Optimal Metastability-Containing Sorting Networks

TL;DR

This paper introduces a metastability-containing sorting network that sorts Gray code inputs with metastable bits, achieving asymptotically optimal depth and size, and significantly improving delay and area over previous solutions.

Contribution

It presents the first metastability-containing sorting network with optimal asymptotic depth and size, outperforming prior work in delay and area for specific configurations.

Findings

48.46% delay improvement for 10-channel, 16-bit inputs

71.58% area reduction compared to previous solutions

Simulations suggest transistor-level optimization can match standard solutions

Abstract

When setup/hold times of bistable elements are violated, they may become metastable, i.e., enter a transient state that is neither digital 0 nor 1. In general, metastability cannot be avoided, a problem that manifests whenever taking discrete measurements of analog values. Metastability of the output then reflects uncertainty as to whether a measurement should be rounded up or down to the next possible measurement outcome. Surprisingly, Lenzen and Medina (ASYNC 2016) showed that metastability can be contained, i.e., measurement values can be correctly sorted without resolving metastability first. However, both their work and the state of the art by Bund et al. (DATE 2017) leave open whether such a solution can be as small and fast as standard sorting networks. We show that this is indeed possible, by providing a circuit that sorts Gray code inputs (possibly containing a metastable bit) and has asymptotically optimal depth and size. Concretely, for 10-channel sorting networks and 16-bit wide inputs, we improve by 48.46% in delay and by 71.58% in area over Bund et al. Our simulations indicate that straightforward transistor-level optimization is likely to result in performance on par with standard (non-containing) solutions.