Faster Deterministic Communication in Radio Networks

We study the communication primitives of broadcasting (one-to-all communication) and gossiping (all-to-all communication) in known topology radio networks, i.e., where for each primitive the schedule of transmissions is precomputed based on full knowledge about the size and the topology of the network. We show that gossiping can be completed in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(D+\frac{\varDelta\log n}{\log{\varDelta}-\log{\log n}})$\end{document} time units in any radio network of size n, diameter D, and maximum degree Δ=Ω(log n). This is an almost optimal schedule in the sense that there exists a radio network topology, specifically a Δ-regular tree, in which the radio gossiping cannot be completed in less than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varOmega(D+\frac{\varDelta\log n}{\log{\varDelta}})$\end{document} units of time. Moreover, we show a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D+O(\frac{\log^{3}n}{\log{\log n}})$\end{document} schedule for the broadcast task. Both our transmission schemes significantly improve upon the currently best known schedules by Gąsieniec, Peleg, and Xin (Proceedings of the 24th Annual ACM SIGACT-SIGOPS PODC, pp. 129–137, 2005), i.e., a O(D+Δlog n) time schedule for gossiping and a D+O(log 3n) time schedule for broadcast. Our broadcasting schedule also improves, for large D, a very recent O(D+log 2n) time broadcasting schedule by Kowalski and Pelc.