Geodesic Ham-Sandwich Cuts

Authors:
Prosenjit Bose;Erik D. Demaine;Ferran Hurtado;John Iacono;Stefan Langerman;Pat Morin
Affiliations:
School of Computer Science, Carleton University, Ottawa, Ontario, Canada K1S 5B6;Laboratory for Computer Science, MIT, Cambridge, MA 02139, USA;Departament de Matematica Aplicada II, Universitat Politecnica de Catalunya, 08028 Barcelona, Spain;Department of Computer and Information Sciences, Brooklyn Polytechnic University, Six Metro Tech Center, Brooklyn, NY 11201, USA;Chercheur Qualifie du FNRS, Universite Libre de Bruxelles, boulevard du Triomphe, 1050 Bruxelles, Belgium;School of Computer Science, Carleton University, Ottawa, Ontario, Canada K1S 5B6
Venue:
Discrete & Computational Geometry
Year:
2007

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Abstract

Let P be a simple polygon with m vertices, k of which are reflex, and which contains r red points and b blue points in its interior. Let n = m + r + b. A ham-sandwich geodesic is a shortest path in P between two points on the boundary of P that simultaneously bisects the red points and the blue points. We present an O(n log k)-time algorithm for finding a ham-sandwich geodesic. We also show that this algorithm is optimal in the algebraic computation tree model when parameterizing the running time with respect to n and k.