A New Sufficient Condition for 1-Coverage to Imply Connectivity

## Abstract

An effective approach for energy conservation in wireless sensor networks is scheduling sleep intervals for extraneous nodes while the remaining nodes stay active to provide continuous service. For the sensor network to operate successfully the active nodes must maintain both sensing coverage and network connectivity, It proved before if the communication range of nodes is at least twice the sensing range, complete coverage of a convex area implies connectivity among the working set of nodes. In this paper we consider a rectangular region A = a *b , 
such that R a R b s ≤ ,s ≤ **, where** Rs is the sensing range of nodes. and put a constraint on minimum allowed distance between nodes( Rs
). according to this constraint we present a new lower bound for communication range relative to sensing range of sensors( 2 + 3 *Rs
) that complete coverage of considered area implies connectivity among the working set of nodes; also we present a new distribution method , that satisfy our constraint. Keywords: **Sensor Network, Coverage,** 
Connectivity, Sensing Range, Communication Range, Distribution Method 

## 1. Introduction

Recent technological advances have led to the emergence of pervasive networks of small, lowpower devices that integrate sensors and actuators with limited on-board processing and wireless communication capabilities. These sensor networks open new vistas for many potential applications, such as battlefield surveillance, environment monitoring and biological detection [1], [2], [3]. 

Sensing is only one responsibility of a sensor network. To operate successfully, a sensor network must also provide satisfactory connectivity so that nodes can communicate for data fusion and reporting to base stations. the relationship between connectivity Seyed Hossein Khasteh Saeid Bagheri Shouraki Ali Akbar Kiaei 

ACECR Department of Electrical 

Engineering ACECR 

Nasir Branch Sharif University of Technology Nasir Branch 

Tehran, Iran Tehran, Iran Tehran, Iran 

H_khasteh@ce.sharif.edu bagheri-s@sharif.edu kiaei@ce.sharif.edu 

and coverage depends on the ratio of the communication range to the sensing range. However, it is easily seen that a connected network may not guarantee its coverage regardless of the ranges. This is because coverage is concerned with whether *any* location is uncovered while connectivity only requires all locations of active nodes are connected. 

Hence we focus on analyzing the condition for a covered network **to guarantee connectivity.** 
Wang *et al.* **[7], proved that in a convex region if** 
2Rs ≤ Rc then 1-Coverage Imply Connectivity, also Zhang *et al.* **[8], proved that** 2Rs ≤ Rc is a necessary condition for 1-Coverage to Imply Connectivity. In the rest of this paper, we first formulate the problem of coverage and connectivity in Section 2; a new sufficient condition for Complete Coverage to Imply Connectivity is presented in Section 3. We present a new distribution method, in Section 4 and conclude the paper in Section 5. 

## 2. Problem Formulation

We use the same problem formulation that used in [7], several coverage models [4], [5], [6], have been proposed for different application scenarios. In this paper, we assume a point p is *covered* **(monitored)** 
by a node v **if their Euclidian distance is less than** 
the sensing range of v , Rs
, i.e., pv < Rs
. We define the *sensing circle* C(v) of node v **as the** 
boundary of v **'s coverage region. We assume that** 
any point p **on the sensing circle** C(v)
(i.e., pv = Rs
) is not covered by v **. Although this** 
definition has an insignificant practical impact, it simplifies our geometric analysis in following sections. Based on the above coverage model, we define a convex region A **(that contains at least one** 
sensing circle) as having a *coverage degree* of K
(i.e., being *K-covered***) if every location inside** A **is** covered by at least K **nodes. Practically speaking, a** 
network with a higher degree of coverage can achieve 