804 research outputs found
Additive Spanners: A Simple Construction
We consider additive spanners of unweighted undirected graphs. Let be a
graph and a subgraph of . The most na\"ive way to construct an additive
-spanner of is the following: As long as is not an additive
-spanner repeat: Find a pair that violates the
spanner-condition and a shortest path from to in . Add the edges of
this path to .
We show that, with a very simple initial graph , this na\"ive method gives
additive - and -spanners of sizes matching the best known upper bounds.
For additive -spanners we start with and end with
edges in the spanner. For additive -spanners we start with containing
arbitrary edges incident to each node and end with a
spanner of size .Comment: To appear at proceedings of the 14th Scandinavian Symposium and
Workshop on Algorithm Theory (SWAT 2014
Improved Purely Additive Fault-Tolerant Spanners
Let be an unweighted -node undirected graph. A \emph{-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 \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 , , 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 \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 edge failures.Comment: 17 pages, 4 figures, ESA 201
Collective additive tree spanners for circle graphs and polygonal graphs
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
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
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 of the network such that
subsequent to the failure of a single vertex, the surviving part of still
contains an \emph{additive} spanner for (the surviving part of) , satisfying
for every
. Recently, the problem of constructing fault-tolerant additive
spanners resilient to the failure of up to \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 and
edges. Our second result is an FT-spanner with additive stretch and
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 for a given subset of
sources . 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
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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