Journal of Computational Information Systems 10: 16 (2014) 6973–6981
Available at http://www.Jofcis.com
A Lifetime-extended Size-bounded Construction
Algorithm for Connected Dominating Sets in
Heterogeneous Wireless Sensor Networks ⋆
Xiaohong LI, Yunchao YANG, Dong WANG ∗,
Zhu XIAO
College of Information Science and Engineering, Hunan University, Changsha 410082, China
Abstract
The use of the connected dominating set (CDS) has been a well-known approach to construct a virtual
backbone that alleviates broadcasting storms in wireless networks. Current research has focused on
minimizing CDS size, prolonging network lifetime. However, no study to date has developed an algorithm
to handle the tradeoﬀ and joint optimization of these two objectives. Considering the imbalance of
node energy consumption in heterogeneous wireless sensor networks, a novel node weight function is
presented in order to solve the joint optimization problem. We propose a distributed lifetime-extended
CDS algorithm (TCDS) based on the node weight function. Theoretical analysis shows that the size
of the CDS obtained from our algorithm is bounded. The simulation results show that our algorithm
prolong network lifetime more eﬃciently than others.
Keywords: Heterogeneous Wireless Sensor Networks; Connected Dominating Set; Network Lifetime
1
Introduction
Heterogeneous wireless sensor networks (HWSNs) are formed by diﬀerent types of sensor nodes
that communicate via radio without any additional backbone infrastructure [1]. The nodes in
HWSNs are powered by energy-limited batteries, so each node is limited in its active lifetime.
Prolonging network lifetime can be achieved in two ways. One method is by saving energy to
prolong network lifetime. In HWSNs, nodes possess diﬀerent transmission ranges and initial
energies, so the energy consumption of nodes is imbalanced. In this regard, balancing energy
consumption is another eﬀective method to prolong network lifetime [2].
Using connected dominating set (CDS) theory to construct a virtual backbone network is an
eﬀective method to save energy [3]. In CDS-based backbone networks, backbone nodes are in a
⋆
Project supported by the National Nature Science Foundation of China (No. 61272061 and No. 61301148), the
fundamental research funds for the central universities of China (No. 531107040263, 531107040276), the Research
Funds for the Doctoral Program of Higher Education of China (No. 20120161120019 and No. 20130161110002),
Hunan Natural Science Foundations of China (No. 10JJ5069 and No. 14JJ7023) and the Open Fund Project of
Key Laboratory in Hunan Universities (No. 11K017).
∗
Corresponding author.
Email address: [email protected] (Dong WANG).
1553–9105 / Copyright © 2014 Binary Information Press
DOI: 10.12733/jcis11245
Augest 15, 2014
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X. Li et al. /Journal of Computational Information Systems 10: 16 (2014) 6973–6981
working state and are responsible for data forwarding, whereas nonbackbone nodes are periodically
fall asleep to save energy [4]. Building a CDS with a minimized size can reduce the number of
control messages and save energy. However, the problem of ﬁnding a minimal CDS (MCDS) has
already been proven to be NP-hard [5].
Most MCDS algorithms neglect the energy consumption of nodes in constructing a CDS. To
address energy consumption imbalanced problem, Tang et al. proposed an energy eﬃcient MCDS
algorithm [6]. However, the nodes in HWSNs consume energy diﬀerently, so those nodes that
possess the maximum residual energy are not necessarily those with the longest lifetime. Thus,
node lifetime is a better metric of node energy consumption than node residual energy.
Reducing the CDS size and improving energy eﬃciency can decrease the total energy consumption of nodes, but these techniques may not eﬀectively prolong network lifetime because
the energy consumption of nodes is imbalanced. In this study, we consider node lifetime and
CDS size when constructing a CDS. Due to node lifetime is considered when constructing a CDS,
the energy consumption of network is balanced. Therefore, the algorithm eﬃciently prolongs
network lifetime. No distributed algorithm to date has considered CDS size, network lifetime in
constructing a CDS in HWSNs.
The construction of CDS in wireless networks has been extensively studied [7, 8]. Existing algorithms can be categorized into two groups: MCDS algorithms, energy-eﬃcient MCDS algorithms.
Wan et al. ﬁrst examined the MCDS problem in a general graph [7]. A distributed maximal independent set (MIS-based) algorithm was presented. Theoretical analysis proves that the algorithm
has a small CDS size. Wu et al. proposed a DDA algorithm to construct a fault-tolerant CDS [8].
DDA selects the node with the largest degree as CDS nodes in order to minimize the CDS size.
