Shortest Paths in a Hybrid Network Model.

We introduce a communication model for hybrid networks, where nodes have access to two different communication modes: a local mode where (like in traditional networks) communication is only possible between specific pairs of nodes, and a global mode where (like in overlay networks) communication between any pair of nodes is possible. Typically, communication over short-range connections is cheaper and can be done at a much higher rate than communication via the overlay network. Therefore, we are focusing on the LOCAL model for the local connections where nodes can exchange an unbounded amount of information per round. For the global communication we assume the so-called node-capacitated clique model, where in each round every node can exchange O(log n)-bit messages with O(log n) arbitrary nodes. We explore the impact of hybrid communication on the complexity of distributed algorithms by studying the problem of computing shortest paths in the graph given by the local connections. We present the following results. For the all-pairs shortest paths problem, we show that an exact solution can be computed in time Õ(n2/3) and that approximate solutions can be computed in time [MATH HERE] but not faster. For the single-source shortest paths problem an exact solution can be computed in time [MATH HERE], where SPD denotes the shortest path diameter. Furthermore, a (1 + o(1))-approximate solution can be computed in time Õ(n1/3). Finally, we show that for every constant ε > 0, it is possible to compute an O(1)-approximate solution in time Õ(nε).