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 3f≥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. Proofs are omitted in this version; the article with full proofs is provided online at http://arxiv.org/abs/1911.00267.
|Number of pages||14|
|Journal||IEEE Transactions on Computers|
|State||Published - 1 Feb 2020|
Bibliographical noteFunding Information:
We thank Attila Kinali and the anonymous reviewers for valuable input. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement 716562). This article generalizes and extends work presented at DATE 2018 . This research was supported by the Israel Science Foundation under Grant 867/19.
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