Hopsets with Constant Hopbound, and Applications to Approximate Shortest   Paths
  A $(\beta,\epsilon)$-hopset for a weighted undirected $n$-vertex graph $G=(V,E)$ is a set of edges, whose addition to the graph guarantees that every pair of vertices has a path between them that contains at most $\beta$ edges, whose length is within $1+\epsilon$ of the shortest path. In her seminal paper, Cohen \cite[JACM 2000]{C00} introduced the notion of hopsets in the context of parallel computation of approximate shortest paths, and since then it has found numerous applications in various other settings, such as dynamic graph algorithms, distributed computing, and the streaming model.   Cohen \cite{C00} devised efficient algorithms for constructing hopsets with {\em polylogarithmic} in $n$ number of hops. Her constructions remain the state-of-the--art since the publication of her paper in STOC'94, i.e., for more than two decades.   In this paper we exhibit the first construction of sparse hopsets with a {\em constant number of hops}. We also find efficient algorithms for hopsets in various computational settings, improving the best known constructions. Generally, our hopsets strictly outperform the hopsets of \cite{C00}, both in terms of their parameters, and in terms of the resources required to construct them.   We demonstrate the applicability of our results for the fundamental problem of computing approximate shortest paths from $s$ sources. Our results improve the running time for this problem in the parallel, distributed and streaming models, for a vast range of $s$. 
Michael Elkin and Ofer Neiman
cs.DS