Near-Linear Time Construction of Sparse Neighborhood Covers

This paper introduces a near-linear time sequential algorithm for constructing a sparse neighborhood cover. This implies analogous improvements (from quadratic to near-linear time) for any problem whose solution relies on network decompositions, including small edge cuts in planar graphs, approximate shortest paths, and weight- and distance-preserving graph spanners. In particular, an O(log n) approximation to the k-shortest paths problem on an n-vertex, E-edge graph is obtained that runs in $\soh{n + E + k}$ time.