Discrete & Computational Geometry

, Volume 26, Issue 2, pp 245–265

A Randomized Algorithm for Triangulating a Simple Polygon in Linear Time

  • N. M. Amato
  • M. T. Goodrich
  • E. A. Ramos

DOI: 10.1007/s00454-001-0027-x

Cite this article as:
Amato, N., Goodrich, M. & Ramos, E. Discrete Comput Geom (2001) 26: 245. doi:10.1007/s00454-001-0027-x


We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle’s [3] celebrated optimal deterministic algorithm. The new algorithm can be viewed as a combination of Chazelle’s algorithm and of simple nonoptimal randomized algorithms due to Clarkson et al. [6], [7], [9] and to Seidel [20]. As in Chazelle’s algorithm, it is indispensable to include a bottom-up preprocessing phase, in addition to the actual top-down construction. An essential new idea is the use of random sampling on subchains of the initial polygonal chain, rather than on individual edges as is normally done.

Copyright information

© Springer-Verlag New York Inc. 2001

Authors and Affiliations

  • N. M. Amato
    • 1
  • M. T. Goodrich
    • 2
  • E. A. Ramos
    • 3
  1. 1.Department of Computer Science, Texas A&M University, College Station, TX 77843, USA amato@cs.tamu.edu USA
  2. 2.Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218, USA goodrich@jhu.edu USA
  3. 3.Max-Planck-Institut für Informatik, Im Stadtwald, D-66401 Saarbrücken, Germany ramos@mpi-sb.mpg.deGermany