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Query Complexity of Sampling and Small Geometric Partitions

Published online by Cambridge University Press:  21 October 2014

NAVIN GOYAL ,
LUIS RADEMACHER  and
SANTOSH VEMPALA
NAVIN GOYAL
Affiliation:
Microsoft Research India, Vigyan, 9 Lavelle Road, Bangalore 560001, India (e-mail: navingo@microsoft.com)
LUIS RADEMACHER
Affiliation:
Computer Science and Engineering, Ohio State University, Dreese Labs 495 2015 Neil Avenue, Columbus, OH 43210, USA (e-mail: lrademac@cse.ohio-state.edu)
SANTOSH VEMPALA
Affiliation:
College of Computing, Georgia Institute of Technology, 801 Atlantic Drive, Atlanta, GA 30332, USA (e-mail: vempala@cc.gatech.edu)
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Abstract

In this paper we study the following problem.

Discrete partitioning problem (DPP). Let $\mathbb{F}_q$Pn denote the n-dimensional finite projective space over $\mathbb{F}_q$. For positive integer k ⩽ n, let {Ai}i = 1N be a partition of ($\mathbb{F}_q$Pn)k such that:

  1. (1) for all i ⩽ N, Ai = ∏j=1kAji (partition into product sets),

  2. (2) for all i ⩽ N, there is a (k − 1)-dimensional subspace Li ⊆ $\mathbb{F}_q$Pn such that Ai ⊆ (Li)k.

What is the minimum value of N as a function of q, n, k? We will be mainly interested in the case k = n.

DPP arises in an approach that we propose for proving lower bounds for the query complexity of generating random points from convex bodies. It is also related to other partitioning problems in combinatorics and complexity theory. We conjecture an asymptotically optimal partition for DPP and show that it is optimal in two cases: when the dimension is low (k = n = 2) and when the factors of the parts are structured, namely factors of a part are close to being a subspace. These structured partitions arise naturally as partitions induced by query algorithms. Our problem does not seem to be directly amenable to previous techniques for partitioning lower bounds such as rank arguments, although rank arguments do lie at the core of our techniques.

Keywords

Primary 68Q17Secondary 68R0551E20
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 5 , September 2015 , pp. 733 - 753
Copyright
Copyright © Cambridge University Press 2014 

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