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

Computing \emph{all best swap edges} (ABSE) of a spanning tree $T$ of a given $n$-vertex and $m$-edge undirected and weighted graph $G$ means to select, for each edge $e$ of $T$, a corresponding non-tree edge $f$, in such a way that the tree obtained by replacing $e$ with $f$ enjoys some optimality criterion (which is naturally defined according to some objective function originally addressed by $T$). Solving efficiently an ABSE problem is by now a classic algorithmic issue, since it conveys a very successful way of coping with a (transient) \emph{edge failure} in tree-based communication networks: just replace the failing edge with its respective swap edge, so as that the connectivity is promptly reestablished by minimizing the rerouting and set-up costs. In this paper, we solve the ABSE problem for the case in which $T$ is a \emph{single-source shortest-path tree} of $G$, and our two selected swap criteria aim to minimize either the \emph{maximum} or the \emph{average stretch} in the swap tree of all the paths emanating from the source. Having these criteria in mind, the obtained structures can then be reviewed as \emph{edge-fault-tolerant single-source spanners}. For them, we propose two efficient algorithms running in $O(m n +n^2 \log n)$ and $O(m n \log \alpha(m,n))$ time, respectively, and we show that the guaranteed (either maximum or average, respectively) stretch factor is equal to 3, and this is tight. Moreover, for the maximum stretch, we also propose an almost linear $O(m \log \alpha(m,n))$ time algorithm computing a set of \emph{good} swap edges, each of which will guarantee a relative approximation factor on the maximum stretch of $3/2$ (tight) as opposed to that provided by the corresponding BSE. Surprisingly, no previous results were known for these two very natural swap problems.Comment: 15 pages, 4 figures, SIROCCO 201

Linear Time Distributed Swap Edge Algorithms

In this paper, we consider the all best swap edges problem in a distributed environment. We are given a 2-edge connected positively weighted network X, where all communication is routed through a rooted spanning tree T of X. If one tree edge e = {x, y} fails, the communication network will be disconnected. However, since X is 2-edge connected, communication can be restored by replacing e by non-tree edge e′, called a swap edge of e, whose ends lie in different components of T − e. Of all possible swap edges of e, we would like to choose the best, as defined by the application. The all best swap edges problem is to identify the best swap edge for every tree edge, so that in case of any edge failure, the best swap edge can be activated quickly. There are solutions to this problem for a number of cases in the literature. A major concern for all these solutions is to minimize the number of messages. However, especially in fault-transient environments, time is a crucial factor. In this paper we present a novel technique that addresses this problem from a time perspective; in fact, we present a distributed solution that works in linear time with respect to the height h of T for a number of differentcriteria, while retaining the optimal number of messages. To the best of our knowledge, all previous solutions solve the problem in O(h^2) time in the cases we consider

A Distributed Algorithm for Finding All Best Swap Edges Of a Minimum Diameter Spanning Tree

ABSTRACT Communication in networks suffers if a link fails. When the links are edge of a tree that has been chose

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

Given a 2-edge connected, unweighted, and undirected graph $G$ with $n$ vertices and $m$ edges, a $\sigma$-tree spanner is a spanning tree $T$ of $G$ in which the ratio between the distance in $T$ of any pair of vertices and the corresponding distance in $G$ is upper bounded by $\sigma$. The minimum value of $\sigma$ for which $T$ is a $\sigma$-tree spanner of $G$ is also called the {\em stretch factor} of $T$. We address the fault-tolerant scenario in which each edge $e$ of a given tree spanner may temporarily fail and has to be replaced by a {\em best swap edge}, i.e. an edge that reconnects $T-e$ at a minimum stretch factor. More precisely, we design an $O(n^2)$ time and space algorithm that computes a best swap edge of every tree edge. Previously, an $O(n^2 \log^4 n)$ time and $O(n^2+m\log^2n)$ space algorithm was known for edge-weighted graphs [Bil\`o et al., ISAAC 2017]. Even if our improvements on both the time and space complexities are of a polylogarithmic factor, we stress the fact that the design of a $o(n^2)$ time and space algorithm would be considered a breakthrough.Comment: The paper has been accepted for publication at the 29th International Symposium on Algorithms and Computation (ISAAC 2018). 12 pages, 3 figure

Quantum Error Correcting Codes and Fault-Tolerant Quantum Computation over Nice Rings

Quantum error correcting codes play an essential role in protecting quantum information from the noise and the decoherence. Most quantum codes have been constructed based on the Pauli basis indexed by a finite field. With a newly introduced algebraic class called a nice ring, it is possible to construct the quantum codes such that their alphabet sizes are not restricted to powers of a prime. Subsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. We show a generalization of subsystem codes over nice rings. Furthermore, we prove that free subsystem codes over a finite chain ring cannot outperform those over a finite field. We also generalize entanglement-assisted quantum error correcting codes to nice rings. With the help of the entanglement, any classical code can be used to derive the corresponding quantum codes, even if such codes are not self-orthogonal. We prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs using symplectic geometry over rings. So, we can find hyperbolic pairs and commuting generators generating the check matrix of the entanglement-assisted quantum code. Fault-tolerant quantum computation has been also studied over a finite field. Transversal operations are the simplest way to implement fault-tolerant quantum gates. We derive transversal Clifford operations for CSS codes over nice rings, including Fourier transforms, SUM gates, and phase gates. Since transversal operations alone cannot provide a computationally universal set of gates, we add fault-tolerant implementations of doubly-controlled Z gates for triorthogonal stabilizer codes over nice rings. Finally, we investigate optimal key exchange protocols for unconditionally secure key distribution schemes. We prove how many rounds are needed for the key exchange between any pair of the group on star networks, linear-chain networks, and general networks

Dynamic distributed programming and applications to swap edge problem

Link failure is a common reason for disruption in communication networks. If communication between processes of a weighted distributed network is maintained by a spanning tree T, and if one edge e of T fails, communication can be restored by finding a new spanning tree, T’. If the network is 2-edge connected, T’ can always be constructed by replacing e by a single edge, e’, of the network. We refer to e’ as a swap edge of e. The best swap edge problem is to find the best choice of e’, that is, that e which causes the new spanning tree T’ to have the least cost, where cost is measured in a way that is determined by the application. Two examples of such measures are total weight of T‘ and diameter of T’. The all best swap edges problem is the problem of determining, in advance of any failure, the best swap edge for every edge in T. The justification for this problem is that we wish to be ready, when a failure occurs, to quickly activate a replacement for the failed edge. In this thesis, we give algorithms for the all best swap edges problem for six different cost measures. We first present an algorithm which can be adapted to all six measures, and which takes O (d2) time, where d is the diameter of T. This algorithm is essentially a form of distributed dynamic programming, since we compute the answers to sub problems at each node of T. We then present a novel paradigm for speeding up distributed computations under certain conditions. We apply this paradigm to find O(d)-time distributed algorithms for the all best swap edge problem for all but one of our cost measures. Formal algorithms and their correctness proofs will be given