127 research outputs found
Quasirandom Load Balancing
We propose a simple distributed algorithm for balancing indivisible tokens on
graphs. The algorithm is completely deterministic, though it tries to imitate
(and enhance) a random algorithm by keeping the accumulated rounding errors as
small as possible.
Our new algorithm surprisingly closely approximates the idealized process
(where the tokens are divisible) on important network topologies. On
d-dimensional torus graphs with n nodes it deviates from the idealized process
only by an additive constant. In contrast to that, the randomized rounding
approach of Friedrich and Sauerwald (2009) can deviate up to Omega(polylog(n))
and the deterministic algorithm of Rabani, Sinclair and Wanka (1998) has a
deviation of Omega(n^{1/d}). This makes our quasirandom algorithm the first
known algorithm for this setting which is optimal both in time and achieved
smoothness. We further show that also on the hypercube our algorithm has a
smaller deviation from the idealized process than the previous algorithms.Comment: 25 page
Optimal path and cycle decompositions of dense quasirandom graphs
Motivated by longstanding conjectures regarding decompositions of graphs into
paths and cycles, we prove the following optimal decomposition results for
random graphs. Let be constant and let . Let be
the number of odd degree vertices in . Then a.a.s. the following hold:
(i) can be decomposed into cycles and a
matching of size .
(ii) can be decomposed into
paths.
(iii) can be decomposed into linear forests.
Each of these bounds is best possible. We actually derive (i)--(iii) from
`quasirandom' versions of our results. In that context, we also determine the
edge chromatic number of a given dense quasirandom graph of even order. For all
these results, our main tool is a result on Hamilton decompositions of robust
expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte
Rotor-router aggregation on the layered square lattice
In rotor-router aggregation on the square lattice Z^2, particles starting at
the origin perform deterministic analogues of random walks until reaching an
unoccupied site. The limiting shape of the cluster of occupied sites is a disk.
We consider a small change to the routing mechanism for sites on the x- and
y-axes, resulting in a limiting shape which is a diamond instead of a disk. We
show that for a certain choice of initial rotors, the occupied cluster grows as
a perfect diamond.Comment: 11 pages, 3 figures
Quasirandom Rumor Spreading
We propose and analyze a quasirandom analogue of the classical push model for disseminating information in networks (ârandomized rumor spreadingâ). In the classical model, in each round, each informed vertex chooses a neighbor at random and informs it, if it was not informed before. It is known that this simple protocol succeeds in spreading a rumor from one vertex to all others within
O
(log
n
) rounds on complete graphs, hypercubes, random regular graphs, ErdĆs-RĂ©nyi random graphs, and Ramanujan graphs with probability 1 â
o
(1). In the quasirandom model, we assume that each vertex has a (cyclic) list of its neighbors. Once informed, it starts at a random position on the list, but from then on informs its neighbors in the order of the list. Surprisingly, irrespective of the orders of the lists, the above-mentioned bounds still hold. In some cases, even better bounds than for the classical model can be shown.
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Traversals of Infinite Graphs with Random Local Orientations
We introduce the notion of a "random basic walk" on an infinite graph, give
numerous examples, list potential applications, and provide detailed
comparisons between the random basic walk and existing generalizations of
simple random walks. We define analogues in the setting of random basic walks
of the notions of recurrence and transience in the theory of simple random
walks, and we study the question of which graphs have a cycling random basic
walk and which a transient random basic walk.
We prove that cycles of arbitrary length are possible in any regular graph,
but that they are unlikely. We give upper bounds on the expected number of
vertices a random basic walk will visit on the infinite graphs studied and on
their finite analogues of sufficiently large size. We then study random basic
walks on complete graphs, and prove that the class of complete graphs has
random basic walks asymptotically visit a constant fraction of the nodes. We
end with numerous conjectures and problems for future study, as well as ideas
for how to approach these problems.Comment: This is my masters thesis from Wesleyan University. Currently my
advisor and I are selecting a journal where we will submit a shorter version.
We plan to split this work into two papers: one for the case of infinite
graphs and one for the finite case (which is not fully treated here
07391 Abstracts Collection -- Probabilistic Methods in the Design and Analysis of Algorithms
From 23.09.2007 to 28.09.2007, the Dagstuhl Seminar 07391 "Probabilistic Methods in the Design and Analysis of Algorithms\u27\u27was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
The seminar brought together leading researchers in probabilistic
methods to strengthen and foster collaborations among various areas of
Theoretical Computer Science. The interaction between researchers
using randomization in algorithm design and researchers studying known
algorithms and heuristics in probabilistic models enhanced the
research of both groups in developing new complexity frameworks and in
obtaining new algorithmic results.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
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
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