60 research outputs found
A Distributed algorithm to find Hamiltonian cycles in Gnp random graphs
In this paper, we present a distributed algorithm to find Hamiltonian cycles in random binomial graphs Gnp. The algorithm works on a synchronous distributed setting by first creating a small cycle, then covering almost all vertices in the graph with several disjoint paths, and finally patching these paths and the uncovered vertices to the cycle. Our analysis shows that, with high probability, our algorithm is able to find a Hamiltonian cycle in Gnp when p_n=omega(sqrt{log n}/n^{1/4}). Moreover, we conduct an average case complexity analysis that shows that our algorithm terminates in expected sub-linear time, namely in O(n^{3/4+epsilon}) pulses.Postprint (published version
Critical Branching Random Walks with Small Drift
We study critical branching random walks (BRWs) on~
where for each , the displacement of an offspring from its parent has
drift~ towards the origin and reflection at the origin. We
prove that for any~, conditional on survival to
generation~, the maximal displacement is asymptotically
equivalent to . We further show that for a
sequence of critical BRWs with such displacement distributions, if the number
of initial particles grows like~ for some and ,
and the particles are concentrated in~ then the
measure-valued processes associated with the BRWs, under suitable scaling
converge to a measure-valued process, which, at any time~ distributes its
mass over~ like an exponential distribution
Hamiltonian cycles in Cayley graphs of imprimitive complex reflection groups
Generalizing a result of Conway, Sloane, and Wilkes for real reflection
groups, we show the Cayley graph of an imprimitive complex reflection group
with respect to standard generating reflections has a Hamiltonian cycle. This
is consistent with the long-standing conjecture that for every finite group, G,
and every set of generators, S, of G the undirected Cayley graph of G with
respect to S has a Hamiltonian cycle.Comment: 15 pages, 4 figures; minor revisions according to referee comments,
to appear in Discrete Mathematic
Efficient diagnosis of multiprocessor systems under probabilistic models
The problem of fault diagnosis in multiprocessor systems is considered under a probabilistic fault model. The focus is on minimizing the number of tests that must be conducted in order to correctly diagnose the state of every processor in the system with high probability. A diagnosis algorithm that can correctly diagnose the state of every processor with probability approaching one in a class of systems performing slightly greater than a linear number of tests is presented. A nearly matching lower bound on the number of tests required to achieve correct diagnosis in arbitrary systems is also proven. Lower and upper bounds on the number of tests required for regular systems are also presented. A class of regular systems which includes hypercubes is shown to be correctly diagnosable with high probability. In all cases, the number of tests required under this probabilistic model is shown to be significantly less than under a bounded-size fault set model. Because the number of tests that must be conducted is a measure of the diagnosis overhead, these results represent a dramatic improvement in the performance of system-level diagnosis techniques
- …