Isolating an Odd Number of Elements and Applications in Complexity Theory

Abstract

Naor and Naor [11] implicitly isolate an odd number of elements of a nonempty set of n -bit vectors. We perform a tighter analysis of their construction and obtain better probability bounds. Using this construction, we improve bounds on several results in complexity theory that originally used a construction due to Valiant and Vazirani [18]. In particular, we obtain better bounds on polynomials which approximate boolean functions; improve bounds on the running time of the ⊕ P machine in Toda's result [16]; and improve bounds on the size of threshold circuits accepting languages accepted by AC⁰ circuits.