The MCDS algorithm does not consider energy in constructing a CDS, so the network lifetime is
short. Kim et al. presented an energy-eﬃcient distributed algorithm called CDS-BD-D [9], which
considers the residual energy and node degree in constructing an MCDS. Network lifetime is associated with both residual energy and energy consumption rate, so the lifetime of the network
constructed by the CDS-BD-D algorithm is not maximal.
All the aforementioned algorithms are for homogeneous WSNs. Thai [10] introduced the disk
graph with bidirectional links (DGB) model to study the MCDS problem in HWSNs. However,
some HWSNs algorithms focus on the MCDS problem neglecting network lifetime. An interesting
combination prolonging network lifetime, and minimizing the CDS size in HWSNs is investigated
in this study.
2
2.1
Models and the Problem Statement
Network model
Heterogeneous networks composed of diﬀerent types of sensor nodes possess diﬀerent transmission
ranges and initial energies. Each node possesses a ﬁxed transmission range. We deﬁne that every
node has two statuses, namely, working and sleep. In this study, we use DGB G(V, E) to model
the heterogeneous network, where V is the node set and E is the edge set. Let r(vi ) and d(vi , vj ) be
the vi and the Euclidian distance between vi and vj , respectively. An edge (ui , uj ) may exist only if
both incident nodes can send a message over (ui , uj ), particularly if min(r(ui ), r(uj )) ≥ d(ui , uj ).
X. Li et al. /Journal of Computational Information Systems 10: 16 (2014) 6973–6981
2.2
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Data aggreation model
Assume that the operation of the algorithm involves two stages: backbone network construction
and data collection. In the data collection stage, a round is considered the time interval between
the production of data by sensor nodes and the receipt of data from all nodes by the sink node. In
the ﬁrst stage, the algorithm is executed, and a backbone network is formed. In the second stage,
nonbackbone nodes periodically send data to backbone nodes and fall asleep to save energy; each
backbone node receives data, aggregates them, and forwards them to other backbone nodes. In
this study, assume that data come from diﬀerent nodes can be merged into a single packet of ﬁxed
sized [11]. The energy consumed by aggreating one packet is given as follows. EDA represents
the energy/bit consumed by aggregation, p represents the size of one packet.
EDm (p) = EDA × p
2.3
(1)
Node energy model
We adopt the ﬁrst-order radio model, which is widely used as a node energy model in wireless
networks [12]. In the ﬁrst-order radio model, the energy consumption of a transceiver is mainly
accounted for by the transmitted signal being ampliﬁed and the other circuitry of the radio
interface. The energy consumed by transmitting one packet can be calculated as:
ET x (p, r) = Eelec × p + Emp × p × rα
(2)
where Eelec represents the energy/bit consumed by the circuit, Emp is the energy/bit consumed by
the transmitter RF ampliﬁer, r is the transmission range of node, and α is the path loss exponent.
The energy consumed by receiving one packet is given as:
ERx (p) = Eelec × p
2.4
(3)
Definitions
Definition 1 Given a graph G(V, E), a dominating set (DS) of G is a subset C ⊂ V , such that
each node either belongs to Cor is adjacent to at least one node in C. A connected dominating
set Cof G is a DS of G, which induces a connected subgraph of G.
Definition 2 The independent set I is a subset of V , such that for any two nodes, u, v ∈ I,
uv ∈ E. A maximal independent set (MIS) is an independent set in which no additional nodes
can be added to retain the nonadjacency property. This definition indicates that an MIS is also
a dominating set. In this study, the nodes in MIS are called dominators, and the nodes in CDS
but not in MIS are called connectors; otherwise, they are called dominatees.
