Skip to main content

Routing Among Convex Polygonal Obstacles in the Plane

  • Conference paper
  • First Online:
Combinatorial Optimization and Applications (COCOA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13135))

  • 848 Accesses

Abstract

Given a set \(\mathcal P\) of h pairwise disjoint convex polygonal obstacles in the plane, defined with n vertices, we preprocess \(\mathcal P\) and compute one routing table at each vertex in a subset of vertices of \(\mathcal P\). For routing a packet from any vertex \(s \in \mathcal P\) to any vertex \(t \in \mathcal P\), our scheme computes a routing path with a multiplicative stretch \(1+\epsilon \) and an additive stretch \(2 k \ell \), by consulting routing tables at only a subset of vertices along that path. Here, k is the number of obstacles of \(\mathcal P\) the routing path intersects, and \(\ell \) depends on the geometry of obstacles in \(\mathcal P\). During the preprocessing phase, we construct routing tables of size \(O(n + \frac{h^3}{\epsilon ^2} \mathrm {polylog}{(\frac{h}{\epsilon })})\) in \(O(n+\frac{h^3}{\epsilon ^2}\mathrm {polylog}{(\frac{h}{\epsilon })})\) time, where \(\epsilon < 1\) is an input parameter.

R. Inkulu—This research is supported in part by SERB MATRICS grant MTR/2017/000474.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abraham, I., Gavoille, C.: On approximate distance labels and routing schemes with affine stretch. In: Peleg, D. (ed.) DISC 2011. LNCS, vol. 6950, pp. 404–415. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-24100-0_39

    Chapter  Google Scholar 

  2. Arikati, S., Chen, D.Z., Chew, L.P., Das, G., Smid, M., Zaroliagis, C.D.: Planar spanners and approximate shortest path queries among obstacles in the plane. In: Diaz, J., Serna, M. (eds.) ESA 1996. LNCS, vol. 1136, pp. 514–528. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61680-2_79

    Chapter  Google Scholar 

  3. Asano, T., Asano, T., Guibas, L.J., Hershberger, J., Imai, H.: Visibility of disjoint polygons. Algorithmica 1(1), 49–63 (1986). https://doi.org/10.1007/BF01840436

    Article  MathSciNet  MATH  Google Scholar 

  4. Awerbuch, B., Bar-Noy, A., Linial, N., Peleg, D.: Improved routing strategies with succinct tables. J. Algorithms 11(3), 307–341 (1990)

    Article  MathSciNet  Google Scholar 

  5. Banyassady, B., et al.: Routing in polygonal domains. Comput. Geom. 87, 101593 (2020)

    Article  MathSciNet  Google Scholar 

  6. Bar-Yehuda, R., Chazelle, B.: Triangulating disjoint Jordan chains. Int. J. Comput. Geom. Appl. 4(4), 475–481 (1994)

    Article  MathSciNet  Google Scholar 

  7. Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: Optimal local routing on Delaunay triangulations defined by empty equilateral triangles. SIAM J. Comput. 44, 1626–1649 (2015)

    Article  MathSciNet  Google Scholar 

  8. Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: Competitive local routing with constraints. J. Comput. Geom. 8(1), 125–152 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Bose, P., Korman, M., van Renssen, A., Verdonschot, S.: Routing on the visibility graph. In: Proceedings of International Symposium on Algorithms and Computation, pp. 18:1–18:2 (2017)

    Google Scholar 

  10. Bose, P., Korman, M., van Renssen, A., Verdonschot, S.: Constrained routing between non-visible vertices. Theor. Comput. Sci. 861, 144–154 (2021)

