%AFanti, Giulia [Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213,]%AFanti, Giulia%AHolden, Nina%AHolden, Nina [Institute for Theoretical Studies, Swiss Federal Institute of Technology (ETH) Zürich, 8092 Zürich, Switzerland,]%APeres, Yuval [Microsoft Research, Redmond, WA 98052,]%APeres, Yuval%ARanade, Gireeja [Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720]%ARanade, Gireeja%BJournal Name: Proceedings of the National Academy of Sciences; Journal Volume: 117; Journal Issue: 11; Related Information: CHORUS Timestamp: 2023-09-27 23:40:14
%D2020%IProceedings of the National Academy of Sciences
%JJournal Name: Proceedings of the National Academy of Sciences; Journal Volume: 117; Journal Issue: 11; Related Information: CHORUS Timestamp: 2023-09-27 23:40:14
%K
%MOSTI ID: 10137950
%PMedium: X
%TCommunication cost of consensus for nodes with limited memory
%XMotivated by applications in wireless networks and the Internet of Things, we consider a model of$n$nodes trying to reach consensus with high probability on their majority bit. Each node$i$is assigned a bit at time 0 and is a finite automaton with$m$bits of memory (i.e.,${2}^{m}$states) and a Poisson clock. When the clock of$i$rings,$i$can choose to communicate and is then matched to a uniformly chosen node$j$. The nodes$j$and$i$may update their states based on the state of the other node. Previous work has focused on minimizing the time to consensus and the probability of error, while our goal is minimizing the number of communications. We show that, when$m>3\mathrm{log}\mathrm{log}\mathrm{log}\left(n\right)$, consensus can be reached with linear communication cost, but this is impossible if$m<\mathrm{log}\mathrm{log}\mathrm{log}\left(n\right)$. A key step is to distinguish when nodes can become aware of knowing the majority bit and stop communicating. We show that this is impossible if their memory is too low.

%0Journal Article