Journal of Sensors

Volume 2015, Article ID 565983, 12 pages

http://dx.doi.org/10.1155/2015/565983

## Linear Time Approximation Algorithms for the Relay Node Placement Problem in Wireless Sensor Networks with Hexagon Tessellation

^{1}Department of Information Engineering, I-Shou University, Kaohsiung 84001, Taiwan^{2}Department of Electronic Engineering, National Chin-Yi University of Technology, Taichung 41170, Taiwan

Received 26 June 2014; Revised 9 September 2014; Accepted 20 March 2015

Academic Editor: Chenzhong Li

Copyright © 2015 Chi-Chang Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The relay node placement problem in wireless sensor network (WSN) aims at deploying the minimum number of relay nodes over the network so that each sensor can communicate with at least one relay node. When the deployed relay nodes are homogeneous and their communication ranges are circular, one way to solve the WSN relay node placement problem is to solve the minimum geometric disk cover (MGDC) problem first and place the relay nodes at the centers of the covering disks and then, if necessary, deploy additional relay nodes to meet the connection requirement of relay nodes. It is known that the MGDC problem is NP-complete. A novel linear time approximation algorithm for the MGDC problem is proposed, which identifies covering disks using the regular hexagon tessellation of the plane with bounded area. The approximation ratio of the proposed algorithm is (), where . Experimental results show that the worst case is rare, and on average the proposed algorithm uses less than 1.7 times the optimal disks of the MGDC problem. In cases where quick deployment is necessary, this study provides a fast 7-approximation algorithm which uses on average less than twice the optimal number of relay nodes in the simulation.

#### 1. Introduction

Given a set of points on the Euclidean plane and a prescribed radius , disk covers a point if the distance between the center of and is not greater than . The minimum geometric disk cover (MGDC) problem is to identify a set of disks with minimal cardinality covering all points in . If radius equals 1, the MGDC problem is called the minimum geometric unit disk cover (MGUDC) problem. To simplify calculations, this study focuses on solving the MGUDC problem. However, all MGUDC problem algorithms can easily be extended to the MGDC problem by changing the radius from 1 to any fixed real number . Solutions to the MGDC problem can be used to solve the relay node placement problems in wireless sensor networks (WSNs).

A WSN consists of spatially distributed autonomous sensors that cooperatively monitor a region for physical or environmental conditions, such as temperature, sound, vibration, pressure, motion, and pollution. The transmission power consumed by a wireless radio is proportional to the distance squared or even higher in the presence of obstacles. Thus, multihop routing (instead of direct communication) is usually used for sending collected data to the sink.

A method of prolonging the lifetime of a WSN and preserving network connectivity is to deploy a few costly, but powerful, relay nodes to communicate with the other sensors or relay nodes. The WSN relay node placement problem studies how to deploy the fewest relay nodes so that each sensor can communicate with at least one relay node. In this study, we assume that the relay nodes in the WSN are homogeneous and their communication ranges are circles with radius . (If the communication range is irregular, the can be taken as the minimum radius among the inscribed circles of the irregular ranges.) Therefore, the relay node placement problem of WSNs can be treated as solving the MGDC problem and then placing the relay nodes at the centers of the chosen disks. In case the resulting relay nodes are not connected, additional relay nodes can be deployed to comply with the connection requirement among the relay nodes.

This study proposes two simple and effective approximation algorithms for the MGDC problem that identify covering disks using the regular hexagon tessellation of the plane. Tessellation is the process of creating a 2D plane by repeating a geometric shape with no overlaps or gaps. A regular tessellation is a tessellation that covers a 2D plane with regular polygons of the same shape and size. Only three regular tessellations (composed of the hexagon, square, and triangle) exist in the Euclidean plane [1].

A regular polygon enclosed by a unit disk (i.e., a circle with a radius of 1) is a unit polygon. If the polygons in a regular tessellation are all unit polygons, it is called a unit polygon tessellation. The three unit polygon tessellations are as follows: a unit triangle tessellation, unit square tessellation, and unit hexagon tessellation (Figure 1). Let us check how many unit polygons are needed to tessellate a given plane.