405 research outputs found
On The De Bruijn Torus Problem
A (kn;n)k-de Bruijn Cycle is a cyclic k-ary sequence with the property that every k-ary n-tuple appears exactly once contiguously on the cycle. A (kr, ks; m, n)k-de Bruijn Torus is a k-ary krXks toroidal array with the property that every k-ary m x n matrix appears exactly once contiguously on the torus. As is the case with de Bruijn cycles, the 2-dimensional version has many interesting applications, from coding and communications to pseudo-random arrays, spectral imaging, and robot self-location. J.C. Cock proved the existence of such tori for all m, n, and k, and Chung, Diaconis, and Graham asked if it were possible that r = s and m -= n for n even. Fan, Fan, Ma and Siu showed this was possible for k - 2. Combining new techniques with old, we prove the result for k \u3e 2 and show that actually much more is possible. The cases in 3 or more dimensions remain
On the Cost of Participating in a Peer-to-Peer Network
In this paper, we model the cost incurred by each peer participating in a
peer-to-peer network. Such a cost model allows to gauge potential disincentives
for peers to collaborate, and provides a measure of the ``total cost'' of a
network, which is a possible benchmark to distinguish between proposals. We
characterize the cost imposed on a node as a function of the experienced load
and the node connectivity, and show how our model applies to a few proposed
routing geometries for distributed hash tables (DHTs). We further outline a
number of open questions this research has raised.Comment: 17 pages, 4 figures. Short version to be published in the Proceedings
of the Third International Workshop on Peer-to-Peer Systems (IPTPS'04). San
Diego, CA. February 200
The asymptotical error of broadcast gossip averaging algorithms
In problems of estimation and control which involve a network, efficient
distributed computation of averages is a key issue. This paper presents
theoretical and simulation results about the accumulation of errors during the
computation of averages by means of iterative "broadcast gossip" algorithms.
Using martingale theory, we prove that the expectation of the accumulated error
can be bounded from above by a quantity which only depends on the mixing
parameter of the algorithm and on few properties of the network: its size, its
maximum degree and its spectral gap. Both analytical results and computer
simulations show that in several network topologies of applicative interest the
accumulated error goes to zero as the size of the network grows large.Comment: 10 pages, 3 figures. Based on a draft submitted to IFACWC201
Using alternating de Bruijn sequences to construct de Bruijn tori
A de Bruijn torus is the two dimensional generalization of a de Bruijn
sequence. While some methods exist to generate these tori, only a few methods
of construction are known. We present a novel method to generate de Bruijn tori
with rectangular windows by combining two variants de Bruijn sequences called
`Alternating de Bruijn sequences' and `De Bruijn families'.Comment: 21 pages, comments welcom
Enumeration of points, lines, planes, etc
One of the earliest results in enumerative combinatorial geometry is the
following theorem of de Bruijn and Erd\H{o}s: Every set of points in a
projective plane determines at least lines, unless all the points are
contained in a line. Motzkin and others extended the result to higher
dimensions, who showed that every set of points in a projective space
determines at least hyperplanes, unless all the points are contained in a
hyperplane. Let be a spanning subset of a -dimensional vector space. We
show that, in the partially ordered set of subspaces spanned by subsets of ,
there are at least as many -dimensional subspaces as there are
-dimensional subspaces, for every at most . This confirms the
"top-heavy" conjecture of Dowling and Wilson for all matroids realizable over
some field. The proof relies on the decomposition theorem package for
-adic intersection complexes.Comment: 18 pages, major revisio
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