Math @ Duke

Publications [#235402] of Pankaj K. Agarwal
Papers Published
 Agarwal, PK; Schwarzkopf, O; Sharir, M, The overlay of lower envelopes and its applications,
Discrete & Computational Geometry, vol. 15 no. 1
(1996),
pp. 113, ISSN 01795376
(last updated on 2017/12/16)
Abstract: Let ℱ and G be two collections of a total of n (possibly partially defined) bivariate, algebraic functions of constant maximum degree. The minimization diagrams of ℱ, G are the planar maps obtained by the xyprojections of the lower envelopes of ℱ, G, respectively. We show that the combinatorial complexity of the overlay of the minimization diagrams of ℱ and of G is O(n2+ε), for any ε > 0. This result has several applications: (i) a nearquadratic upper bound on the complexity of the region in 3space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an efficient and simple divideandconquer algorithm for constructing lower envelopes in three dimensions; and (iii) a nearquadratic upper bound on the complexity of the space of all plane transversals of a collection of simply shaped convex sets in three dimensions.


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