Document

# Hop-Spanners for Geometric Intersection Graphs

## File

LIPIcs.SoCG.2022.30.pdf
• Filesize: 0.91 MB
• 17 pages

## Acknowledgements

We thank Sujoy Bhore for helpful discussions on geometric intersections graphs.

## Cite As

Jonathan B. Conroy and Csaba D. Tóth. Hop-Spanners for Geometric Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.30

## Abstract

A t-spanner of a graph G = (V,E) is a subgraph H = (V,E') that contains a uv-path of length at most t for every uv ∈ E. It is known that every n-vertex graph admits a (2k-1)-spanner with O(n^{1+1/k}) edges for k ≥ 1. This bound is the best possible for 1 ≤ k ≤ 9 and is conjectured to be optimal due to Erdős' girth conjecture. We study t-spanners for t ∈ {2,3} for geometric intersection graphs in the plane. These spanners are also known as t-hop spanners to emphasize the use of graph-theoretic distances (as opposed to Euclidean distances between the geometric objects or their centers). We obtain the following results: (1) Every n-vertex unit disk graph (UDG) admits a 2-hop spanner with O(n) edges; improving upon the previous bound of O(nlog n). (2) The intersection graph of n axis-aligned fat rectangles admits a 2-hop spanner with O(nlog n) edges, and this bound is the best possible. (3) The intersection graph of n fat convex bodies in the plane admits a 3-hop spanner with O(nlog n) edges. (4) The intersection graph of n axis-aligned rectangles admits a 3-hop spanner with O(nlog² n) edges.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Discrete mathematics
• Mathematics of computing → Paths and connectivity problems
• Theory of computation → Computational geometry
##### Keywords
• geometric intersection graph
• unit disk graph
• hop-spanner

