Parity Tests with Ties

TL;DR

This paper extends the Ting--Yao maximum-finding algorithm to handle inputs with ties, increasing the depth complexity while maintaining a low failure probability.

Contribution

It introduces a method to simulate parity tests with ties using polynomial tests, improving the algorithm's applicability and depth complexity.

Findings

Depth increases from O((log n)^2) to O((log n)^3)

Maintains O(n^{-c}) failure probability for fixed c

Handles inputs with ties in maximum-finding algorithms

Abstract

We extend the Ting--Yao randomized maximum-finding algorithm [TY94] to inputs that need not be pairwise distinct: each parity test $P(i,B)=\prod_{a\in B}(x_i-x_a):0$ on $B\subseteq[n]\setminus\{i\}$ is simulated by $O(\log |B|)$ ordinary polynomial tests, raising depth from $O((\log n)^2)$ to $O((\log n)^3)$ while preserving the $O(n^{-c})$ failure probability for every fixed $c>0$.