20,621 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
Faster Algorithms for the Maximum Common Subtree Isomorphism Problem
The maximum common subtree isomorphism problem asks for the largest possible
isomorphism between subtrees of two given input trees. This problem is a
natural restriction of the maximum common subgraph problem, which is -hard in general graphs. Confining to trees renders polynomial time
algorithms possible and is of fundamental importance for approaches on more
general graph classes. Various variants of this problem in trees have been
intensively studied. We consider the general case, where trees are neither
rooted nor ordered and the isomorphism is maximum w.r.t. a weight function on
the mapped vertices and edges. For trees of order and maximum degree
our algorithm achieves a running time of by
exploiting the structure of the matching instances arising as subproblems. Thus
our algorithm outperforms the best previously known approaches. No faster
algorithm is possible for trees of bounded degree and for trees of unbounded
degree we show that a further reduction of the running time would directly
improve the best known approach to the assignment problem. Combining a
polynomial-delay algorithm for the enumeration of all maximum common subtree
isomorphisms with central ideas of our new algorithm leads to an improvement of
its running time from to ,
where is the order of the larger tree, is the number of different
solutions, and is the minimum of the maximum degrees of the input
trees. Our theoretical results are supplemented by an experimental evaluation
on synthetic and real-world instances
A Local Computation Approximation Scheme to Maximum Matching
We present a polylogarithmic local computation matching algorithm which
guarantees a (1-\eps)-approximation to the maximum matching in graphs of
bounded degree.Comment: Appears in Approx 201
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