A Distributed Approximation for Multi-Hop
Clustering Problem in Wireless Sensor Networks
Xudong Zhu
∗
, Jun Li
∗
, Xiaofeng Gao
∗§
, Fan Wu
∗
, Guihai Chen
∗
, Athanasios V. Vasilakos
†
∗
Shanghai Key Laboratory of Scalable Computing and Systems,
Department of Computer Science and Engineering,
Shanghai Jiao Tong University, Shanghai, 200240, China
†
Dept of Computer Science, Electrical and Space Engineering,
Lulea University of Technology, Lulea, 97187, Sweden
xudongzhu42@gmail.com, lijun2009@sjtu.edu.cn, {gao-xf, fwu, gchen}@cs.sjtu.edu.cn, vasilako@ath.forthnet.gr
Abstract—In wireless sensor networks (WSNs), there is no pre-
defined infrastructure. Nodes need to frequently flood messages to
discover routes, which badly decreases the network performance.
To overcome such drawbacks, WSNs are often grouped into
several disjointed clusters, each with a representative cluster head
(CH) in charge of the routing process. In order to further improve
the efficiency of WSNs, it is crucial to find a cluster partition
with minimum number of clusters and the distance between each
node to its corresponding CH can be bounded by a constant
number of hops. Finding such a partition is defined as minimum
d-hop cluster head set (d-MCHS) problem, which is proved to be
NP-hard. In this paper, we propose a distributed approximation
algorithm, named d
2
-Cluster, to address d-MCHS problem and
prove that the approximation ratio of d
2
-Cluster under unit disk
graph (UDG) is a constant factor λ which is related to d. To
the best of our knowledge, it is the first constant approximation
ratio for d-MDS problem in UDG.
I. INTRODUCTION
Wireless sensor networks (WSNs) is a kind of autonomous
communication systems with lots of small-sized, inexpensive
and battery-powered sensors. Since such sensors are cheap
and can be deployed in various environments, WSNs have
been widely used in many applications, such as monitoring,
disaster management, battlefield surveillance, etc. [1]–[3].
Usually the most important task for a sensor is to gather
its surrounding information and send its data to the nearest
sink. However, in WSNs, there is no fixed or predefined in-
frastructure. If a sensor wants to communicate with a specified
peer, it needs to discover the route between them by flooding
messages. This mechanism causes a lot of serious problems,
such as traffic collision, energy consumption, etc.
In order to overcome these shortcomings, an efficient ap-
proach named clustering is widely pursued by the research
§
X.Gao is the corresponding author.
This work has been supported in part by the National Natural Sci-
ence Foundation of China (Grant number 61202024, 61472252, 61133006,
61422208), China 973 project (2012CB316200), the Natural Science Founda-
tion of Shanghai (Grant No.12ZR1445000), Shanghai Educational Develop-
ment Foundation (Chenguang Grant No.12CG09), Shanghai Pujiang Program
13PJ1403900, and in part by Jiangsu Future Network Research Project No.
BY2013095-1-10 and CCF-Tencent Open Fund.
community. We can partition a given WSN into several dis-
jointed clusters, each of which is a small-scale autonomous
communication system. Specially, each cluster has a leader,
called cluster head (CH). A CH is the interface for ordinary
sensors in the corresponding cluster to contact the outsides.
With clustering technique, an ordinary sensor only needs to
focus on gathering information and sending the data to its
CH, and its CH will forward the data to the sink outside.
Obviously, this mechanism can save a lot of energy and reduce
the number of collisions by avoiding many redundant message
forwardings.
In the literature, many clustering algorithms were proposed
for WSNs. In [4], Younis et al. devised a clustering algorithm
to achieve equal-sized clusters to balance the work load of each
cluster. In [5], Demirbas et al. presented a clustering algorithm,
named FLOC, to partition a WSN into non-overlapping and
approximately equal-sized clusters. Since each cluster is equal-
sized, CHs take charge for a same amount of sensors. However,
there still exists a serious problem. Some ordinary sensors
in a cluster may be too far away from the CH, which will
take a long route when transmitting data to the CH. As a
consequence, it downgrades the performance of the cluster
strategy. On contrast, Wang et al. studied the problem of
minimizing the average hop distance from any node to its
cluster head in 2D sensor networks in [6]. However, they did
not consider to control the sizes of clusters when partitioning a
WSN, which will causes load imbalance among cluster heads.
To solve the conflicts above, an effective approach is to
set a parameter d, and for each node in a cluster, the hop
distance between it to its CH is upper bounded by d. Since
every sensor in a cluster is within d hops from its CH, the
CH can manage this cluster efficiently. Besides, since sensors
are usually deployed randomly in lots of applications, this
approach can also effectively obtain approximately equal-sized
cluster. Moreover, the average size of clusters can be controlled
by setting different values of d.
In this paper, we focus on partitioning a WSN into the
minimum number of d-hop clusters. Here, a d-hop cluster
means each sensor in it is either a CH or within d hops