Finding 4-Additive Spanners: Faster, Stronger, and Simpler
Chuhan Qi
TL;DR
This work tackles the problem of distance-preserving graph sparsification using $4$-additive spanners. It replaces prior randomized techniques with a deterministic, domination-based derandomization inspired by high-degree elimination and BFS-augmentation, and then reduces the problem to constructing a $5$-additive spanner followed by a bipartite reduction. The authors obtain a deterministic algorithm that outputs a $4$-additive spanner with $ ilde{O}(n^{7/5})$ edges in $ ilde{O}( ilde{min\{m n^{3/5}, n^{11/5}\}})$ time, matching the known edge bound while improving running time and simplifying the approach. This contributes to practical graph sparsification for distance oracles and network design, and motivates further work to close gaps to lower bounds and to improve time bounds further. Overall, the paper delivers a simpler, fully deterministic construction with strong theoretical guarantees for 4-additive spanners.
Abstract
Additive spanners are fundamental graph structures with wide applications in network design, graph sparsification, and distance approximation. In particular, a $4$-additive spanner is a subgraph that preserves all pairwise distances up to an additive error of $4$. In this paper, we present a new deterministic algorithm for constructing $4$-additive spanners that matches the best known edge bound of $\tilde{O}(n^{7/5})$ (up to polylogarithmic factors), while improving the running time to $\tilde{O}(\min\{mn^{3/5}, n^{11/5}\})$, compared to the previous $\tilde{O}(mn^{3/5})$ randomized construction. Our algorithm is not only faster in the dense regime but also fully deterministic, conceptually simpler, and easier to implement and analyze.