64,396 research outputs found
Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading
We study gossip algorithms for the rumor spreading problem which asks one
node to deliver a rumor to all nodes in an unknown network. We present the
first protocol for any expander graph with nodes such that, the
protocol informs every node in rounds with high probability, and
uses random bits in total. The runtime of our protocol is
tight, and the randomness requirement of random bits almost
matches the lower bound of random bits for dense graphs. We
further show that, for many graph families, polylogarithmic number of random
bits in total suffice to spread the rumor in rounds.
These results together give us an almost complete understanding of the
randomness requirement of this fundamental gossip process.
Our analysis relies on unexpectedly tight connections among gossip processes,
Markov chains, and branching programs. First, we establish a connection between
rumor spreading processes and Markov chains, which is used to approximate the
rumor spreading time by the mixing time of Markov chains. Second, we show a
reduction from rumor spreading processes to branching programs, and this
reduction provides a general framework to derandomize gossip processes. In
addition to designing rumor spreading protocols, these novel techniques may
have applications in studying parallel and multiple random walks, and
randomness complexity of distributed algorithms.Comment: 41 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:1304.135
Topological Soliton with Nonzero Hopf Invariant in Yang-Mills-Higgs Model
We propose a topological soliton or instanton solution with nonzero Hopf
invariant to the 3+1D non-Abelian gauge theory coupled with scalar fields. This
solution, which we call Hopf soliton, represents a spacetime event that makes a
rotation of the monopole. Although the action of this Hopf soliton is
logarithmically divergent, it may still give relevant contributions in a
finite-sized system. Since the Chern-Simons term for the unbroken gauge
field may appear in the low energy effective theory, the Hopf soliton may
possibly generate fractional statistics for the monopoles.Comment: 16 pages, 1 figure
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