Finding 4-Additive Spanners: Faster, Stronger, and Simpler

TL;DR

This paper introduces a faster, deterministic algorithm for constructing 4-additive spanners in graphs, matching the best known size bounds while improving efficiency and simplicity over previous randomized methods.

Contribution

It presents a new deterministic algorithm for 4-additive spanners that improves running time and simplicity while maintaining optimal size bounds.

Findings

Matches the best known edge bound of n^{7/5} with polylog factors

Improves running time to (mn^{3/5}, n^{11/5})

Provides a fully deterministic and simpler construction method

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.