7,763 research outputs found
Observability of Lattice Graphs
We consider a graph observability problem: how many edge colors are needed
for an unlabeled graph so that an agent, walking from node to node, can
uniquely determine its location from just the observed color sequence of the
walk?
Specifically, let G(n,d) be an edge-colored subgraph of d-dimensional
(directed or undirected) lattice of size n^d = n * n * ... * n. We say that
G(n,d) is t-observable if an agent can uniquely determine its current position
in the graph from the color sequence of any t-dimensional walk, where the
dimension is the number of different directions spanned by the edges of the
walk. A walk in an undirected lattice G(n,d) has dimension between 1 and d, but
a directed walk can have dimension between 1 and 2d because of two different
orientations for each axis.
We derive bounds on the number of colors needed for t-observability. Our main
result is that Theta(n^(d/t)) colors are both necessary and sufficient for
t-observability of G(n,d), where d is considered a constant.
This shows an interesting dependence of graph observability on the ratio
between the dimension of the lattice and that of the walk. In particular, the
number of colors for full-dimensional walks is Theta(n^(1/2)) in the directed
case, and Theta(n) in the undirected case, independent of the lattice
dimension.
All of our results extend easily to non-square lattices: given a lattice
graph of size N = n_1 * n_2 * ... * n_d, the number of colors for
t-observability is Theta (N^(1/t))
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
Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable
There has been significant recent interest in parallel graph processing due
to the need to quickly analyze the large graphs available today. Many graph
codes have been designed for distributed memory or external memory. However,
today even the largest publicly-available real-world graph (the Hyperlink Web
graph with over 3.5 billion vertices and 128 billion edges) can fit in the
memory of a single commodity multicore server. Nevertheless, most experimental
work in the literature report results on much smaller graphs, and the ones for
the Hyperlink graph use distributed or external memory. Therefore, it is
natural to ask whether we can efficiently solve a broad class of graph problems
on this graph in memory.
This paper shows that theoretically-efficient parallel graph algorithms can
scale to the largest publicly-available graphs using a single machine with a
terabyte of RAM, processing them in minutes. We give implementations of
theoretically-efficient parallel algorithms for 20 important graph problems. We
also present the optimizations and techniques that we used in our
implementations, which were crucial in enabling us to process these large
graphs quickly. We show that the running times of our implementations
outperform existing state-of-the-art implementations on the largest real-world
graphs. For many of the problems that we consider, this is the first time they
have been solved on graphs at this scale. We have made the implementations
developed in this work publicly-available as the Graph-Based Benchmark Suite
(GBBS).Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
On Packing Colorings of Distance Graphs
The {\em packing chromatic number} of a graph is the
least integer for which there exists a mapping from to
such that any two vertices of color are at distance at
least . This paper studies the packing chromatic number of infinite
distance graphs , i.e. graphs with the set of
integers as vertex set, with two distinct vertices being
adjacent if and only if . We present lower and upper bounds for
, showing that for finite , the packing
chromatic number is finite. Our main result concerns distance graphs with
for which we prove some upper bounds on their packing chromatic
numbers, the smaller ones being for :
if is odd and
if is even
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