5,888 research outputs found
Super-Fast 3-Ruling Sets
A -ruling set of a graph is a vertex-subset
that is independent and satisfies the property that every vertex is
at a distance of at most from some vertex in . A \textit{maximal
independent set (MIS)} is a 1-ruling set. The problem of computing an MIS on a
network is a fundamental problem in distributed algorithms and the fastest
algorithm for this problem is the -round algorithm due to Luby
(SICOMP 1986) and Alon et al. (J. Algorithms 1986) from more than 25 years ago.
Since then the problem has resisted all efforts to yield to a sub-logarithmic
algorithm. There has been recent progress on this problem, most importantly an
-round algorithm on graphs with
vertices and maximum degree , due to Barenboim et al. (Barenboim,
Elkin, Pettie, and Schneider, April 2012, arxiv 1202.1983; to appear FOCS
2012).
We approach the MIS problem from a different angle and ask if O(1)-ruling
sets can be computed much more efficiently than an MIS? As an answer to this
question, we show how to compute a 2-ruling set of an -vertex graph in
rounds. We also show that the above result can be improved
for special classes of graphs such as graphs with high girth, trees, and graphs
of bounded arboricity.
Our main technique involves randomized sparsification that rapidly reduces
the graph degree while ensuring that every deleted vertex is close to some
vertex that remains. This technique may have further applications in other
contexts, e.g., in designing sub-logarithmic distributed approximation
algorithms. Our results raise intriguing questions about how quickly an MIS (or
1-ruling sets) can be computed, given that 2-ruling sets can be computed in
sub-logarithmic rounds
Super-Fast Distributed Algorithms for Metric Facility Location
This paper presents a distributed O(1)-approximation algorithm, with
expected- running time, in the model for
the metric facility location problem on a size- clique network. Though
metric facility location has been considered by a number of researchers in
low-diameter settings, this is the first sub-logarithmic-round algorithm for
the problem that yields an O(1)-approximation in the setting of non-uniform
facility opening costs. In order to obtain this result, our paper makes three
main technical contributions. First, we show a new lower bound for metric
facility location, extending the lower bound of B\u{a}doiu et al. (ICALP 2005)
that applies only to the special case of uniform facility opening costs. Next,
we demonstrate a reduction of the distributed metric facility location problem
to the problem of computing an O(1)-ruling set of an appropriate spanning
subgraph. Finally, we present a sub-logarithmic-round (in expectation)
algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our
algorithm accomplishes this by using a combination of randomized and
deterministic sparsification.Comment: 15 pages, 2 figures. This is the full version of a paper that
appeared in ICALP 201
Local Approximation Schemes for Ad Hoc and Sensor Networks
We present two local approaches that yield polynomial-time approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1+ε)-approximation to the problems at hand for any given ε > 0. The time complexity of both algorithms is O(TMIS + log*! n/εO(1)), where TMIS is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pair-wise independent nodes in every r-neighborhood is at most polynomial in r. Such graphs of polynomially bounded growth are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs
Distributed -Coloring in Sublogarithmic Rounds
We give a new randomized distributed algorithm for -coloring in
the LOCAL model, running in
rounds in a graph of maximum degree~. This implies that the
-coloring problem is easier than the maximal independent set
problem and the maximal matching problem, due to their lower bounds of by Kuhn, Moscibroda, and Wattenhofer [PODC'04].
Our algorithm also extends to list-coloring where the palette of each node
contains colors. We extend the set of distributed symmetry-breaking
techniques by performing a decomposition of graphs into dense and sparse parts
Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence
We study the {edge-coloring} problem in the message-passing model of
distributed computing. This is one of the most fundamental and well-studied
problems in this area. Currently, the best-known deterministic algorithms for
(2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01},
where Delta is the maximum degree of the input graph. Also, recent results of
\cite{BE10} for vertex-coloring imply that one can get an
O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an
O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an
arbitrarily small constant epsilon > 0.
In this paper we devise a drastically faster deterministic edge-coloring
algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in
O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 +
epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves
the previous state-of-the-art {exponentially} in a wide range of Delta,
specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In
addition, for small values of Delta our deterministic algorithm outperforms all
the existing {randomized} algorithms for this problem.
On our way to these results we study the {vertex-coloring} problem on the
family of graphs with bounded {neighborhood independence}. This is a large
family, which strictly includes line graphs of r-hypergraphs for any r = O(1),
and graphs of bounded growth. We devise a very fast deterministic algorithm for
vertex-coloring graphs with bounded neighborhood independence. This algorithm
directly gives rise to our edge-coloring algorithms, which apply to {general}
graphs.
Our main technical contribution is a subroutine that computes an
O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood
independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq
Delta
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