Dispersion of Mass and the Complexity of Randomized Geometric Algorithms

TL;DR

This paper investigates how randomness influences computational complexity in geometric algorithms, establishing lower bounds for volume computation and connecting dispersion measures to convex geometry and matrix problems.

Contribution

It introduces a new notion of dispersion related to randomized algorithms and derives nearly quadratic lower bounds for volume computation complexity.

Findings

Nearly quadratic lower bound on randomized volume algorithms

Dispersion measures relate to convex geometry and matrix problems

Tools applicable to broader computational problems

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.