^{4/3}) time. One thus has to strongly rely on the non-intersecting property of S and Q to perform updates and queries in O(polylog(n)) (amortized) time each. Using these results on nearest-neighbor (NN) searching for disjoint segments, we show that a DFS tree (or forest) of G_r(S) can be computed in O^*(n) time. We also obtain an O^*(n)-time algorithm for constructing a minimum spanning tree of G_r(S). Finally, we present an O^*(n^{4/3})-time algorithm for computing a single-source shortest-path tree in G_r(S). This is the only result that does not exploit the disjointness of the input segments.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • segment proximity graphs
  • nearest neighbor searching
  • dynamic data structures
  • BFS
  • DFS
  • unit-disk graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. P. K. Agarwal, Simplex range searching and its variants: A review,in Journey through Discrete Mathematics: A Tribute to Jiří Matoušek,M. Loebl, J. Nešetřil and R. Thomas, editors,Springer Verlag, Berlin-Heidelberg, 2017, pages 1-30. Google Scholar
  2. P. K. Agarwal, A. Efrat and M. Sharir,Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications,SIAM J. Comput. 29(3) (1999), 912-953. Google Scholar
  3. P. K. Agarwal, M. J. Katz, and M. Sharir,On reverse shortest paths in geometric proximity graphs,Comput. Geom. Theory Appls. 117 (2024), Art. 102053.Also in Proc. 33rd Internat. Sympos. on Algorithms and Computation, 2022, 42:1-42:19. Google Scholar
  4. P. K. Agarwal and J. Matoušek,Dynamic half-space range reporting and its applications,Algorithmica 13 (1995), 325-345. Google Scholar
  5. P. K. Agarwal, M. H. Overmars, and M. Sharir,Computing maximally separated sets in the plane,SIAM J. Comput., 36(3) (2006), 815-834. Google Scholar
  6. N. Alon, J. Pach, R. Pinchasi, R. Radoicic and M. Sharir, Crossing patterns of semi-algebraic sets,J. Combin. Theory Ser. A 111 (2005), 310-326. Google Scholar
  7. F. Aurenhammer, R. Klein and D.-T. Lee,Voronoi Diagrams and Delaunay Triangulations,World Scientific, Singapore 2013. Google Scholar
  8. R. Ben Avraham, O. Filtser, H. Kaplan, M. J. Katz and M. Sharir,The discrete and semicontinuous Fréchet distance with shortcuts via approximate distance counting and selection,ACM Trans. Algorithms 11 (2015), Art. 29. Google Scholar
  9. J. L. Bentley and J. B. Saxe,Decomposable searching problems I: Static-to-dynamic transformation,J. Algorithms 1(4) (1980), 301-–358. Google Scholar
  10. M. de Berg, O. Cheong, M. J. van Kreveld, and M. H. Overmars,Computational Geometry: Algorithms and Applications, 3rd Edition,Springer Verlag, Berlin, 2008. Google Scholar
  11. S. Bespamyatnikh,Computing closest points for segments,Int. J. Comput. Geom. Appl. 13(5) (2003), 419-438. Google Scholar
  12. S. Bespamyatnikh and J. Snoeyink,Queries with segments in Voronoi diagrams,Comput. Geom. Theory Appls. 16 (2000), 23-33. Google Scholar
  13. N. Blum and K. Mehlhorn,On the average number of rebalancing operations in weight-balanced trees,Theor. Comput. Sci. 11 (1980), 303-320. Google Scholar
  14. H. Breu and D. G. Kirkpatrick,Unit disk graph recognition is NP-hard,Comput. Geom. Theory Appls., 9(1-2) (1998), 3-24. Google Scholar
  15. S. Cabello and M. Jejčič, Shortest paths in intersection graphs of unit disks, Comput. Geom. Theory Appls. 48 (2015), 360-367. Google Scholar
  16. T. M. Chan, Optimal partition trees, Discrete Comput. Geom., 47 (2012), 661-690. Google Scholar
  17. T. M. Chan,Dynamic generalized closest pair: Revisiting Eppstein’s technique, Proc. 3rd Sympos. Simplicity in Algorithms (SOSA), 2020, 33-37. Google Scholar
  18. T. M. Chan,Finding triangles and other small subgraphs in geometric intersection graphs,Proc. 34th ACM-SIAM Symposium on Discrete Algorithms, 2023, 1777-1805. Google Scholar
  19. T. M. Chan and D. Skrepetos,All-pairs shortest paths in unit-disk graphs in slightly subquadratic time,Proc. 27th Internat. Sympos. on Algorithms and Computation, pages 24:1-24:13, 2016. Google Scholar
  20. T. M. Chan and D. Skrepetos,All-pairs shortest paths in geometric intersection graphs,J. Comput. Geom. Theory Appls. 10(1) (2019), 27-41. Google Scholar
  21. T. M. Chan and D. Skrepetos,Approximate shortest paths and distance oracles in weighted unit-disk graphs,J. Comput. Geom. Theory Appls. 10(2) (2019), 3-20. Google Scholar
  22. B. Chazelle, H. Edelsbrunner, L. Guibas and M. Sharir,Algorithms for bichromatic line-segment problems and polyhedral terrains,Algorithmica 11 (1994), 116-132. Google Scholar
  23. B. N. Clark, C. J. Colbourn, and D. S. Johnson,Unit disk graphs,Discrete Math. 86(1-3) (1990), 165-177. Google Scholar
  24. D. Conlon, J. Fox, J. Pach, B. Sudakov, and A. Suk, Ramsey-type results for semi-algebraic hypergraphs, Trans. Amer. Math. Soc. 366 (2014), 5043-5065. Google Scholar
