804 research outputs found

    Additive Spanners: A Simple Construction

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    We consider additive spanners of unweighted undirected graphs. Let GG be a graph and HH a subgraph of GG. The most na\"ive way to construct an additive kk-spanner of GG is the following: As long as HH is not an additive kk-spanner repeat: Find a pair (u,v)H(u,v) \in H that violates the spanner-condition and a shortest path from uu to vv in GG. Add the edges of this path to HH. We show that, with a very simple initial graph HH, this na\"ive method gives additive 66- and 22-spanners of sizes matching the best known upper bounds. For additive 22-spanners we start with H=H=\emptyset and end with O(n3/2)O(n^{3/2}) edges in the spanner. For additive 66-spanners we start with HH containing n1/3\lfloor n^{1/3} \rfloor arbitrary edges incident to each node and end with a spanner of size O(n4/3)O(n^{4/3}).Comment: To appear at proceedings of the 14th Scandinavian Symposium and Workshop on Algorithm Theory (SWAT 2014

    Improved Purely Additive Fault-Tolerant Spanners

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    Let GG be an unweighted nn-node undirected graph. A \emph{β\beta-additive spanner} of GG is a spanning subgraph HH of GG such that distances in HH are stretched at most by an additive term β\beta w.r.t. the corresponding distances in GG. A natural research goal related with spanners is that of designing \emph{sparse} spanners with \emph{low} stretch. In this paper, we focus on \emph{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 66, 2828, and 3838, and reduce the stretch to 44, 1010, and 1414, 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 \emph{small} number of edges) a fault-tolerant additive \emph{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 ff edge failures.Comment: 17 pages, 4 figures, ESA 201

    Collective additive tree spanners for circle graphs and polygonal graphs

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    AbstractA graph G=(V,E) is said to admit a system of μ collective additive tree r-spanners if there is a system T(G) of at most μ spanning trees of G such that for any two vertices u,v of G a spanning tree T∈T(G) exists such that the distance in T between u and v is at most r plus their distance in G. In this paper, we examine the problem of finding “small” systems of collective additive tree r-spanners for small values of r on circle graphs and on polygonal graphs. Among other results, we show that every n-vertex circle graph admits a system of at most 2log32n collective additive tree 2-spanners and every n-vertex k-polygonal graph admits a system of at most 2log32k+7 collective additive tree 2-spanners. Moreover, we show that every n-vertex k-polygonal graph admits an additive (k+6)-spanner with at most 6n−6 edges and every n-vertex 3-polygonal graph admits a system of at most three collective additive tree 2-spanners and an additive tree 6-spanner. All our collective tree spanners as well as all sparse spanners are constructible in polynomial time

    An FPT Algorithm for Minimum Additive Spanner Problem

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    For a positive integer t and a graph G, an additive t-spanner of G is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus t. The Minimum Additive t-Spanner Problem is to find an additive t-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs when we deal with additive t-spanners, the Minimum Additive t-Spanner Problem is hard to handle and hence only few results are known for it. In this paper, we study the Minimum Additive t-Spanner Problem from the viewpoint of parameterized complexity. We formulate a parameterized version of the problem in which the number of removed edges is regarded as a parameter, and give a fixed-parameter algorithm for it. We also extend our result to the case with both a multiplicative approximation factor ? and an additive approximation parameter ?, which we call (?, ?)-spanners

    Vertex Fault Tolerant Additive Spanners

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    A {\em fault-tolerant} structure for a network is required to continue functioning following the failure of some of the network's edges or vertices. In this paper, we address the problem of designing a {\em fault-tolerant} additive spanner, namely, a subgraph HH of the network GG such that subsequent to the failure of a single vertex, the surviving part of HH still contains an \emph{additive} spanner for (the surviving part of) GG, satisfying dist(s,t,H{v})dist(s,t,G{v})+βdist(s,t,H\setminus \{v\}) \leq dist(s,t,G\setminus \{v\})+\beta for every s,t,vVs,t,v \in V. Recently, the problem of constructing fault-tolerant additive spanners resilient to the failure of up to ff \emph{edges} has been considered by Braunschvig et. al. The problem of handling \emph{vertex} failures was left open therein. In this paper we develop new techniques for constructing additive FT-spanners overcoming the failure of a single vertex in the graph. Our first result is an FT-spanner with additive stretch 22 and O~(n5/3)\widetilde{O}(n^{5/3}) edges. Our second result is an FT-spanner with additive stretch 66 and O~(n3/2)\widetilde{O}(n^{3/2}) edges. The construction algorithm consists of two main components: (a) constructing an FT-clustering graph and (b) applying a modified path-buying procedure suitably adopted to failure prone settings. Finally, we also describe two constructions for {\em fault-tolerant multi-source additive spanners}, aiming to guarantee a bounded additive stretch following a vertex failure, for every pair of vertices in S×VS \times V for a given subset of sources SVS\subseteq V. The additive stretch bounds of our constructions are 4 and 8 (using a different number of edges)

    The Sparsest Additive Spanner via Multiple Weighted BFS Trees

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    Spanners are fundamental graph structures that sparsify graphs at the cost of small stretch. In particular, in recent years, many sequential algorithms constructing additive all-pairs spanners were designed, providing very sparse small-stretch subgraphs. Remarkably, it was then shown that the known (+6)-spanner constructions are essentially the sparsest possible, that is, larger additive stretch cannot guarantee a sparser spanner, which brought the stretch-sparsity trade-off to its limit. Distributed constructions of spanners are also abundant. However, for additive spanners, while there were algorithms constructing (+2) and (+4)-all-pairs spanners, the sparsest case of (+6)-spanners remained elusive. We remedy this by designing a new sequential algorithm for constructing a (+6)-spanner with the essentially-optimal sparsity of O~(n^{4/3}) edges. We then show a distributed implementation of our algorithm, answering an open problem in [Keren Censor{-}Hillel et al., 2016]. A main ingredient in our distributed algorithm is an efficient construction of multiple weighted BFS trees. A weighted BFS tree is a BFS tree in a weighted graph, that consists of the lightest among all shortest paths from the root to each node. We present a distributed algorithm in the CONGEST model, that constructs multiple weighted BFS trees in |S|+D-1 rounds, where S is the set of sources and D is the diameter of the network graph
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