2.5
Node lifetime model
Network lifetime is the time when the ﬁrst node dies in the network. Therefore, network lifetime
is decided by node energy consumption. In this study, energy consumption consists of the energy consumed in the stages of backbone network construction and data collection. The energy
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consumed in the construction of backbone network stage Econstruct consists of the energy consumption of sending data and receiving data. Etransf er represents the energy consumed in the
data collection stage of one round, which includes sending data, receiving data, and aggregating
data. Therefore, the node lifetime can be expressed as follows:
Eresidual − Econstruct
(4)
Etransf er
Eresidual represents the residual energy of node u. Combining Formulas (1), (2), (3)and (4), we
can deduce node lifetime as follows:

Eresidual −(Ndomtx (Emp dα +Eelec )rcon +Ndomrx Eelec rcon )


 (Emp dα +Eelec )rdata +(Nu −1)Eelec rdata +(Nu −1)EDA rdata , u ∈ Do min ator
T (u) =
T (u) =
Eresidual −Ncontx (Emp dα +Eelec )rcon −Nconrx Eelec rcon
, u ∈ Connector
α +E
(E
d
mp
elec )rdata +(Nudom −1)Eelec rdata +(Nudom −1)EDA rdata


Eresidual −Nnontx (Emp dα +Eelec )rcon −Nnonrx Eelec rcon

, u ∈ Do min atee
(Emp dα +Eelec )rdata +Eelec rdata
(5)
rcon and rdata represent the length of the control message and data message, respectively. The
number of messages sent by a dominator, a connector, and a dominatee in the stage of backbone
network construction is Ndomtx , Ncontx , and Nnontx , respectively. The number of messages received
by a dominator, a connector, and a dominatee in the same stage is Ndomrx , Nconrx , and Nnonrx ,
respectively. Nu is the number of neighbors of node u, and Nudom is the number of dominator
neighbors of node u.
2.6
Problem statement
Given a DGB G(V, E), ﬁnd a subset C ⊆ V satisﬁes the following two conditions: (1) the
subgraph induced by C, the size of C is bounded; and (2) the lifetime of network is extended.
3
TCDS Algorithm
In order to balance energy consumption and reduce CDS size, we present a node weight function.
We then propose a lifetime-extended distributed algorithm (TCDS) to solve the aforementioned
problem. In TCDS, a BFS tree is built ﬁrst. An MIS based on the BFS tree is then determined,
and the MIS nodes are connected to form a CDS.
3.1
Node weight function
Compared with other energy-eﬃcient algorithms that consider the residual energy of nodes, dominator node selection in TCDS is based on a method of node lifetime forecast. In this study, we
consider network lifetime and CDS size when constructing a CDS. By combining node lifetime
with node degree when selecting a dominator node, we obtain the following selection criteria of
weight:
Nu
T (u)
+ (1 − a) ×
(6)
W eight(u) = a ×
Tavg
Navg
Navg represents the average degree of network, Tavg represents the maximum network lifetime
that is based on the average degree, and a is a coeﬃcient of T (u). We can obtain the optimal
value by adjusting a in diﬀerent networks.
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One the one hand, choosing a node with the longest lifetime among its neighbors as a dominator
can maximize node lifetime and prolong network lifetime. On the other hand, selecting a node
with the largest node degree as a dominator can minimize CDS size, and reduce total energy
consumption and prolong network lifetime [13].
3.2
TCDS algorithm
We ﬁrst introduce a new term. Color Notiﬁcation Message (CNM), which can be WHITE, GRAY,
BLUE, or BLACK, is transmitted when a node wants to advertise its state (color) to its neighbors.
Every node initially has a WHITE state. TCDS consists of three phases as follows:
Phase 1: Building a BFS tree and exchanging neighbor information
1. Each node learns its height from the Sink (i.e., using the distributed BFS tree algorithm)
[14].
2. Each node u exchanges its information (e.g., IDu , Nu , Heightu , W eight(u)) with neighbors.
Phase 2: Constructing a dominating set
1. The Sink node colors itself black ﬁrst and broadcasts a BLACK CNM with ID. Any white
node u receiving BLACK CNM from v colors itself gray and broadcasts a GRAY CNM. Node u
selects v as its dominator. If a white node u receives a GRAY message the ﬁrst time, it becomes
active. If an active white node u ﬁnds that all parents and siblings with more weight than u are
gray, then u colors itself black.
Phase 3: Selecting connector nodes
1. First, each gray node v counts the number of black neighbors NconnDom (v). If a gray node v
is adjacent to more than one black node, the lifetime Tconnect (v) is calculated and broadcasted as
the connector node.
2. Second, each black node checks if it has a blue parent. If so, the black node chooses it as a
connector. If not, a gray node v with the highest weight W (Tconnect (v)) is selected from its parents
or siblings, and is connected to the Sink node (broadcasted CONNECTED message) and sends
it a CONNECT message. When a gray node receives a CONNECT message, it colors itself blue
and broadcasts a BLUE CNM. Other gray nodes that receive a CONNECT message broadcast a
CONNECTED message.
3. Finally, the set of blue and black nodes is a CDS, and the set of gray nodes is a dominatee
set.
