Effective Edge-Fault-Tolerant Single-Source Spanners via Best (or Good) Swap Edges

Abstract

Nowadays there is an increasing demand for an efficient and resilient information exchange in communication networks. This means to design on one hand a logical structure onto a given communication infrastructure, which optimizes some sought routing protocol in the absence of failures, and on the other hand, to make such a structure resistant against possible link/node malfunctioning, by suitably adding to it a set of redundant links, which will enter into operation as soon as a failure takes place.More formally, the depicted situation can be modeled as follows: the underlying communication network is an n-vertex and m-edge undirected input graph G = (V (G), E(G), w), with positive real edge weights defined by w, the logical (or primary) structure is a (spanning) subgraph H of G, and finally the additional links is a set of edges A in E(G) \ E(H). Under normal circumstances, communication takes place on H, by following a certain protocol, but as soon as an edge in H fails, then one or more edges in A come into play, and the communication protocol is suitably adjusted.In particular, if the primary structure is a (spanning) tree of G, then a very effective way of defining the set of additional edges is the following: with each tree edge, say e, we associate a so-called best swap edge, namely a non-tree edge that will replace e once it (transiently) fails, in such a way that the resulting swap tree enjoys some nice property in terms of the currently implemented communication protocol. By doing in this way, rerouting and set-up costs will be minimized, in general, and the quality of the post-failure service remains guaranteed. Then, an all best swap edges (ABSE) problem is that of finding efficiently (in term of time complexity) a best swap edge for each tree edge.Due to their fault-tolerance application counterpart, ABSE problems received a large attention by the algorithmic community. In such a framework, a key role has been played by the Shortest-Path Tree (SPT) structure, which is commonly used for implementing efficiently the broadcasting communication primitive. Indeed, it is was shown already in [15] that an effective post-swap broadcast protocol can be put in place just after the original SPT undergoes an edge failure. Not surprisingly then, several ABSE problems w.r.t. an SPT have been studied in the literature, for many different swap criteria.

An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner

“…Concerning the SPT, the most prominent swap criteria are those aiming to minimize either the maximum or the average distance from the root, and the corresponding ABSE problems can be addressed in O(m log α(m, n)) time [6] and O(m α(n, n) log 2 n) time [21], respectively. Recently, in [4], the authors proposed two new criteria for swapping in a SPT, which are in a sense related with this paper, namely the minimization of the maximum and the average stretch factor from the root, for which they proposed an efficient O(mn + n 2 log n) and O(mn log α(m, n)) time solution, respectively.…”

“…Procedure FindCriticalCandidate(f = (v, u), Ψ, Λ) 1 if V (Λ) ∩ De = ∅ then return ⊥; 2 if V (Ψ) = {v} then return Qe(f, Ψ, Λ); 3 b ← Centroid of Ψ; 4 Let j be the unique index in {0, 1, 2, 3} such that v ∈ V (τ j b );…”

A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners

“…Concerning the single-source shortest-path tree, several criteria for measuring the quality of a swap edge have been considered. The most important ones are: the maximum or the average distance from the root; here the corresponding ABSE problems can be solved in O(m log α(m, n)) time (see [6]) and O(m α(n, n) log 2 n) time (see [17]), respectively; the maximum and the average stretch factor from the root for which the corresponding ABSE problems have been solved in O(mn + n 2 log n) and O(mn log α(m, n)) time, respectively [3]. Finally, the ABSE problems have also been studied in a distributed setting [8,9,10].…”

“…Thanks to Lemma 3, once both a x and b x are known, and since all tree edges have length equal to 1, we can compute γ x in constant time using a constant number of least common ancestor and level ancestor queries [1]. 3 Finally, for each x ∈ X, we show how to compute the vertex y x that is closest to γ x in constant time using range-minimum-query data structures [1,11].…”

“…Figure3 An example showing how the set Z(x, 1), Z(x, 2) e Z(x, 3) are defined. Tree edges are solid, while triangles are subtrees.…”