Computing Shortest Paths and Diameter in the Hybrid Network Model

The $\mathsf{HYBRID}$ model, introduced in [Augustine et al., SODA '20], provides a theoretical foundation for networks that allow multiple communication modes. The model follows the principles of synchronous message passing, whereas nodes are allowed to use \textit{two} fundamentally different communication modes. First, a local mode where nodes may exchange arbitrary information per round over edges of a local communication graph $G$ (akin to the $\mathsf{LOCAL}$ model). Second, a global mode where every node may exchange $O(\log n)$ messages of size $O(\log n)$ bits per round with arbitrary nodes in the network. The $\mathsf{HYBRID}$ model intends to reflect the conditions of many real hybrid networks, where high-bandwidth but inherently local communication is combined with highly flexible global communication with restricted bandwidth. We continue to explore the power and limitations of the $\mathsf{HYBRID}$ model by investigating the complexity of computing shortest paths and diameter of the local communication graph $G$. We improve on the known upper bound for the exact all pairs shortest paths problem (APSP) from [Augustine et al., SODA '20] and provide algorithms to approximate solutions for the $k$ source shortest paths problem ($k$-SSSP). We demonstrate that our results for APSP and $k$-SSP are almost tight (up to poly-logarithmic factors). Furthermore, we give an improved algorithm for the exact single source shortest paths problem for graphs with large diameter. For the problem of approximating the diameter of the local communication network we give the first non-trivial upper bound. This upper bound is complemented by a lower bound for the exact diameter problem.