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Parks and Recreation: Color Fault-Tolerant Spanners Made Local

Merav Parter, Asaf Petruschka, Shay Sapir, Elad Tzalik

TL;DR

This work provides new algorithms for constructing spanners of arbitrarily edge- or vertex-colored graphs, that can endure up to up to $f$ failures of entire color classes, and produces color fault-tolerant spanners of size $\tilde{O}_k (s)$, which are near-optimal for any fixed $k$.

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 $s=Θ(f n^{1+1/k})$ or $s=Θ(f^{1-1/k} n^{1+1/k})$ 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.

Parks and Recreation: Color Fault-Tolerant Spanners Made Local

TL;DR

This work provides new algorithms for constructing spanners of arbitrarily edge- or vertex-colored graphs, that can endure up to up to failures of entire color classes, and produces color fault-tolerant spanners of size , which are near-optimal for any fixed .

Abstract

We provide new algorithms for constructing spanners of arbitrarily edge- or vertex-colored graphs, that can endure up to 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 of such spanners, where or for spanners with stretch that are resilient to at most edge- or vertex-color faults, respectively. Additionally, they showed an algorithm for computing spanners of size , running in 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 (hence near-optimal for any fixed ), have optimal locality (i.e., take rounds in the LOCAL model), can be implemented in rounds in CONGEST, and take 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.

Paper Structure

This paper contains 55 sections, 15 theorems, 44 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. On the Color Faults Model
  3. The Quest for Optimal-Size FT Spanners
  4. Our Contribution
  5. Discussion and Open Problems.
  6. Technical Highlights
  7. From Vertex Clusters to an Edge-Centric Perspective.
  8. From (Vertex) Disjointness to (Color) Spreadness.
  9. Preliminaries
  10. Notations and Terminology
  11. An Alternative View of the Baswana-Sen Algorithm
  12. Warm-Up: $f$-ECFT $3$-Spanner
  13. The First Level ($i=0$).
  14. The Last Level ($i=1$).
  15. The Size Argument.
  16. ...and 40 more sections

Key Result

Theorem 1.2

Let $G$ be a weighted edge-colored or vertex-colored graph with $n$ vertices and $m$ edges. Denote by $s$ the (conditionally) optimal worst-case bound on the size of an $f$-CFT $(2k-1)$-spanner for an $n$-vertex graph, namely: There is a randomized algorithm that, with high probability, constructs an $f$-CFT $(2k-1)$-spanner $H$ of $G$, with $|E(H)| \leq s \cdot \log n \cdot 2^{O(k^2)} = \tilde{O

Figures (1)

  • Figure 1: Deciding on an edge $e = \{v,u\}\in E_i(v)$. The left figure illustrates that $v$ maintains a touristic park $\widehat{\mathcal{P}}_v$ (w.r.t. $\mathsf{l}\widehat{\mathsf{s}}\mathsf{c}^i, \mathsf{g}\widehat{\mathsf{s}}\mathsf{c}^i$) of $(i+1)$-paths ending at $S_i$, and $e$ comes with a touristic park $\mathcal{P}_u$ (w.r.t. $\mathsf{lsc}^i, \mathsf{gsc}^i$), as guaranteed by Invariant \ref{['inv:I2']}, of $i$-paths ending at $S_i$. The middle figure illustrates a special case of a $\mathsf{safe}$ vote of a path $P$ ending at $s\in S_i$, since there are $\Theta_k(1)$ paths (depicted as a single path) in $\widehat{\mathcal{P}}_{v,s}$ with the same color set as $P$, and thus $\widehat{\mathcal{P}}_{v,s}$ is $J$-full for $J=c(P)$. The figure on the right illustrates a $\mathsf{pstpn}$ vote of a path $P$, caused since $\widehat{\mathcal{P}}_v$ contains $\Theta_k(\tfrac{\log n}{p})$ paths having the same color set as $P$, and thus $\widehat{\mathcal{P}}_v$ is $J$-full for $J=c(P)$.

Theorems & Definitions (62)

  • Definition 1.1: $f$-CFT $t$-Spanners
  • Theorem 1.2
  • Definition 2.1: Link
  • Definition 4.1: Score Function
  • Definition 4.2: Park
  • Lemma 4.3: Fault-Tolerance of Parks
  • proof
  • Definition 4.4: $J$-Full Park
  • Definition 4.5: Touristic Park
  • Lemma 5.0: Last Level Lemma
  • ...and 52 more