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The Cost of Global Broadcast in Dynamic Radio Networks

We study the single-message broadcast problem in dynamic radio networks. We show that the time complexity of the problem depends on the amount of stability and connectivity of the dynamic network topology and on the adaptiveness of the adversary providing the dynamic topology. More formally, we model communication using the standard graph-based radio network model. To model the dynamic network, we use a generalization of the synchronous dynamic graph model introduced in [Kuhn et al., STOC 2010]. For integer parameters $T\geq 1$ and $k\geq 1$, we call a dynamic graph $T$-interval $k$-connected if for every interval of $T$ consecutive rounds, there exists a $k$-vertex-connected stable subgraph. Further, for an integer parameter $τ\geq 0$, we say that the adversary providing the dynamic network is $τ$-oblivious if for constructing the graph of some round $t$, the adversary has access to all the randomness (and states) of the algorithm up to round $t-τ$. As our main result, we show that for any $T\geq 1$, any $k\geq 1$, and any $τ\geq 1$, for a $τ$-oblivious adversary, there is a distributed algorithm to broadcast a single message in time $O\big(\big(1+\frac{n}{k\cdot\min\left\{τ,T\right\}}\big)\cdot n\log^3 n\big)$. We further show that even for large interval $k$-connectivity, efficient broadcast is not possible for the usual adaptive adversaries. For a $1$-oblivious adversary, we show that even for any $T\leq (n/k)^{1-\varepsilon}$ (for any constant $\varepsilon>0$) and for any $k\geq 1$, global broadcast in $T$-interval $k$-connected networks requires at least $Ω(n^2/(k^2\log n))$ time. Further, for a $0$ oblivious adversary, broadcast cannot be solved in $T$-interval $k$-connected networks as long as $T<n-k$.