Finding large sticks and potatoes in polygons

  • Authors:

  • Olaf Hall-Holt;Matthew J. Katz;Piyush Kumar;Joseph S. B. Mitchell;Arik Sityon

  • Affiliations:

  • St. Olaf College, Northfield, MN;Ben-Gurion University, Beer-Sheva, Israel;Florida State University, Tallahassee, FL;Stony Brook University, Stony Brook, NY;Ben-Gurion University, Beer-Sheva, Israel

  • Venue:
  • SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
  • Year:
  • 2006

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Abstract

We study a class of optimization problems in polygons that seek to compute the "largest" subset of a prescribed type, e.g., a longest line segment ("stick") or a maximum-area triangle or convex body ("potato"). Exact polynomial-time algorithms are known for some of these problems, but their time bounds are high (e.g., O(n7) for the largest convex polygon in a simple n-gon). We devise efficient approximation algorithms for these problems. In particular, we give near-linear time algorithms for a (1 - ∈)-approximation of the biggest stick, an O(1)-approximation of the maximum-area convex body, and a (1 - ∈)-approximation of the maximum-area fat triangle or rectangle. In addition, we give efficient methods for computing large ellipses inside a polygon (whose vertices are a dense sampling of a closed smooth curve). Our algorithms include both deterministic and randomized methods, one of which has been implemented (for computing large area ellipses in a well sampled closed smooth curve).