Tight time bounds for the minimum local convex partition problem
JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
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We provide a simple and powerful combinatorial method for proving lower bounds for algebraic computation trees over algebraically closed fields of characteristic p0. We apply our method to prove, for example, an /spl Omega/(n log n) lower bound for the n element distinctness problem, an /spl Omega/(n log(n/k)) lower bound to the "k-equal problem"-that is deciding whether there are k identical elements out of n input elements, and more. The proof of the main theorem relies on the deep work of B.M. Dwork, P. Deligne, and E. Bombieri on the Weil conjectures. In particular we make use of Bombieri's bound on the degree of the Zeta function of algebraic varieties over finite fields. Our bounds provide a natural extension to the recent topological lower bounds obtained by A. Bjorner, L. Lovasz and A.C. Yao for algebraic computation trees over the real numbers. For the special cases of real subspace arrangements and general complex varieties we can reformulate their specific results using our combinatorial approach without mentioning any topological invariants.