On overlays and minimization diagrams

Authors:
Vladlen Koltun;Micha Sharir
Affiliations:
Stanford University, Stanford, CA;Tel Aviv University, Tel-Aviv, Israel and New York University, New York, NY
Venue:
Proceedings of the twenty-second annual symposium on Computational geometry
Year:
2006

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Abstract

The overlay of 2≤m≤d minimization diagrams of n surfaces in Rd is isomorphic to a substructure of a suitably constructed minimization diagram of mn surfaces in Rd+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 compleity of the overlay of minimization diagrams of collections of hyperplanes, and faster algorithms for constructing such overlays. Further algoithmic implications are discussed.