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 density-aware approach that allows us to simulate a set of 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 exact weighted shortest paths from Õ (n1/3) sources which completes in Õ(n1/3) 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 Õ (n2/5) rounds w.h.p. We additionally show a 2-approximation for weighted diameter and a (1 - ϵ)-approximation for unweighted diameter, both in Õ (n1/3) rounds w.h.p., which is comparable to the Ω(n1/3) lower bound of [Kuhn and Schneider, PODC '20] for a (2 - ϵ)-approximation for weighted diameter and an exact unweighted diameter. We also provide fast distance approximations from multiple sources and fast approximations for eccentricities.