  25. D. Eppstein,Fast hierarchical clustering and other applications of dynamic closest pairs, Discrete Comput. Geom. 13 (1995), 111-122. Google Scholar
  26. J. Erickson,New lower bounds for Hopcroft’s problem,Discrete Comput. Geom. 16 (1996), 389-418. Google Scholar
  27. A. V. Fishkin,Disk graphs: A short survey,Proc. 1st Internat. Workshop on Approximation and Online Algorithms, volume 2909 of Lecture Notes in Computer Science, pages 260-264, 2003. Google Scholar
  28. G. D. da Fonseca, V. G. P. de Sá, and C. M. H. de Figueiredo,Shifting coresets: Obtaining linear-time approximations for unit disk graphs and other geometric intersection graphs,Int. J. Comput. Geom. Appl. 27(4) (2017), 255-276. Google Scholar
  29. J. Fox, J. Pach, and A. Suk, Density and regularity theorems for semi-algebraic hypergraphs, Proc. 26th ACM-SIAM Symposium on Discrete Algorithms, 2015, pp. 1517-1530. Google Scholar
  30. W. Gálvez, A. Khan, M. Mari, T. Mömke, M. R. Pittu and A. Wiese,A 3-approximation algorithm for maximum independent set of rectangles, Proc. 33rd ACM-SIAM Symposium on Discrete Algorithms, 2022, 894-905. Google Scholar
  31. J. Gao and L. Zhang,Well-separated pair decomposition for the unit-disk graph metric and its applications,SIAM J. Comput., 35(1) (2005), 151-169. Google Scholar
  32. H. Kaplan, M. J. Katz, R. Saban and M. Sharir,The unweighted and weighted reverse shortest path problem for disk graphs,Proc. 31st European Sympos. Algorithms (2023), 67:1-67:14. Also in arXiv:2307.14663. Google Scholar
  33. H. Kaplan, W. Mulzer, L. Roditty, P. Seiferth and M. Sharir,Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications,Discrete Comput. Geom. 64 (2020), 838-904. Google Scholar
  34. H. Kaplan, R. E. Tarjan and K. Tsioutsiouliklis,Faster kinetic heaps and their use in broadcast scheduling,Proc. 12th ACM-SIAM Sympos. on Discrete Algorithms, 2001, 836-844. Google Scholar
  35. K. Klost,An algorithmic framework for the single source shortest path problem with applications to disk graphs,Comput. Geom. Theory Appls. 111 (2023), Art. 101979. Google Scholar
  36. J. Kratochvíl and J. Matoušek,Intersection graphs of segments,J. Combinat. Theory, Ser. B 62 (1994), 289-–315. Google Scholar
  37. C.-H. Liu,Nearly optimal planar k nearest neighbors queries under general distance functions,SIAM J. Comput. 51(3) (2022), 723-765. Google Scholar
  38. K. Mehlhorn,Data Structures and Algorithms 1: Sorting and Searching, Springer, 1984. Google Scholar
  39. K. Mehlhorn,Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry, Springer, 1984. Google Scholar
  40. K. Mehlhorn, S. Näher,Dynamic fractional cascading, Algorithmica 5 (1990), 215-241. Google Scholar
  41. A. Meyerson,Online facility location, Proc. 42nd IEEE Sympos. Foundations of Computer Science (FOCS), 2001, 426-–431. Google Scholar
  42. J. S. B. Mitchell,Approximating maximum independent set for rectangles in the plane,Proc. 62nd IEEE Sympos. on Foundations of Computer Science, 2021, 339-350. Google Scholar
  43. J. Nievergelt and E. M. Reingold,Binary search trees of bounded balance, SIAM J. Comput. 2 (1973), 33-43. Google Scholar
  44. M. H. Overmars,The Design of Dynamic Data Structures,Lecture Notes in Computer Science 156, Springer, 1983. Google Scholar
  45. M. H. Overmars and J. van Leeuwen,Maintenance of configurations in the plane,J. Comput. Syst. Sci. 23 (1981), 166-204. Google Scholar
  46. F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer, Berlin, 1985. Google Scholar
  47. L. Roditty and M. Segal,On bounded leg shortest paths problems,Algorithmica 59(4) (2011), 583-600. Google Scholar
  48. M. Schaefer and D. Štefankovič, Decidability of string graphs, Proc. Sympos. on Theory of Computing (STOC), 2001, 241-246. Google Scholar
  49. M. Sharir and P.K. Agarwal,Davenport-Schinzel Sequences and Their Geometric Applications,Cambridge University Press, Cambridge-New York-Melbourne, 1995. Google Scholar
  50. H. Wang,Algorithms for computing closest points for segments,Proc. 41st Internat. Sympos. on Theoretical Aspects of Computer Science (STACS), 2024, 58:1-58:17. Google Scholar
  51. H. Wang and J. Xue,Near-optimal algorithms for shortest paths in weighted unit-disk graphs,Discrete Comput. Geom. 64(4) (2020), 1141-1166. Google Scholar
  52. H. Wang and Y. Zhao,Reverse shortest path problem for unit-disk graphs,Proc. 17th Internat. Sympos. on Algorithms and Data Structures, 2021, 655-668. Google Scholar
  53. H. Wang and Y. Zhao,Reverse shortest path problem in weighted unit-disk graphs,Proc. 16th Internat. Conf. on Algorithms and Computation, 2022, 135-146. Google Scholar
  54. D. E. Willard and George S. Lueker,Adding range restriction capability to dynamic data structures, J. ACM 32 (1985), 597-617. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact