Fault Tolerant BFS Structures: A Reinforcement-Backup Tradeoff

TL;DR

This paper explores the trade-off between backup and reinforcement mechanisms in fault-tolerant BFS network structures, proposing a new $(b,r)$ FT-BFS model and establishing a fundamental relationship between the number of fault-resistant edges and backup edges needed for resilience.

Contribution

It introduces the $(b,r)$ FT-BFS model to analyze fault resilience trade-offs and derives tight bounds linking reinforcement and backup edge quantities.

Findings

A tradeoff between reinforcement and backup edges is established.

Necessary and sufficient conditions for fault-tolerant BFS structures are derived.

The relationship scales with network size and fault tolerance parameters.

Abstract

This paper initiates the study of fault resilient network structures that mix two orthogonal protection mechanisms: (a) {\em backup}, namely, augmenting the structure with many (redundant) low-cost but fault-prone components, and (b) {\em reinforcement}, namely, acquiring high-cost but fault-resistant components. To study the trade-off between these two mechanisms in a concrete setting, we address the problem of designing a $(b,r)$ {\em fault-tolerant} BFS (or $(b,r)$ FT-BFS for short) structure, namely, a subgraph $H$ of the network $G$ consisting of two types of edges: a set $E' \subseteq E$ of $r(n)$ fault-resistant {\em reinforcement} edges, which are assumed to never fail, and a (larger) set $E(H) \setminus E'$ of $b(n)$ fault-prone {\em backup} edges, such that subsequent to the failure of a single fault-prone backup edge $e \in E \setminus E'$, the surviving part of $H$ still contains an BFS spanning tree for (the surviving part of) $G$, satisfying $dist(s,v,H\setminus \{e\}) \leq dist(s,v,G\setminus \{e\})$ for every $v \in V$ and $e \in E \setminus E'$. We establish the following tradeoff between $b(n)$ and $r(n)$: For every real $\epsilon \in (0,1]$, if $r(n) = {\tilde\Theta}(n^{1-\epsilon})$, then $b(n) = {\tilde\Theta}(n^{1+\epsilon})$ is necessary and sufficient.