On Overlays and Minimization Diagrams

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 complexity of the overlay of minimization diagrams of collections of hyperplanes, and faster algorithms for constructing such overlays. Further algorithmic implications are discussed.

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