Near-linear cost sequential and distributed constructions of sparse neighborhood covers

This paper introduces the first near-linear (specifically, O(Elog n+nlog² n)) time algorithm for constructing a sparse neighborhood cover in sequential and distributed environments. This automatically implies analogous improvements (from quadratic to near-linear) to all the results in the literature that rely on network decompositions, both in sequential and distributed domains, including adaptive routing schemes with O˜(1) stretch and memory, small edge cuts in planar graphs, sequential algorithms for dynamic approximate shortest paths with O˜(E) cost for edge insertion/deletion and O˜(1) time to answer shortest-path queries, weight and distance-preserving graph spanners with O˜(E) running time and space, and distributed asynchronous “from-scratch” breadth-first-search and network synchronizer constructions with O˜(1) message and space overhead (down from O(n))