Distance Computations in the Hybrid Network Model via Oracle Simulations

The Hybrid network model was introduced in [Augustine et al., SODA '20] for laying down a theoretical foundation for networks which combine two possible modes of communication: One mode allows high-bandwidth communication with neighboring nodes, and the other allows low-bandwidth communication over few long-range connections at a time. This fundamentally abstracts networks such as hybrid data centers, and class-based software-defined networks. Our technical contribution is a \emph{density-aware} approach that allows us to simulate a set of \emph{oracles} for an overlay skeleton graph over a Hybrid network. As applications of our oracle simulations, with additional machinery that we provide, we derive fast algorithms for fundamental distance-related tasks. One of our core contributions is an algorithm in the Hybrid model for computing \emph{exact} weighted shortest paths from $\tilde O(n^{1/3})$ sources which completes in $\tilde O(n^{1/3})$ rounds w.h.p. This improves, in both the runtime and the number of sources, upon the algorithm of [Kuhn and Schneider, PODC '20], which computes shortest paths from a single source in $\tilde O(n^{2/5})$ rounds w.h.p. We additionally show a 2-approximation for weighted diameter and a $(1+\epsilon)$-approximation for unweighted diameter, both in $\tilde O(n^{1/3})$ rounds w.h.p., which is comparable to the $\tilde \Omega(n^{1/3})$ lower bound of [Kuhn and Schneider, PODC '20] for a $(2-\epsilon)$-approximation for weighted diameter and an exact unweighted diameter. We also provide fast distance \emph{approximations} from multiple sources and fast approximations for eccentricities.