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SIAM J. on Discrete Mathematics / Volume 21 / Issue 3

SIAM J. Discrete Math. 21, pp. 637-646 (10 pages)

Locating Servers for Reliability and Affine Embeddings

Kenneth A. Berman

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Consider the problem of locating servers in a network for the purpose of storing data, performing an application, etc., so that at least one server will be available to clients even if up to $k$ component failures occur throughout the network. Letting $G = (V,E)$ be the graph with vertex set $V$ and edge set $E$ representing the topology of the network, and letting $L \subseteq V$ be a set of potential locations for the servers, a fundamental problem is to determine a minimum-size set $S \subseteq L$ such that the network remains connected to $S$ even if up to $k$ component failures occur throughout the network. We say that such a set $S$ is $k$-fault-tolerant. In this paper we present an algebraic characterization of $k$-fault-tolerant sets in terms of affine embeddings of $G$ in $k$-dimensional Euclidean space. Employing this characterization, we present a polynomial-time Monte Carlo algorithm for computing a minimum-size $k$-fault-tolerant subset $S$ of $L$. In fact, we solve the following more general problem for directed networks: given a digraph $G = (V,E)$ (an undirected graph is equivalent to a symmetric digraph) and a subset $L \subseteq V$, we find a $k$-fault-tolerant subset $S$ of $L$ having minimum cost, where a unary integer cost $c(v)$ is associated with locating a server at vertex $v \in V$.

© 2007 Society for Industrial and Applied Mathematics

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KEYWORDS

AMS Subject Headings

05C85, 05C50, 05C10, 68M10

PUBLICATION DATA

ISSN

0895-4801 (print)  
1095-7146 (online)

Publisher

$abstract.society.value is a Member of CrossRef Society for Industrial and Applied Mathematics

ARTICLE DATA

History
Received September 18, 2006
Accepted February 06, 2007
Published online July 25, 2007
Digital Object Identifier
http://dx.doi.org/10.1137/060670158

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