Connectivity Labeling in Faulty Colored Graphs

TL;DR

This paper considers the color faults model, recently introduced in the context of spanners, and gives a randomized labeling scheme with $\tilde{O}(n^{1-1/2^f})$-bit labels, along with a lower bound of $\Omega(n^{1-1/(f+1)})$ bits.

Abstract

Fault-tolerant connectivity labelings are schemes that, given an $n$-vertex graph $G=(V,E)$ and $f\geq 1$, produce succinct yet informative labels for the elements of the graph. Given only the labels of two vertices $u,v$ and of the elements in a faulty-set $F$ with $|F|\leq f$, one can determine if $u,v$ are connected in $G-F$, the surviving graph after removing $F$. For the edge or vertex faults models, i.e., $F\subseteq E$ or $F\subseteq V$, a sequence of recent work established schemes with $poly(f,\log n)$-bit labels. This paper considers the color faults model, recently introduced in the context of spanners [Petruschka, Sapir and Tzalik, ITCS'24], which accounts for known correlations between failures. Here, the edges (or vertices) of the input $G$ are arbitrarily colored, and the faulty elements in $F$ are colors; a failing color causes all edges (vertices) of that color to crash. Our main contribution is settling the label length complexity for connectivity under one color fault ($f=1$). The existing implicit solution, by applying the state-of-the-art scheme for edge faults of [Dory and Parter, PODC'21], might yield labels of $Ω(n)$ bits. We provide a deterministic scheme with labels of $\tilde{O}(\sqrt{n})$ bits in the worst case, and a matching lower bound. Moreover, our scheme is universally optimal: even schemes tailored to handle only colorings of one specific graph topology cannot produce asymptotically smaller labels. We extend our labeling approach to yield a routing scheme avoiding a single forbidden color. We also consider the centralized setting, and show an $\tilde{O}(n)$-space oracle, answering connectivity queries under one color fault in $\tilde{O}(1)$ time. Turning to $f\geq 2$ color faults, we give a randomized labeling scheme with $\tilde{O}(n^{1-1/2^f})$-bit labels, along with a lower bound of $Ω(n^{1-1/(f+1)})$ bits.

Figures (1)

• Figure 1: Illustration of the proof of \ref{['thm:f-faults-lower-bound']}. Left: The "$f$-thick spider" graph topology $G$. Right: The coloring procedure for $E_{k,l}$.

Theorems & Definitions (44)

• Theorem 1.1: $f=1$, informal
• Theorem 1.2: Forbidden color routing, informal
• Theorem 1.3: $f \geq 2$ upper bound, informal
• Theorem 1.4: $f \geq 2$ lower bound, informal
• Theorem 1.5: $f=2$ upper bound, informal
• Theorem 1.6: Single-source reduction, informal
• Lemma 2.1: Indexing Lower Bound KremerNR99
• Definition 3.1: Proper $r$-ball
• Definition 3.3: Ball-packing number
• Theorem 3.5