122 research outputs found
Upper Bound on the Number of Vertices of Polyhedra with -Constraint Matrices
In this note we show that the maximum number of vertices in any polyhedron
with -constraint matrix and a
real vector is at most .Comment: 3 page
Distributed Computing on Core-Periphery Networks: Axiom-based Design
Inspired by social networks and complex systems, we propose a core-periphery
network architecture that supports fast computation for many distributed
algorithms and is robust and efficient in number of links. Rather than
providing a concrete network model, we take an axiom-based design approach. We
provide three intuitive (and independent) algorithmic axioms and prove that any
network that satisfies all axioms enjoys an efficient algorithm for a range of
tasks (e.g., MST, sparse matrix multiplication, etc.). We also show the
minimality of our axiom set: for networks that satisfy any subset of the
axioms, the same efficiency cannot be guaranteed for any deterministic
algorithm
A note on uniform power connectivity in the SINR model
In this paper we study the connectivity problem for wireless networks under
the Signal to Interference plus Noise Ratio (SINR) model. Given a set of radio
transmitters distributed in some area, we seek to build a directed strongly
connected communication graph, and compute an edge coloring of this graph such
that the transmitter-receiver pairs in each color class can communicate
simultaneously. Depending on the interference model, more or less colors,
corresponding to the number of frequencies or time slots, are necessary. We
consider the SINR model that compares the received power of a signal at a
receiver to the sum of the strength of other signals plus ambient noise . The
strength of a signal is assumed to fade polynomially with the distance from the
sender, depending on the so-called path-loss exponent .
We show that, when all transmitters use the same power, the number of colors
needed is constant in one-dimensional grids if as well as in
two-dimensional grids if . For smaller path-loss exponents and
two-dimensional grids we prove upper and lower bounds in the order of
and for and
for respectively. If nodes are distributed
uniformly at random on the interval , a \emph{regular} coloring of
colors guarantees connectivity, while colors are required for any coloring.Comment: 13 page
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