Optimal Metastability-Containing Sorting via Parallel Prefix Computation

TL;DR

This paper presents a novel circuit design for sorting Gray code inputs with metastability, achieving asymptotically optimal size, depth, and fan-out using parallel prefix computation, and verifies its effectiveness through simulations.

Contribution

It introduces the first metastability-containing sorting circuits with optimal size, depth, and constant fan-out based on parallel prefix computation.

Findings

Circuit sorts metastable Gray code inputs correctly.

Achieves asymptotically optimal size and depth.

Verifies correctness and efficiency via simulations.

Abstract

Friedrichs et al. (TC 2018) showed that metastability can be contained when sorting inputs arising from time-to-digital converters, i.e., measurement values can be correctly sorted without resolving metastability using synchronizers first. However, this work left open whether this can be done by small circuits. 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. Our solution utilizes the parallel prefix computation (PPC) framework (JACM 1980). We improve this construction by bounding its fan-out by an arbitrary $f \geq 3$, without affecting depth and increasing circuit size by a small constant factor only. Thus, we obtain the first PPC circuits with asymptotically optimal size, constant fan-out, and optimal depth. To show that applying the PPC framework to the sorting task is feasible, we prove that the latter can, despite potential metastability, be decomposed such that the core operation is associative. We obtain asymptotically optimal metastability-containing sorting networks. We complement these results with simulations, independently verifying the correctness as well as small size and delay of our circuits.