1,026 research outputs found
Rigidity of spherical codes
A packing of spherical caps on the surface of a sphere (that is, a spherical
code) is called rigid or jammed if it is isolated within the space of packings.
In other words, aside from applying a global isometry, the packing cannot be
deformed. In this paper, we systematically study the rigidity of spherical
codes, particularly kissing configurations. One surprise is that the kissing
configuration of the Coxeter-Todd lattice is not jammed, despite being locally
jammed (each individual cap is held in place if its neighbors are fixed); in
this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic
lattice in three dimensions. By contrast, we find that many other packings have
jammed kissing configurations, including the Barnes-Wall lattice and all of the
best kissing configurations known in four through twelve dimensions. Jamming
seems to become much less common for large kissing configurations in higher
dimensions, and in particular it fails for the best kissing configurations
known in 25 through 31 dimensions. Motivated by this phenomenon, we find new
kissing configurations in these dimensions, which improve on the records set in
1982 by the laminated lattices.Comment: 39 pages, 8 figure
Geometry of Rounding
Rounding has proven to be a fundamental tool in theoretical computer science.
By observing that rounding and partitioning of are equivalent,
we introduce the following natural partition problem which we call the {\em
secluded hypercube partition problem}: Given (ideally small)
and (ideally large), is there a partition of with
unit hypercubes such that for every point , its closed
-neighborhood (in the norm) intersects at most
hypercubes?
We undertake a comprehensive study of this partition problem. We prove that
for every , there is an explicit (and efficiently computable)
hypercube partition of with and . We complement this construction by proving that the value of
is the best possible (for any ) for a broad class of
``reasonable'' partitions including hypercube partitions. We also investigate
the optimality of the parameter and prove that any partition in this
broad class that has , must have .
These bounds imply limitations of certain deterministic rounding schemes
existing in the literature. Furthermore, this general bound is based on the
currently known lower bounds for the dissection number of the cube, and
improvements to this bound will yield improvements to our bounds.
While our work is motivated by the desire to understand rounding algorithms,
one of our main conceptual contributions is the introduction of the {\em
secluded hypercube partition problem}, which fits well with a long history of
investigations by mathematicians on various hypercube partitions/tilings of
Euclidean space
Partitions of R^n with Maximal Seclusion and their Applications to Reproducible Computation
We introduce and investigate a natural problem regarding unit cube tilings/partitions of Euclidean space and also consider broad generalizations of this problem. The problem fits well within a historical context of similar problems and also has applications to the study of reproducibility in randomized computation.
Given and , we define a -secluded unit cube partition of to be a unit cube partition of such that for every point , the closed -ball around intersects at most cubes. The problem is to construct such partitions for each dimension with the primary goal of minimizing and the secondary goal of maximizing .
We prove that for every dimension , there is an explicit and efficiently computable -secluded axis-aligned unit cube partition of with and . We complement this construction by proving that for axis-aligned unit cube partitions, the value of is the minimum possible, and when is minimized at , the value is the maximum possible. This demonstrates that our constructions are the best possible.
We also consider the much broader class of partitions in which every member has at most unit volume and show that is still the minimum possible. We also show that for any reasonable (i.e. ), it must be that . This demonstrates that when is minimized at , our unit cube constructions are optimal to within a logarithmic factor even for this broad class of partitions. In fact, they are even optimal in up to a logarithmic factor when is allowed to be polynomial in .
We extend the techniques used above to introduce and prove a variant of the KKM lemma, the Lebesgue covering theorem, and Sperner\u27s lemma on the cube which says that for every , and every proper coloring of , there is a translate of the -ball which contains points of least different colors.
Advisers: N. V. Vinodchandran & Jamie Radcliff
On the Coloring of Grid Wireless Sensor Networks: the Vector-Based Coloring Method
Graph coloring is used in wireless networks to optimize network resources:
bandwidth and energy. Nodes access the medium according to their color. It is
the responsibility of the coloring algorithm to ensure that interfering nodes
do not have the same color. In this research report, we focus on wireless
sensor networks with grid topologies. How does a coloring algorithm take
advantage of the regularity of grid topology to provide an optimal periodic
coloring, that is a coloring with the minimum number of colors? We propose the
Vector-Based Coloring Method, denoted VCM, a new method that is able to provide
an optimal periodic coloring for any radio transmission range and for any h-hop
coloring, h>=1. This method consists in determining at which grid nodes a color
can be reproduced without creating interferences between these nodes while
minimizing the number of colors used. We compare the number of colors provided
by VCM with the number of colors obtained by a distributed coloring algorithm
with line and column priority assignments. We also provide bounds on the number
of colors of optimal general colorings of the infinite grid, and show that
periodic colorings (and thus VCM) are asymptotically optimal. Finally, we
discuss the applicability of this method to a real wireless network
- …