Reducing Randomness via Irrational Numbers

TL;DR

This paper introduces a methodology that reduces randomness in polynomial identity testing by using approximations of irrational points, leading to more efficient randomized algorithms with fewer random bits.

Contribution

The paper presents a novel approach to polynomial identity testing that decreases error probability through increased approximation precision, improving existing randomized algorithms.

Findings

Fewer random bits needed in polynomial testing algorithms.

Improved NC algorithm for perfect matching detection.

Enhanced algorithms for multiset equality testing.

Abstract

We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodology, we discuss two nontrivial applications. The first is to decide whether a graph has a perfect matching in parallel. Our new NC algorithm uses fewer random bits while doing less work than the previously best NC algorithm by Chari, Rohatgi, and Srinivasan. The second application is to test the equality of two multisets of integers. Our new algorithm improves upon the previously best algorithms by Blum and Kannan and can speed up their checking algorithm for sorting programs on a large range of inputs.