Center for Algorithms and Theory of Computation

CS 269S, Spring 2021: Theory Seminar


May 21, 2021, 1:00 – 1:50:

On Additive Spanners in Weighted Graphs with Local Error

Hadi Khodabandeh

An additive +β spanner of a graph G is a subgraph which preserves distances up to an additive +β error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al. 2019 and 2020, Ahmed et al. 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al. 2020] provided constructions of sparse spanners with global error β=cW, where W is the maximum edge weight in G and c is constant. We improve these to local error by giving spanners with additive error +cW(s,t) for each vertex pair (s,t), where W(s,t) is the maximum edge weight along the shortest st path in G. These include pairwise +(2+ε)W(,) and +(6+ε)W(,) spanners over vertex pairs PV×V on Oε(n|P|1/3) and Oε(n|P|1/4) edges for all ε>0, which extend previously known unweighted results up to ε dependence, as well as an all-pairs +4W(,) spanner on O~(n7/5) edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its lightness, defined as the total edge weight of the spanner divided by the weight of a minimum spanning tree of G. We provide a +εW(,) spanner with Oε(n) lightness, and a +(4+ε)W(,) spanner with Oε(n2/3) lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.

Based on a paper by Reyan Ahmed, Greg Bodwin, Keaton Hamm, Stephen Kobourov, and Richard Spence, arXiv:2103.09731.