Computing Largest Circles Separating Two Sets of Segments

Jean-Daniel Boissonnat (1) , Jurek Czyzowicz (2) , Olivier Devillers (1) , Jorge Urrutia (3) , Mariette Yvinec (1)

1 PRISME - Geometry, Algorithms and Robotics
2 DII - Département d'Informatique et d'Ingénierie
3 University of Ottawa [Ottawa]

A circle C separates two planar sets if it encloses one of the sets and its open interior disk does not meet the other set. A separating circle is a largest one if it cannot be locally increased while still separating the two given sets. An Theta(n log n) optimal algorithm is proposed to find all largest circles separating two given sets of line segments when line segments are allowed to meet only at their endpoints. In the general case, when line segments may intersect Omega(n^2) times, our algorithm can be adapted to work in O(n alpha(n) log n) time and O(n alpha(n)) space, where alpha(n) represents the extremely slowly growing inverse of the Ackermann function.

Domaines: Géométrie algorithmique [cs.CG]

Jean-Daniel Boissonnat, Jurek Czyzowicz, Olivier Devillers, Jorge Urrutia, Mariette Yvinec. Computing Largest Circles Separating Two Sets of Segments. International Journal of Computational Geometry and Applications, 2000, 10, pp.41--54. ⟨10.1142/S0218195900000036⟩. ⟨inria-00338701⟩