On Overlays and Minimization Diagrams

Authors:
Vladlen Koltun;Micha Sharir
Affiliations:
Stanford University, Computer Science Department, 353 Serra Mall, 94305, Stanford, CA, USA;Tel Aviv University, School of Computer Science, 69978, Tel Aviv, Israel and New York University, Courant Institute of Mathematical Sciences, New York, NY 10012, USA
Venue:
Discrete & Computational Geometry
Year:
2009

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Geometric optimization and sums of algebraic functions

SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Geometric optimization and sums of algebraic functions

ACM Transactions on Algorithms (TALG)

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Abstract

The overlay of 2≤m≤d minimization diagrams of n surfaces in ℝ d is isomorphic to a substructure of a suitably constructed minimization diagram of mn surfaces in ℝ d+m−1. This elementary observation leads to a new bound on the complexity of the overlay of minimization diagrams of collections of d-variate semi-algebraic surfaces, a tight bound on the complexity of the overlay of minimization diagrams of collections of hyperplanes, and faster algorithms for constructing such overlays. Further algorithmic implications are discussed.