263,932 research outputs found
Finding Even Cycles Faster via Capped k-Walks
In this paper, we consider the problem of finding a cycle of length (a
) in an undirected graph with nodes and edges for constant
. A classic result by Bondy and Simonovits [J.Comb.Th.'74] implies that
if , then contains a , further implying that
one needs to consider only graphs with .
Previously the best known algorithms were an algorithm due to Yuster
and Zwick [J.Disc.Math'97] as well as a algorithm by Alon et al. [Algorithmica'97].
We present an algorithm that uses time and finds a
if one exists. This bound is exactly when . For
-cycles our new bound coincides with Alon et al., while for every our
bound yields a polynomial improvement in .
Yuster and Zwick noted that it is "plausible to conjecture that is
the best possible bound in terms of ". We show "conditional optimality": if
this hypothesis holds then our algorithm is tight as well.
Furthermore, a folklore reduction implies that no combinatorial algorithm can
determine if a graph contains a -cycle in time for any
under the widely believed combinatorial BMM conjecture. Coupled
with our main result, this gives tight bounds for finding -cycles
combinatorially and also separates the complexity of finding - and
-cycles giving evidence that the exponent of in the running time should
indeed increase with .
The key ingredient in our algorithm is a new notion of capped -walks,
which are walks of length that visit only nodes according to a fixed
ordering. Our main technical contribution is an involved analysis proving
several properties of such walks which may be of independent interest.Comment: To appear at STOC'1
Finding Simple Shortest Paths and Cycles
The problem of finding multiple simple shortest paths in a weighted directed
graph has many applications, and is considerably more difficult than
the corresponding problem when cycles are allowed in the paths. Even for a
single source-sink pair, it is known that two simple shortest paths cannot be
found in time polynomially smaller than (where ) unless the
All-Pairs Shortest Paths problem can be solved in a similar time bound. The
latter is a well-known open problem in algorithm design. We consider the
all-pairs version of the problem, and we give a new algorithm to find
simple shortest paths for all pairs of vertices. For , our algorithm runs
in time (where ), which is almost the same bound as
for the single pair case, and for we improve earlier bounds. Our approach
is based on forming suitable path extensions to find simple shortest paths;
this method is different from the `detour finding' technique used in most of
the prior work on simple shortest paths, replacement paths, and distance
sensitivity oracles.
Enumerating simple cycles is a well-studied classical problem. We present new
algorithms for generating simple cycles and simple paths in in
non-decreasing order of their weights; the algorithm for generating simple
paths is much faster, and uses another variant of path extensions. We also give
hardness results for sparse graphs, relative to the complexity of computing a
minimum weight cycle in a graph, for several variants of problems related to
finding simple paths and cycles.Comment: The current version includes new results for undirected graphs. In
Section 4, the notion of an (m,n) reduction is generalized to an f(m,n)
reductio
On the properties of cycles of simple Boolean networks
We study two types of simple Boolean networks, namely two loops with a
cross-link and one loop with an additional internal link. Such networks occur
as relevant components of critical K=2 Kauffman networks. We determine mostly
analytically the numbers and lengths of cycles of these networks and find many
of the features that have been observed in Kauffman networks. In particular,
the mean number and length of cycles can diverge faster than any power law.Comment: 10 pages, 8 figure
Fast Computation of Small Cuts via Cycle Space Sampling
We describe a new sampling-based method to determine cuts in an undirected
graph. For a graph (V, E), its cycle space is the family of all subsets of E
that have even degree at each vertex. We prove that with high probability,
sampling the cycle space identifies the cuts of a graph. This leads to simple
new linear-time sequential algorithms for finding all cut edges and cut pairs
(a set of 2 edges that form a cut) of a graph.
In the model of distributed computing in a graph G=(V, E) with O(log V)-bit
messages, our approach yields faster algorithms for several problems. The
diameter of G is denoted by Diam, and the maximum degree by Delta. We obtain
simple O(Diam)-time distributed algorithms to find all cut edges,
2-edge-connected components, and cut pairs, matching or improving upon previous
time bounds. Under natural conditions these new algorithms are universally
optimal --- i.e. a Omega(Diam)-time lower bound holds on every graph. We obtain
a O(Diam+Delta/log V)-time distributed algorithm for finding cut vertices; this
is faster than the best previous algorithm when Delta, Diam = O(sqrt(V)). A
simple extension of our work yields the first distributed algorithm with
sub-linear time for 3-edge-connected components. The basic distributed
algorithms are Monte Carlo, but they can be made Las Vegas without increasing
the asymptotic complexity.
In the model of parallel computing on the EREW PRAM our approach yields a
simple algorithm with optimal time complexity O(log V) for finding cut pairs
and 3-edge-connected components.Comment: Previous version appeared in Proc. 35th ICALP, pages 145--160, 200
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