Searcharxiv⌕ Search

arXiv subjects

Rafal Kapelko

Publications and source records attributed to Rafal Kapelko.

5 recordsLinked to original sources
arXiv2022

Analysis of the Threshold for Energy Consumption in Displacement of Random Sensors

The fundamental problem of energy-efficient reallocation of mobile random sensors to provide full coverage without interference is addressed in this paper. We consider $n$ mobile sensors with identical sensing range placed randomly on the unit interval and on the unit square. The main contribution is summarized as follows: If the sensors are placed on the unit interval we explain the sharp increase around the sensing radius equal to $\frac{1}{2n}$ and the interference distance equal to $\frac{1}{n}$ for the expected minimal $a$-total displacement, If the sensors are placed on the unit square we explain the sharp increase around the square sensing radius equal to $\frac{1}{2 \sqrt{n}}$ and the interference distance equal to $\frac{1}{\sqrt{n}}$ for the expected minimal $a$-total displacement.

cs.NI↗
arXiv2018

On the expected moments between two identical random processes with application to sensor network

We give a closed analytical formula for expected distance to the power $a$ between two identical general random processes, when $a$ is an even positive number. As an application to sensor network we prove that the optimal transportation cost to the power $b>0$ of the maximal random bicolored matching with edges $\{X_k,Y_k\}$ is in $\frac{Θ\left(n^{\frac{b}{2}+1}\right)}{λ^b}$ when $b \ge 2,$ and in $\frac{O\left(n^{\frac{b}{2}+1}\right)}{λ^b}$ when $0< b < 2.$

cs.DM↗
arXiv2016

On Leader Green Election

We investigate the number of survivors in the Leader Green Election (LGE) algorithm introduced by P. Jacquet, D. Milioris and P. Muhlethaler in 2013. Our method is based on the Rice method and gives quite precise formulas. We derive upper bounds on the number of survivors in this algorithm and we propose a proper use of LGE. Finally, we discuss one property of a general urns and balls problem and show a lower bound for a required number of rounds for a large class of distributed leader election protocols.

cs.DS↗
arXiv2015

On the Displacement for Covering a $d-$dimensional Cube with Randomly Placed Sensors

Consider $n$ sensors placed randomly and independently with the uniform distribution in a $d-$dimensional unit cube ($d\ge 2$). The sensors have identical sensing range equal to $r$, for some $r >0$. We are interested in moving the sensors from their initial positions to new positions so as to ensure that the $d-$dimensional unit cube is completely covered, i.e., every point in the $d-$dimensional cube is within the range of a sensor. If the $i$-th sensor is displaced a distance $d_i$, what is a displacement of minimum cost? As cost measure for the displacement of the team of sensors we consider the $a$-total movement defined as the sum $M_a:= \sum_{i=1}^n d_i^a$, for some constant $a>0$. We assume that $r$ and $n$ are chosen so as to allow full coverage of the $d-$dimensional unit cube and $a > 0$. The main contribution of the paper is to show the existence of a tradeoff between the $d-$dimensional cube, sensing radius and $a$-total movement. The main results can be summarized as follows for the case of the $d-$dimensional cube. If the $d-$dimensional cube sensing radius is $\frac{1}{2n^{1/d}}$ and $n=m^d$, for some $m\in N$, then we present an algorithm that uses $O\left(n^{1-\frac{a}{2d}}\right)$ total expected movement (see Algorithm 2 and Theorem 5). If the $d-$dimensional cube sensing radius is greater than $\frac{3^{3/d}}{(3^{1/d}-1)(3^{1/d}-1)}\frac{1}{2n^{1/d}}$ and $n$ is a natural number then the total expected movement is $O\left(n^{1-\frac{a}{2d}}\left(\frac{\ln n}{n}\right)^{\frac{a}{2d}}\right)$ (see Algorithm 3 and Theorem 7). In addition, we simulate Algorithm 2 and discuss the results of our simulations.

cs.DS↗