On the density of sets of the Euclidean plane avoiding distance 1

A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall x,y \in A, \left\| x-y \right\|_2 \neq 1.$ In this paper we study the number $m_1(\mathbb R^2)$ which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, $m_1(\mathbb R^2)$ represents the highest proportion of the plane that can be filled by a set avoiding distance 1. This parameter is related to the fractional chromatic number $\chi_f(\mathbb R^2)$ of the plane. We establish that $m_1(\mathbb R^2) \leq 0.25646$ and $\chi_f(\mathbb R^2) \geq 3.8992$.Comment: 11 pages, 5 figure

Algorithms for Fast Aggregated Convergecast in Sensor Networks

Fast and periodic collection of aggregated data is of considerable interest for mission-critical and continuous monitoring applications in sensor networks. In the many-to-one communication paradigm, referred to as convergecast, we focus on applications wherein data packets are aggregated at each hop en-route to the sink along a tree-based routing topology, and address the problem of minimizing the convergecast schedule length by utilizing multiple frequency channels. The primary hindrance in minimizing the schedule length is the presence of interfering links. We prove that it is NP-complete to determine whether all the interfering links in an arbitrary network can be removed using at most a constant number of frequencies. We give a sufficient condition on the number of frequencies for which all the interfering links can be removed, and propose a polynomial time algorithm that minimizes the schedule length in this case. We also prove that minimizing the schedule length for a given number of frequencies on an arbitrary network is NP-complete, and describe a greedy scheme that gives a constant factor approximation on unit disk graphs. When the routing tree is not given as an input to the problem, we prove that a constant factor approximation is still achievable for degree-bounded trees. Finally, we evaluate our algorithms through simulations and compare their performance under different network parameters