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Gossiping with bounded size messages in ad hoc radio networks (extended abstract). (English)
Widmayer, Peter (ed.) et al., Automata, languages and programming. 29th international colloquium, ICALP 2002, Málaga, Spain, July 8‒13, 2002. Proceedings. Berlin: Springer (ISBN 3-540-43864-5). Lect. Notes Comput. Sci. 2380, 377-389 (2002).
Summary: We study deterministic algorithms for the gossiping problem in ad hoc radio networks under the assumption that each combined message contains at most $b(n)$ single messages or bits of auxiliary information, where $b$ is an integer function and $n$ is the number of nodes in the network. We term such a restricted gossiping problem $b(n)$-gossiping. We show that $\sqrt n$-gossiping in an ad hoc radio network on $n$ nodes can be done deterministically in time $\tilde O(n^{3/2})$ which asymptotically matches the best known upper bound on the time complexity of unrestricted deterministic gossiping. Notation $\tilde O(f(n))$ stands for $O(f(n)\cdot\log^c n)$, for any constant $c>0$. Our upper bound on $\sqrt n$-gossiping is tight up to a poly-logarithmic factor and it implies similarly tight upper bounds on $b(n)$-gossiping where function $b$ is computable and $1\le b(n)\le \sqrt n$ holds. For symmetric ad hoc radio networks, we show that even $1$-gossiping can be done deterministically in time $\tilde O(n^{3/2})$. We also demonstrate that $O(n^t)$-gossiping in a symmetric ad hoc radio network on $n$ nodes can be done in time $O(n^{2-t})$. Note that the latter upper bound is $o(n^{3/2})$ when the size of a combined message is $ω(n^{1/2})$. Furthermore, by adopting known results on repeated randomized broadcasting in symmetric ad hoc radio networks, we derive a randomized protocol for 1-gossiping in these networks running in time $\tilde O(n)$ on the average. Finally, we observe that when a collision detection mechanism is available, even deterministic 1-gossiping in symmetric ad hoc radio networks can be performed in time $\tilde O(n)$.