Constrained Minimum Enclosing Circle with Center on a Query Line Segment

  • Sasanka Roy
  • Arindam Karmakar
  • Sandip Das
  • Subhas C. Nandy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)


In this paper, we will study the problem of locating the center of smallest enclosing circle of a set P of n points, where the center is constrained to lie on a query line segment. The preprocessing time and space complexities of our proposed algorithm are O(n logn) and O(n) respectively; the query time complexity is O(log2 n). We will use this method for solving the following problem proposed by Bose and Wang [3] – given r simple polygons with a total of m vertices along with the point set P, compute the smallest enclosing circle of P whose center lies in one of the r polygons. This can be solved in O( nlogn+mlog2 n) time using our method in a much simpler way than [3]; the time complexity of the problem is also being improved.


Intersection Point Leaf Node Space Complexity Voronoi Diagram Convex Polygon 
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.
    Aggarwal, A., Guibas, L.J., Saxe, J., Shor, P.W.: A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete Computational Geometry 4, 591–604 (1989)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bose, P., Toussaint, G.: Computing the constrained Euclidean, geodesic and link center of a simple polygon with applications. Studies of facility location analysis 15, 37–66 (2000)MATHMathSciNetGoogle Scholar
  3. 3.
    Bose, P., Wang, Q.: Facility location constrained to a polygonal domain. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 153–164. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Daescu, O., Mi, N., Shin, C., Wolff, A.: Furthest-point queries with geometric and combinatorial constraints. In: Computational Geometry: Theory and Applications (2006) (to appear)Google Scholar
  5. 5.
    Elzinga, J., Hearn, D.W.: Geometrical solutions to some minimax location problems. Transpotation Science 6, 379–394 (1972)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Hurtado, F., Sacristan, V., Toussaint, G.: Facility location problems with constraints. Studies in Locational Analysis 15, 17–35 (2000)MATHMathSciNetGoogle Scholar
  7. 7.
    Lee, D.T., Ching, Y.T.: The power of geometric duality revised. Information Processing Letters 21, 117–122 (1985)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lee, D.T.: Furthest neighbour Voronoi diagrams and applications, Report 80-11-FC-04, Dept. Elect. Engrg. Comput. Sci., Northwestern Univ., Evanston, IL (1980)Google Scholar
  9. 9.
    Megiddo, N.: Linear-time algorithms for linear programming in R 3 and related problems. SIAM J. Comput. 12, 759–776 (1983)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Preparata, F.P.: Minimum spanning circle, Technical report, Univ. Illinois, Urbana, IL, in Steps into Computational Geometry (1977)Google Scholar
  11. 11.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction, 2nd edn. Springer, Heidelberg (1990)Google Scholar
  12. 12.
    Shamos, M.I., Hoey, D.: Closest-point problem. In: Proc. 16th Annual IEEE Sympos. Found. Comput. Sci., pp. 151–162 (1975)Google Scholar
  13. 13.
    Sylvester, J.J.: A question in the geometry of situation. Quarterly Journal of Mathematices, 1–79 (1857)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sasanka Roy
    • 1
  • Arindam Karmakar
    • 1
  • Sandip Das
    • 2
  • Subhas C. Nandy
    • 1
  1. 1.Indian Statistical InstituteCalcuttaIndia
  2. 2.Institut de Mathématiques de BourgogneDijonFrance

Personalised recommendations