This is an announcement for the paper "Dispersion of mass and the complexity of randomized geometric algorithms" by Luis Rademacher, Santosh Vempala.
Abstract: How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R^n (the current best algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing.
Archive classification: Computational Complexity; Computational Geometry; Data Structures; Functional Analysis
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http://arXiv.org/abs/cs.CC/0608054
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