Improved Purely AdditiveFault-Tolerant SpannersThis work was partially supported by the Research Grant PRIN 2010 “ARS TechnoMedia", funded by the Italian Ministry of Education, University, and Research, and by the ERC Starting Grant “New Approaches to Network Design".

# Improved Purely Additive Fault-Tolerant Spanners††thanks: This work was partially supported by the Research Grant PRIN 2010 “ARS TechnoMedia", funded by the Italian Ministry of Education, University, and Research, and by the ERC Starting Grant “New Approaches to Network Design".

Davide Bilò Dipartimento di Scienze Umanistiche e Sociali, Università di Sassari, Italy    Fabrizio Grandoni IDSIA, University of Lugano, Switzerland    Luciano Gualà Dipartimento di Ingegneria dell’Impresa, Università di Roma “Tor Vergata", Italy
Stefano Leucci
DISIM, Università degli Studi dell’Aquila, Italy
Guido Proietti DISIM, Università degli Studi dell’Aquila, Italy Istituto di Analisi dei Sistemi ed Informatica, CNR, Roma, Italy
E-mail: davide.bilo@uniss.it; fabrizio@idsia.ch; guala@mat.uniroma2.it; stefano.leucci@univaq.it; guido.proietti@univaq.it
###### Abstract

Let be an unweighted -node undirected graph. A -additive spanner of is a spanning subgraph of such that distances in are stretched at most by an additive term w.r.t. the corresponding distances in . A natural research goal related with spanners is that of designing sparse spanners with low stretch.

In this paper, we focus on fault-tolerant additive spanners, namely additive spanners which are able to preserve their additive stretch even when one edge fails. We are able to improve all known such spanners, in terms of either sparsity or stretch. In particular, we consider the sparsest known spanners with stretch , , and , and reduce the stretch to , , and , respectively (while keeping the same sparsity).

Our results are based on two different constructions. On one hand, we show how to augment (by adding a small number of edges) a fault-tolerant additive sourcewise spanner (that approximately preserves distances only from a given set of source nodes) into one such spanner that preserves all pairwise distances. On the other hand, we show how to augment some known fault-tolerant additive spanners, based on clustering techniques. This way we decrease the additive stretch without any asymptotic increase in their size. We also obtain improved fault-tolerant additive spanners for the case of one vertex failure, and for the case of edge failures.

## 1 Introduction

We are given an unweighted, undirected -node graph . Let denote the shortest path distance between nodes and in . A spanner of is a spanning subgraph such that for all , , where is the so-called stretch or distortion function of the spanner. In particular, when , for constants , the spanner is named an (,) spanner. If , the spanner is called (purely) additive or also -additive. If , the spanner is called -multiplicative.

Finding sparse (i.e., with a small number of edges) spanners is a key task in many network applications, since they allow for a small-size infrastructure onto which an efficient (in terms of paths’ length) point-to-point communication can be performed. Due to this important feature, spanners were the subject of an intensive research effort, aiming at designing increasingly sparser spanners with lower stretch.

However, as any sparse structure, a spanner is very sensitive to possible failures of components (i.e., edges or nodes), which may drastically affect its performances, or even disconnect it! Thus, to deal with this drawback, a more robust concept of fault-tolerant spanner is naturally conceivable, in which the distortion must be guaranteed even after a subset of components of fails.

More formally, for a subset of edges (resp., vertices) of , let be the graph obtained by removing from the edges (resp., vertices and incident edges) in . When , we will simply write . Then, an -edge fault-tolerant (-EFT) spanner with distortion , is a subgraph of such that, for every set of at most failed edges, we have111Note that in this definition we allow to become infinite (if the removal of disconnects from ). In that case we assume the inequality to be trivially satisfied.

 dH−F(s,t)≤α⋅dG−F(s,t)+β∀s,t∈V(G).

We define similarly an -vertex fault-tolerant (-VFT) spanner. For , we simply call the spanner edge/vertex fault-tolerant (EFT/VFT).

Chechik et al. [10] show how to construct a -multiplicative -EFT spanner of size , for any integer . Their approach also works for weighted graphs and for vertex-failures, returning a -multiplicative -VFT spanner of size .222The notation hides poly-logarithmic factors in . This latter result has been finally improved through a randomized construction in [14], where the expected size was reduced to . For a comparison, the sparsest known -multiplicative standard (non fault-tolerant) spanners have size [2], and this is believed to be asymptotically tight due to the girth conjecture of Erdős [15].