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Pankaj K. Agarwal and Jiangwei Pan. Near-linear algorithms for geometric hitting sets and set covers. Discret. Comput. Geom., 63(2):460-482, 2020. URL: https://doi.org/10.1007/s00454-019-00099-6.
2. Abu Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen G. Kobourov, and Richard Spence. Graph spanners: A tutorial review. Comput. Sci. Rev., 37:100253, 2020. URL: https://doi.org/10.1016/j.cosrev.2020.100253.
3. Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9(1):81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
4. Shinwoo An and Eunjin Oh. Feedback vertex set on geometric intersection graphs. In Hee-Kap Ahn and Kunihiko Sadakane, editors, 32nd International Symposium on Algorithms and Computation, (ISAAC), volume 212 of LIPIcs, pages 47:1-47:12. Schloss Dagstuhl, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.47.
5. Roel Apfelbaum and Micha Sharir. Large complete bipartite subgraphs in incidence graphs of points and hyperplanes. SIAM J. Discret. Math., 21(3):707-725, 2007. URL: https://doi.org/10.1137/050641375.
6. Boris Aronov, Esther Ezra, and Micha Sharir. Small-size ε-nets for axis-parallel rectangles and boxes. SIAM J. Comput., 39(7):3248-3282, 2010. URL: https://doi.org/10.1137/090762968.
7. Baruch Awerbuch. Communication-time trade-offs in network synchronization. In Proc. 4th ACM Symposium on Principles of Distributed Computing (PODC), pages 272-276, 1985. URL: https://doi.org/10.1145/323596.323621.
8. Julien Baste and Dimitrios M. Thilikos. Contraction-bidimensionality of geometric intersection graphs. In Proc. 12th International Symposium on Parameterized and Exact Computation (IPEC), volume 89 of LIPIcs, pages 5:1-5:13. Schloss Dagstuhl, 2017. URL: https://doi.org/10.4230/LIPIcs.IPEC.2017.5.
9. Ahmad Biniaz. Plane hop spanners for unit disk graphs: Simpler and better. Comput. Geom., 89:101622, 2020. URL: https://doi.org/10.1016/j.comgeo.2020.101622.
10. Ulrik Brandes and Dagmar Handke. NP-completness results for minimum planar spanners. In Rolf H. Möhring, editor, Proc. 23rd Workshop on Graph-Theoretic Concepts in Computer Science (WG), volume 1335 of LNCS, pages 85-99. Springer, 1997. URL: https://doi.org/10.1007/BFb0024490.
11. Peter Braß and Christian Knauer. On counting point-hyperplane incidences. Comput. Geom., 25(1-2):13-20, 2003. URL: https://doi.org/10.1016/S0925-7721(02)00127-X.
12. Heinz Breu and David G. Kirkpatrick. Unit disk graph recognition is np-hard. Comput. Geom., 9(1-2):3-24, 1998. URL: https://doi.org/10.1016/S0925-7721(97)00014-X.
13. Norbert Bus, Shashwat Garg, Nabil H. Mustafa, and Saurabh Ray. Tighter estimates for ε-nets for disks. Comput. Geom., 53:27-35, 2016. URL: https://doi.org/10.1016/j.comgeo.2015.12.002.
14. Norbert Bus, Shashwat Garg, Nabil H. Mustafa, and Saurabh Ray. Limits of local search: Quality and efficiency. Discret. Comput. Geom., 57(3):607-624, 2017. URL: https://doi.org/10.1007/s00454-016-9819-x.
15. Sergio Cabello and Miha Jejcic. Shortest paths in intersection graphs of unit disks. Comput. Geom., 48(4):360-367, 2015. URL: https://doi.org/10.1016/j.comgeo.2014.12.003.
16. Leizhen Cai. Np-completeness of minimum spanner problems. Discret. Appl. Math., 48(2):187-194, 1994. URL: https://doi.org/10.1016/0166-218X(94)90073-6.
17. Leizhen Cai and J. Mark Keil. Spanners in graphs of bounded degree. Networks, 24(4):233-249, 1994. URL: https://doi.org/10.1002/net.3230240406.
18. Jean Cardinal, John Iacono, and Grigorios Koumoutsos. Worst-case efficient dynamic geometric independent set. In Proc. 29th European Symposium on Algorithms (ESA), volume 204 of LIPIcs, pages 25:1-25:15. Schloss Dagstuhl, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.25.
19. Nicolas Catusse, Victor Chepoi, and Yann Vaxès. Planar hop spanners for unit disk graphs. In Christian Scheideler, editor, Proc. 6th (ALGOSENSORS), volume 6451 of LNCS, pages 16-30. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-16988-5_2.
20. Keren Censor-Hillel and Michal Dory. Distributed spanner approximation. SIAM J. Comput., 50(3):1103-1147, 2021. URL: https://doi.org/10.1137/20M1312630.
21. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discret. Comput. Geom., 48(2):373-392, 2012. URL: https://doi.org/10.1007/s00454-012-9417-5.
22. Timothy M. Chan and Dimitrios Skrepetos. All-pairs shortest paths in geometric intersection graphs. J. Comput. Geom., 10(1):27-41, 2019. URL: https://doi.org/10.20382/jocg.v10i1a2.
23. Mirela Damian, Saurav Pandit, and Sriram V. Pemmaraju. Local approximation schemes for topology control. In Eric Ruppert and Dahlia Malkhi, editors, Proceedings of the Twenty-Fifth Annual ACM Symposium on Principles of Distributed Computing, PODC 2006, Denver, CO, USA, July 23-26, 2006, pages 208-217. ACM, 2006. URL: https://doi.org/10.1145/1146381.1146413.
24. Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. A framework for exponential-time-hypothesis-tight algorithms and lower bounds in geometric intersection graphs. SIAM J. Comput., 49(6):1291-1331, 2020. URL: https://doi.org/10.1137/20M1320870.
25. Michael Dinitz, Guy Kortsarz, and Ran Raz. Label cover instances with large girth and the hardness of approximating basic k-spanner. ACM Trans. Algorithms, 12(2):25:1-25:16, 2016. URL: https://doi.org/10.1145/2818375.
26. Thao Do. Representation complexities of semialgebraic graphs. SIAM J. Discret. Math., 33(4):1864-1877, 2019. URL: https://doi.org/10.1137/18M1221606.
27. Yevgeniy Dodis and Sanjeev Khanna. Design networks with bounded pairwise distance. In Proc. 31st ACM Symposium on Theory of Computing (STOC), pages 750-759, 1999. URL: https://doi.org/10.1145/301250.301447.
