Improved Weighted Additive Spanners

Abstract

Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount of work on additive spanners, so far little attention was given to weighted graphs. Only very recently \\cite{DBLP:journals/corr/abs-1907-11422,DBLP:journals/corr/abs-2002-07152} extended the classical +2 (respectively, +4) spanners for unweighted graphs of size $O(n^{3/2})$ (resp., $O(n^{7/5})$) to the weighted setting, where the additive error is $+2W$ (resp., $+4W$). This means that for every pair $u,v$, the additive stretch is at most $+2W_{u,v}$, where $W_{u,v}$ is the maximal edge weight on the shortest $u-v$ path (weights are normalized so that the minimum edge weight is 1). In addition, \\cite{DBLP:journals/corr/abs-2002-07152} showed a randomized algorithm yielding a $+8W_{max}$ spanner of size $O(n^{4/3})$, here $W_{max}$ is the maximum edge weight in the entire graph. \nIn this work we improve the latter result by devising a simple deterministic algorithm for a $+(6+\\varepsilon)W$ spanner for weighted graphs with size $O(n^{4/3})$ (for any constant $\\varepsilon>0$), thus nearly matching the classical +6 spanner of size $O(n^{4/3})$ for unweighted graphs. We also show a simple randomized algorithm for a $+4W$ emulator of size $\\tilde{O}(n^{4/3})$.