We consider the problem of computing routing schemes in the $\mathsf{HYBRID}$ model of distributed computing where nodes have access to two fundamentally different communication modes. In this problem nodes have to compute small labels and routing tables that allow for efficient routing of messages in the local network, which typically offers the majority of the throughput. Recent work has shown that using the $\mathsf{HYBRID}$ model admits a significant speed-up compared to what would be possible if either communication mode were used in isolation. Nonetheless, if general graphs are used as the input graph the computation of routing schemes still takes polynomial rounds in the $\mathsf{HYBRID}$ model. We bypass this lower bound by restricting the local graph to unit-disc-graphs and solve the problem deterministically with running time $O(|\mathcal H|^2 \!+\! \log n)$, label size $O(\log n)$, and size of routing tables $O(|\mathcal H|^2 \!\cdot\! \log n)$ where $|\mathcal H|$ is the number of ``radio holes'' in the network. Our work builds on recent work by Coy et al., who obtain this result in the much simpler setting where the input graph has no radio holes. We develop new techniques to achieve this, including a decomposition of the local graph into path-convex regions, where each region contains a shortest path for any pair of nodes in it.
翻译:我们考虑在 $\ mathsf{ Hybreid} 分布式计算模型中计算路由方案的问题, 因为节点可以使用两种截然不同的通信模式。 在此问题上, 问题节点必须计算小标签和路由表, 以便本地网络中的信息高效路由, 通常提供大部分输送量。 最近的工作显示, 使用 $\ mathsf{ Hybriid} 模式可以大大加快速度 。 与 如果在孤立状态中使用更简化的平面计算 。 然而, 如果使用普通图表作为输入图, 则 路由普通图表计算仍然使用 $\ mathfsf{ Hybrid} 模式中的多盘路段。 我们绕过此小路段, 将本地图限制为单位分解图, 并用运行时间 $( mathcal H<unk> 2\\\\\\\\\\\\\\\\\\\\\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ lax lax lax lax lax \ \ \ \ \ \ \ \ lax lax \ \ lax \ lax lax \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ lax \ \ \ \ \ \ lax \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ la \ \ \ \ \ \ la \ \ la la la la</s>