    Article  MathSciNet  Google Scholar 

  11. Bose, P., Morin, P.: Competitive online routing in geometric graphs. Theor. Comput. Sci. 324(2), 273–288 (2004)

    Article  MathSciNet  Google Scholar 

  12. Bose, P., Smid, M.: On plane geometric spanners: a survey and open problems. Comput. Geom. 46(7), 818–830 (2013)

    Article  MathSciNet  Google Scholar 

  13. Chechik, S.: Compact routing schemes with improved stretch. In: Proceedings of Symposium on Principles of Distributed Computing, pp. 33–41 (2013)

    Google Scholar 

  14. Chen, D.Z.: On the all-pairs Euclidean short path problem. In: Proceedings of Symposium on Discrete Algorithms, pp. 292–301 (1995)

    Google Scholar 

  15. Chen, D.Z., Wang, H.: Visibility and ray shooting queries in polygonal domains. Comput. Geom. 48(2), 31–41 (2015)

    Article  MathSciNet  Google Scholar 

  16. Chiang, Y.-J., Mitchell, J.S.B.: Two-point Euclidean shortest path queries in the plane. In: Proceedings of Symposium on Discrete Algorithms, pp. 215–224 (1999)

    Google Scholar 

  17. Clarkson, K.L., Kapoor, S., Vaidya, P.M.: Rectilinear shortest paths through polygonal obstacles in \(O(n(\lg {n})^2)\) time. In: Proceedings of Symposium on Computational Geometry, pp. 251–257 (1987)

    Google Scholar 

  18. Cowen, L.: Compact routing with minimum stretch. J. Algorithms 38, 170–183 (1999)

    Article  MathSciNet  Google Scholar 

  19. de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-77974-2

    Book  MATH  Google Scholar 

  20. Eilam, T., Gavoille, C., Peleg, D.: Compact routing schemes with low stretch factor. J. Algorithms 46(2), 97–114 (2003)

    Article  MathSciNet  Google Scholar 

  21. Fraigniaud, P., Gavoille, C.: Routing in trees. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 757–772. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-48224-5_62

    Chapter  Google Scholar 

  22. Ghosh, S.K.: Visibility Algorithms in the Plane. Cambridge University Press, New York (2007)

    Book  Google Scholar 

  23. Guibas, L.J., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.E.: Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2, 209–233 (1987). https://doi.org/10.1007/BF01840360

    Article  MathSciNet  MATH  Google Scholar 

  24. Hershberger, J., Suri, S.: An optimal algorithm for Euclidean shortest paths in the plane. SIAM J. Comput. 28(6), 2215–2256 (1999)

    Article  MathSciNet  Google Scholar 

  25. Inkulu, R., Kapoor, S.: Planar rectilinear shortest path computation using corridors. Comput. Geom. 42(9), 873–884 (2009)

    Article  MathSciNet  Google Scholar 

  26. Inkulu, R., Kapoor, S.: Approximate Euclidean shortest paths amid polygonal obstacles. In: Proceedings of Symposium on Algorithms and Computation (2019)

    Google Scholar 

  27. Inkulu, R., Kapoor, S., Maheshwari, S.N.: A near optimal algorithm for finding Euclidean shortest path in polygonal domain. CoRR 1011.6481 (2010)

    Google Scholar 

  28. Kaplan, H., Mulzer, W., Roditty, L., Seiferth, P.: Routing in unit disk graphs. Algorithmica 80(3), 830–848 (2018). https://doi.org/10.1007/s00453-017-0308-2

    Article  MathSciNet  MATH  Google Scholar 

  29. Kapoor, S.: Efficient computation of geodesic shortest paths. In: Proceedings of Symposium on Theory of Computing, pp. 770–779 (1999)

    Google Scholar 

  30. Kapoor, S., Maheshwari, S.N.: Efficient algorithms for Euclidean shortest path and visibility problems with polygonal obstacles. In: Proceedings of Symposium on Computational Geometry, pp. 172–182 (1988)

    Google Scholar 

  31. Kapoor, S., Maheshwari, S.N.: Efficiently constructing the visibility graph of a simple polygon with obstacles. SIAM J. Comput. 30(3), 847–871 (2000)

