Advertisement

Encoding Homotopy of Paths in the Plane

  • Sergei Bespamyatnikh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)

Abstract

We study the problem of encoding homotopy of simple paths in the plane. We show that the homotopy of a simple path with k edges in the presence of n obstacles can be encoded using O(n log(n + k)) bits. The bound is tight if k=Ω(n 1 + ε). We present an efficient algorithm for encoding the homotopy of a path. The algorithm can be applied to find homotopic paths among a set of simple paths. We show that the homotopy of a general (not necessary simple) path can be encoded using O(k logn) bits. The bound is tight. The code is based on a homotopic minimum-link path and we present output-sensitive algorithms for computing a path and the code.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bar-Yehuda, R., Chazelle, B.: Triangulating disjoint Jordan chains. Internat. J. Comput. Geom. Appl. 4(4), 475–481 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bespamyatnikh, S.: Computing homotopic shortest paths in the plane. J. Algorithms 49(2), 284–303 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cabello, S., Liu, Y., Mantler, A., Snoeyink, J.: Testing homotopy for paths in the plane. In: Proc. 18th Annu. ACM Sympos. Comput. Geom., pp. 160–169 (2002)Google Scholar
  4. 4.
    Chazelle, B.: A theorem on polygon cutting with applications. In: Proc. 23rd Annu. IEEE Sympos. Found. Comput. Sci., pp. 339–349 (1982)Google Scholar
  5. 5.
    Chazelle, B.: An algorithm for segment-dragging and its implementation. Algorithmica 3, 205–221 (1988)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Efrat, A., Kobourov, S.G., Lubiw, A.: Computing homotopic shortest paths efficiently. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 411–423. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Ghosh, S.K.: Computing visibility polygon from a convex set and related problems. J. Algorithms 12, 75–95 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Comput. Geom. Theory Appl. 4, 63–98 (1994)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Keeler, K., Westbrook, J.: Short encodings of planar graphs and maps. Discrete Applied Mathematics 58(3), 239–252 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lee, D.T., Preparata, F.P.: Euclidean shortest paths in the presence of rectilinear barriers. Networks 14, 393–410 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Matoušek, J.: Range searching with efficient hierarchical cuttings. Discrete Comput. Geom. 10(2), 157–182 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 633–701. Elsevier Science Publishers B.V. North-Holland, Amsterdam (2000)CrossRefGoogle Scholar
  13. 13.
    Ian Munro, J., Raman, V.: Succinct representation of balanced parentheses and static trees. SIAM J. Comput. 31(3), 762–776 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Sergei Bespamyatnikh
    • 1
  1. 1.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA

Personalised recommendations