Parks and Recreation: Color Fault-Tolerant Spanners Made Local

Abstract

We provide new algorithms for constructing spanners of arbitrarily edge- or vertex-colored graphs, that can endure up to $f$ failures of entire color classes. The failure of even a single color may cause a linear number of individual edge/vertex faults. In a recent work, Petruschka, Sapir and Tzalik [ITCS `24] gave tight bounds for the (worst-case) size $s$ of such spanners, where $s=\Theta(f n^{1+1/k})$ or $s=\Theta(f^{1-1/k} n^{1+1/k})$ for spanners with stretch $(2k-1)$ that are resilient to at most $f$ edge- or vertex-color faults, respectively. Additionally, they showed an algorithm for computing spanners of size $\tilde{O}(s)$, running in $\tilde{O}(msf)$ sequential time, based on the (FT) greedy spanner algorithm. The problem of providing faster and/or distributed algorithms was left open therein. We address this problem and provide a novel variant of the classical Baswana-Sen algorithm [RSA `07] in the spirit of Parter's algorithm for vertex fault-tolerant spanners [STOC `22]. In a nutshell, our algorithms produce color fault-tolerant spanners of size $\tilde{O}_k (s)$ (hence near-optimal for any fixed $k$), have optimal locality $O(k)$ (i.e., take $O(k)$ rounds in the LOCAL model), can be implemented in $O_k (f^{k-1})$ rounds in CONGEST, and take $\tilde{O}_k (m + sf^{k-1})$ sequential time. To handle the considerably more difficult setting of color faults, our approach differs from [BS07, Par22] by taking a novel edge-centric perspective, instead of (FT)-clustering of vertices; in fact, we demonstrate that this point of view simplifies their algorithms. Another key technical contribution is in constructing and using collections of short paths that are "colorful at all scales", which we call "parks". These are intimately connected with the notion of spread set-systems that found use in recent breakthroughs regarding the famous Sunflower Conjecture.