Vertex coloring is one of the classic symmetry breaking problems studied in distributed computing. In this paper we present a new algorithm for (∆ + 1)-list coloring in the randomized LOCAL model running in O(Detd(poly log n)) time, where Detd(n ) is the deterministic complexity of (deg +1)-list coloring on n -vertex graphs. (In this problem, each v has a palette of size deg(v) + 1.) This improves upon a previous randomized algorithm of Harris, Schneider, and Su [STOC 2016, J. ACM 2018] with complexity O( √ log ∆ + log log n + Detd(poly log n)), and, for some range of ∆, is much faster than the best known deterministic algorithm of Fraigniaud, Heinrich, and Kosowski [FOCS 2016] and Barenboim, Elkin, and Goldenberg [PODC 2018], with complexity O( √ ∆ log ∆ log * ∆ + log * n). Our algorithm appears to be optimal, in view of the Ω(Det(poly log n)) randomized lower bound due to Chang, Kopelowitz, and Pettie [FOCS 2016, SIAM J. Comput. 2019], where Det is the deterministic complexity of (∆ + 1)-list coloring. At present, the best upper bounds on Detd(n ) and Det(n ) are both 2 O √ log n and use a black box application of network decompositions due to Panconesi and Srinivasan [J.Algorithms 1996]. It is quite possible that the true deterministic complexities of both problems are the same, asymptotically, which would imply the randomized optimality of our (∆ + 1)-list coloring algorithm.1 In the case of MIS, the subproblems actually have size poly(∆) log n, but satisfy the additional property that they contain distance-5 dominating sets of size O(log n), which is often just as good as having poly log(n) size. See [9, §3] or [21, §4] for more discussion of this.2 See [33,14,12] for the formal definition of the class of locally checkable labeling (LCL) problems.

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Brain–computer interfaces (BCIs) enable direct and near-instant communication between the brain and electronic devices. One of the biggest remaining challenges is to develop an effective noninvasive BCI that allows the recording electrodes to avoid hair on human skin without the inconveniences and complications of using a conductive gel. In this study, we developed a cost-effective, easily manufacturable, flexible, robust, and gel-free silver nanowire/polyvinyl butyral (PVB)/melamine sponge (AgPMS) electroencephalogram (EEG) electrode that circumvents problems with hair. Because of surface metallization by the silver nanowires (AgNWs), the sponge has a high conductivity of 917 S/m while its weight remains the same. The flexible sponge framework and self-locking AgNWs combine to give the new electrode remarkable mechanical stability (the conductivity remains unchanged after 10 000 cycles at 10% compression) and the ability to bypass hair. A BCI application based on steady-state visual evoked potential (SSVEP) measurements on hairless skin shows that the BCI accuracy of the new electrode (86%) is approximately the same as that of conventional electrodes supported by a conductive gel (88%). Most importantly, the performance of the AgPMS on hairy skin is not significantly reduced, which indicates that the new electrode can replace conventional electrodes for both hairless and hairy skin BCIs and other EEG applications.

A Si/H-rGO composite film was prepared through self-assembly and the “breath figure” method. The electrode exhibits good cycling stability and enhanced rate capability.