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：我们考虑在分布式计算的 $\mathsf{HYBRID}$ 模型中计算路由方案的问题，其中节点具有两种根本不同的通信模式。在该问题中，节点需要计算较小的标签和路由表，以便在本地网络（通常提供大部分吞吐量）中实现消息的高效路由。近期研究表明，与单独使用任一通信模式相比，采用 $\mathsf{HYBRID}$ 模型可显著加速。然而，若输入图为一般图，则在 $\mathsf{HYBRID}$ 模型中计算路由方案仍需多项式轮次。我们通过将局部图限制为单位圆盘图来绕过这一下界，并确定性求解该问题，其运行时间为 $O(|\mathcal H|^2 \!+\! \log n)$，标签大小为 $O(\log n)$，路由表大小为 $O(|\mathcal H|^2 \!\cdot\! \log n)$，其中 $|\mathcal H|$ 为网络中“无线电空洞”的数量。我们的工作基于 Coy 等人近期的工作，该工作在输入图无无线电空洞的简化场景中取得了这一结果。我们开发了新的技术以实现该目标，包括将局部图分解为路径凸区域，其中每个区域包含任意节点对之间的最短路径。