Fault-Tolerant Approximate Shortest-Path Trees

TL;DR

This paper introduces compact fault-tolerant approximate shortest-path trees for networks, balancing resilience and size, with improved stretch factors and sizes for both weighted and unweighted graphs under single node or edge failures.

Contribution

It presents new sparse structures for fault-tolerant approximate shortest-path trees with better size and stretch guarantees than previous solutions, applicable to both weighted and unweighted graphs.

Findings

Structures of size O(n log n / ε^2) guarantee (1+ε)-stretch under single failures.

Compared to prior solutions, these are smaller and have better stretch factors.

Efficient augmentation of spanners for fault-tolerant BFS trees in unweighted graphs.

Abstract

The resiliency of a network is its ability to remain \emph{effectively} functioning also when any of its nodes or links fails. However, to reduce operational and set-up costs, a network should be small in size, and this conflicts with the requirement of being resilient. In this paper we address this trade-off for the prominent case of the {\em broadcasting} routing scheme, and we build efficient (i.e., sparse and fast) \emph{fault-tolerant approximate shortest-path trees}, for both the edge and vertex \emph{single-failure} case. In particular, for an $n$-vertex non-negatively weighted graph, and for any constant $\varepsilon >0$, we design two structures of size $O(\frac{n \log n}{\varepsilon^2})$ which guarantee $(1+\varepsilon)$-stretched paths from the selected source also in the presence of an edge/vertex failure. This favorably compares with the currently best known solutions, which are for the edge-failure case of size $O(n)$ and stretch factor 3, and for the vertex-failure case of size $O(n \log n)$ and stretch factor 3. Moreover, we also focus on the unweighted case, and we prove that an ordinary $(\alpha,\beta)$-spanner can be slightly augmented in order to build efficient fault-tolerant approximate \emph{breadth-first-search trees}.