Dispersion of Mass and the Complexity of Randomized Algorithms

Santosh Vempala
Georgia Tech

Friday, October 13th, 2006, 3:30 pm
Wean Hall 7220

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 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.

This is joint work with Luis Rademacher.