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

This paper introduces the first near-linear (specifically, O(E log n+n log² 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)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)).