    Article  MathSciNet  Google Scholar 

  32. Kapoor, S., Maheshwari, S.N., Mitchell, J.S.B.: An efficient algorithm for Euclidean shortest paths among polygonal obstacles in the plane. Discrete Comput. Geom. 18(4), 377–383 (1997). https://doi.org/10.1007/PL00009323

    Article  MathSciNet  MATH  Google Scholar 

  33. Konjevod, G., Richa, A.W., Xia, D.: Scale-free compact routing schemes in networks of low doubling dimension. ACM Trans. Algorithms 12(3), 1–29 (2016)

    Article  MathSciNet  Google Scholar 

  34. Mitchell, J.S.B.: Shortest paths among obstacles in the plane. Int. J. Comput. Geom. Appl. 6(3), 309–332 (1996)

    Article  MathSciNet  Google Scholar 

  35. Mitchell, J.S.B.: Shortest paths and networks. In: Handbook of Discrete and Computational Geometry, pp. 811–848. CRC Press (2017)

    Google Scholar 

  36. Narasimhan, G., Smid, M.H.M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  37. Overmars, M.H., Welzl, E.: New methods for computing visibility graphs. In: Proceedings of the Fourth Annual Symposium on Computational Geometry, pp. 164–171 (1988)

    Google Scholar 

  38. Peleg, D., Upfal, E.: A trade-off between space and efficiency for routing tables. J. ACM 36(3), 510–530 (1989)

    Article  MathSciNet  Google Scholar 

  39. Roditty, L., Tov, R.: New routing techniques and their applications. In: Proceedings of ACM Symposium on Principles of Distributed Computing, pp. 23–32 (2015)

    Google Scholar 

  40. Roditty, L., Tov, R.: Close to linear space routing schemes. Distrib. Comput. 29(1), 65–74 (2015). https://doi.org/10.1007/s00446-015-0256-5

    Article  MathSciNet  MATH  Google Scholar 

  41. Santoro, N., Khatib, R.: Labelling and implicit routing in networks. Comput. J. 28(1), 5–8 (1985)

    Article  MathSciNet  Google Scholar 

  42. Sharir, M., Schorr, A.: On shortest paths in polyhedral spaces. SIAM J. Comput. 15(1), 193–215 (1986)

    Article  MathSciNet  Google Scholar 

  43. Storer, J.A., Reif, J.H.: Shortest paths in the plane with polygonal obstacles. J. ACM 41(5), 982–1012 (1994)

    Article  MathSciNet  Google Scholar 

  44. Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. J. ACM 51(6), 993–1024 (2004)

    Article  MathSciNet  Google Scholar 

  45. Thorup, M., Zwick, U.: Compact routing schemes. In: Proceedings of Symposium on Parallel Algorithms and Architectures, pp. 1–10 (2001)

    Google Scholar 

  46. Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 1–24 (2005)

    Article  MathSciNet  Google Scholar 

  47. Welzl, E.: Constructing the visibility graph for \(n\)-line segments in \(O(n^2)\) time. Inf. Process. Lett. 20(4), 167–171 (1985)

    Article  Google Scholar 

  48. Yan, C., Xiang, Y., Dragan, F.F.: Compact and low delay routing labeling scheme for unit disk graphs. Comput. Geom. 45(7), 305–325 (2012)

    Article  MathSciNet  Google Scholar 

  49. Yao, A.C.: On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM J. Comput. 11(4), 721–736 (1982)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. Inkulu .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Inkulu, R., Kumar, P. (2021). Routing Among Convex Polygonal Obstacles in the Plane. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Combinatorial Optimization and Applications. COCOA 2021. Lecture Notes in Computer Science(), vol 13135. Springer, Cham. https://doi.org/10.1007/978-3-030-92681-6_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-92681-6_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-92680-9

  • Online ISBN: 978-3-030-92681-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics