Dispersion of Mass and the Complexity of Randomized Geometric Algorithms

Luis Rademacher; Santosh Vempala

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arXiv · cs/0608054

Dispersion of Mass and the Complexity of Randomized Geometric Algorithms

Luis Rademacher·Santosh Vempala

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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.

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BibTeX RIS

Luis Rademacher, Santosh Vempala. 2008-06-17. Dispersion of Mass and the Complexity of Randomized Geometric Algorithms. https://arxiv.org/abs/cs/0608054

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