Querying Two Boundary Points for Shortest Paths in a Polygonal Domain

(Extended Abstract)
  • Sang Won Bae
  • Yoshio Okamoto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5878)


We consider a variant of two-point Euclidean shortest path query problem: given a polygonal domain, build a data structure for two-point shortest path query, provided that query points always lie on the boundary of the domain. As a main result, we show that a logarithmic-time query for shortest paths between boundary points can be performed using \(\tilde{O}(n^5)\) preprocessing time and \(\tilde{O}(n^5)\) space where n is the number of corners of the polygonal domain and the \(\tilde{O}\)-notation suppresses the polylogarithmic factor. This is realized by observing a connection between Davenport-Schinzel sequences and our problem in the parameterized space. We also provide a tradeoff between space and query time; a sublinear time query is possible using O(n 3 + ε ) space. Our approach also extends to the case where query points should lie on a given set of line segments.


Short Path Grid Cell Query Point Surface Patch Algebraic Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwal, P.K., Aronov, B., Sharir, M.: Computing envelopes in four dimensions with applications. SIAM J. Comput. 26(6), 1714–1732 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Agarwal, P.K., Sharir, M.: Arrangements and their applications. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computationaal Geometry, pp. 49–119. Elsevier Science Publishers B.V, Amsterdam (2000)CrossRefGoogle Scholar
  3. 3.
    Chiang, Y.-J., Mitchell, J.S.B.: Two-point Euclidean shortest path queries in the plane. In: Proc. 10th ACM-SIAM Sympos. Discrete Algorithms (SODA), pp. 215–224 (1999)Google Scholar
  4. 4.
    Hershberger, J.: Finding the upper envelope of n line segments in O(n logn) time. Inf. Process. Lett. 33(4), 169–174 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hershberger, J., Suri, S.: An optimal algorithm for Euclidean shortest paths in the plane. SIAM J. Comput. 28(6), 2215–2256 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Mitchell, J.S.B.: Shortest paths among obstacles in the plane. Internat. J. Comput. Geom. Appl. 6(3), 309–331 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Mitchell, J.S.B.: Shortest paths and networks. In: Handbook of Discrete and Computational Geometry, 2nd edn., ch. 27, pp. 607–641. CRC Press, Inc., Boca Raton (2004)Google Scholar
  8. 8.
    Nivasch, G.: Improved bounds and new techniques for Davenport-Schinzel sequences and their generatlizations. In: Proc. 20th ACM-SIAM Sympos. Discrete Algorithms, pp. 1–10 (2009)Google Scholar
  9. 9.
    Sharir, M.: Almost tight upper bounds for lower envelopes in higher dimensions. Discrete Comput. Geom. 12, 327–345 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York (1995)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Sang Won Bae
    • 1
  • Yoshio Okamoto
    • 2
  1. 1.Department of Computer Science and EngineeringPOSTECHPohangKorea
  2. 2.Graduate School of Information Science and EngineeringTokyo Institute of TechnologyTokyoJapan

Personalised recommendations