28. Adrian Dumitrescu, Anirban Ghosh, and Csaba D. Tóth. Sparse hop spanners for unit disk graphs. Computational Geometry, page 101808, 2021. URL: https://doi.org/10.1016/j.comgeo.2021.101808.
29. Michael Elkin and David Peleg. The hardness of approximating spanner problems. Theory Comput. Syst., 41(4):691-729, 2007. URL: https://doi.org/10.1007/s00224-006-1266-2.
30. David Eppstein and Hadi Khodabandeh. Optimal spanners for unit ball graphs in doubling metrics. CoRR, abs/2106.15234, 2021. URL: http://arxiv.org/abs/2106.15234.
31. Paul Erdős. Extremal problems in graph theory. In Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), pages 29-36, Prague, 1964. Publishing House of the Czechoslovak Academy of Sciences. URL: https://old.renyi.hu/~p_erdos/1964-06.pdf.
32. Paul Erdős, A. W. Goodman, and Louis Pósa. The representation of a graph by set intersections. Canadian Journal of Mathematics, 18:106-112, 1966. URL: https://doi.org/10.4153/CJM-1966-014-3.
33. Paul Erdös and László Pyber. Covering a graph by complete bipartite graphs. Discret. Math., 170(1-3):249-251, 1997. URL: https://doi.org/10.1016/S0012-365X(96)00124-0.
34. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Finding, hitting and packing cycles in subexponential time on unit disk graphs. Discret. Comput. Geom., 62(4):879-911, 2019. URL: https://doi.org/10.1007/s00454-018-00054-x.
35. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Bidimensionality and geometric graphs. In Yuval Rabani, editor, Proc. 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1563-1575, 2012. URL: https://doi.org/10.1137/1.9781611973099.124.
36. Jacob Fox and János Pach. Applications of a new separator theorem for string graphs. Comb. Probab. Comput., 23(1):66-74, 2014. URL: https://doi.org/10.1017/S0963548313000412.
37. Martin Fürer and Shiva Prasad Kasiviswanathan. Spanners for geometric intersection graphs with applications. J. Comput. Geom., 3(1):31-64, 2012. URL: https://doi.org/10.20382/jocg.v3i1a3.
38. Jie Gao and Li Zhang. Well-separated pair decomposition for the unit-disk graph metric and its applications. SIAM J. Comput., 35(1):151-169, 2005. URL: https://doi.org/10.1137/S0097539703436357.
39. Petr Hlinený and Jan Kratochvíl. Representing graphs by disks and balls (a survey of recognition-complexity results). Discret. Math., 229(1-3):101-124, 2001. URL: https://doi.org/10.1016/S0012-365X(00)00204-1.
40. Bruno Jartoux and Nabil H. Mustafa. Optimality of geometric local search. In Proc. 34th Symposium on Computational Geometry (SoCG), volume 99 of LIPIcs, pages 48:1-48:15. Schloss Dagstuhl, 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.48.
41. Yusuke Kobayashi. Np-hardness and fixed-parameter tractability of the minimum spanner problem. Theor. Comput. Sci., 746:88-97, 2018. URL: https://doi.org/10.1016/j.tcs.2018.06.031.
42. Guy Kortsarz. On the hardness of approximating spanners. Algorithmica, 30(3):432-450, 2001. URL: https://doi.org/10.1007/s00453-001-0021-y.
43. Guy Kortsarz and David Peleg. Generating low-degree 2-spanners. SIAM J. Comput., 27(5):1438-1456, 1998. URL: https://doi.org/10.1137/S0097539794268753.
44. Jan Kratochvíl. A special planar satisfiability problem and a consequence of its np-completeness. Discret. Appl. Math., 52(3):233-252, 1994. URL: https://doi.org/10.1016/0166-218X(94)90143-0.
45. Hung Le and Shay Solomon. Truly optimal Euclidean spanners. In Proc. 60th IEEE Symposium on Foundations of Computer Science (FOCS), pages 1078-1100, 2019. URL: https://doi.org/10.1109/FOCS.2019.00069.
46. Hung Le and Shay Solomon. Towards a unified theory of light spanners I: fast (yet optimal) constructions. CoRR, abs/2106.15596, 2021. URL: http://arxiv.org/abs/2106.15596.
47. James R. Lee. Separators in region intersection graphs. In Proc. 8th Innovations in Theoretical Computer Science (ITCS), volume 67 of LIPIcs, pages 1:1-1:8. Schloss Dagstuhl, 2017. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.1.
48. Colin McDiarmid and Tobias Müller. Integer realizations of disk and segment graphs. J. Comb. Theory, Ser. B, 103(1):114-143, 2013. URL: https://doi.org/10.1016/j.jctb.2012.09.004.
49. T. S. Michael and Thomas Quint. Sphericity, cubicity, and edge clique covers of graphs. Discret. Appl. Math., 154(8):1309-1313, 2006. URL: https://doi.org/10.1016/j.dam.2006.01.004.
50. Nabil H. Mustafa, Kunal Dutta, and Arijit Ghosh. A simple proof of optimal epsilon nets. Combinatorica, 38(5):1269-1277, 2018. URL: https://doi.org/10.1007/s00493-017-3564-5.
51. Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discret. Comput. Geom., 44(4):883-895, 2010. URL: https://doi.org/10.1007/s00454-010-9285-9.
52. Nabil H. Mustafa and Kasturi R. Varadarajan. Epsilon-approximations and epsilon-nets. In Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, chapter 47. CRC Press, Boca Raton, FL, 3rd edition, 2017.
53. János Pach and Gábor Tardos. Tight lower bounds for the size of epsilon-nets. J. AMS, 26:645-658, 2013. URL: https://doi.org/10.1090/S0894-0347-2012-00759-0.
54. David Peleg and Alejandro A. Schäffer. Graph spanners. Journal of Graph Theory, 13(1):99-116, 1989. URL: https://doi.org/10.1002/jgt.3190130114.
55. Micha Sharir and Noam Solomon. Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances. In Proc. 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2456-2475, 2017. URL: https://doi.org/10.1137/1.9781611974782.163.
56. Zsolt Tuza. Covering of graphs by complete bipartite subgraphs; complexity of 0-1 matrices. Comb., 4(1):111-116, 1984. URL: https://doi.org/10.1007/BF02579163.
57. Chenyu Yan, Yang Xiang, and Feodor F. Dragan. Compact and low delay routing labeling scheme for unit disk graphs. Comput. Geom., 45(7):305-325, 2012. URL: https://doi.org/10.1016/j.comgeo.2012.01.015.
X

Feedback for Dagstuhl Publishing