Let us illustrate phases 1 to 3 with a graph. Assume that each node has a weight value W
and lifetime T if it is selected as a dominator. First, all nodes are white. In Fig. 1(b), the Sink
node 0 colors itself black and broadcasts a BLACK CNM. When nodes 1 and 2 receive it, they
color themselves gray and broadcast a GRAY CNM. Nodes 3, 4, 5, and 6 receive a GRAY CNM.
Given that the weight of node 3 is larger than that of node 4, node 3 colors itself black. Node 6
becomes black, similar to node 3, as shown in Fig. 1(c). White node 8 receives a GRAY CNM
and becomes active, whereas nodes 4, 7 are gray and the weight of node 9 is less than that of node
8, so it colors itself black and broadcasts a BLACK CNM. After phase 2, all nodes are divided
into two categories: black nodes and gray nodes, as shown in Fig. 1(d). Finally, nodes 0, 1, 2,
3, 4, 6, and 8 form a CDS in Fig. 1(e). Since DDA selects the node which has the largest node
degree as dominators and connectors, the CDS constructed by DDA are nodes 0, 1, 2, 3, 4, 5,
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: : : : 7 7 7 7 : : : : : : : : 7 7 7 7 7 7 7 7 : : : : : : 7 7 7 : 7 7 : 7 7 7 : 7 ˄D˅
: 7 : 7 ˄E˅
: 7 ˄G˅
: 7 ˄F˅
: 7 : 7 ˄H˅
: 7 : : : : 7 7 7 7 : : : : :
: : : 7 7 7 7
7 7 7 7 : :
: : : : 7 7 7 : 7 7 : 7 7 7 : 7 : : 7 7 : : : : 7 7 7 7 : : : 7 7 : 7 7 : 7 : 7 : 7 : 7 : 7 : 7 : : 7 7 : 7 : 7 : 7 ˄I˅
: 7 : 7 Fig. 1: Example of TCDS algorithm phases 1 to 3
and 8, as shown in Fig. 1(f). Due to energy consumption of dominators is much more than that
of connectors and dominatees, network lifetime is mainly decided by dominators. The network
lifetime of DDA is 0.22, the network lifetime of TCDS is 0.55. Therefore, TCDS has a longer
network lifetime than DDA.
4
Theoretical Analysis
Theorem 1 After phase 2 of TCDS algorithm, a set of black node I is an MIS.
Proof Let I be the set of black nodes
and Gray be the set of gray nodes. Note that I and G are
∪
an independent set. Then V = I Gray and after phase 2 there is no white node exists in V .
Assume that I is not an MIS. Suppose two nodes u ∈ I and v ∈ I exist, and u, v are adjacent.
u is colored black ﬁrst and v is adjacent to u, so v must be colored gray by u. v cannot be a black
node and all pairs of nodes in I are not adjacent. Thus, I is an MIS.
Theorem 2 After phase 3 of TCDS algorithm, the set of backbone node C is a CDS.
Proof After phase 2, all nodes are black or gray, and all black nodes set I form a DS. Next, the
proof that C (formed by adding more blue nodes to I) is connected is given. In phase 3, each
black node selects a blue node to connect another black node, which is already connected to the
Sink node. Therefore, each black node connects to the Sink node. Every blue node connects to
two black nodes, so it also connects to the Sink node. Thus, C is a CDS.
Lemma 1 In DGB G(V, E), the size of any maximal independent set
⌊ is upper⌋ bounded by
rmax
ln r
K|M CDS| where r = rmin and K = 5 if r = 1; otherwise, K = 10 ln(2×cos
. |M CDS| is the
π
)
5
size of the MCDS [15].
Theorem 3 If r ≥ 1, let C be the CDS constructed by TCDS algorithm. |C| ≤2K|M CDS|-1.
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Proof Let B be the set of connectors. From Lemma 5.1, I≤ K|M CDS|, where I is any MIS.
After phase 3, since the Sink node does not need a connector, we have |B| ≤ I ≤ K|M CDS|-1.
Thus, the total number of nodes in C is |C| = |B| + |I| ≤ 2K|M CDS|-1. Therefore, this theorem
proves that the CDS size is upper bound.
Theorem 4 The message complexity is O(|E| + |V |1.6 + ∆|V |), where ∆ is the maximum node
degree of G [14].
5
Simulation and Performance Evaluation
Through theoretical analysis, we have proven some properties of TCDS algorithm, such as the
CDS size is upper bound. Next, we compare network lifetime using TCDS, CDS-BD-D, and DDA
because they are closely related to our work.
For these simulations, assume that all nodes are uniformly distributed in a 400 × 400m2 2D
virtual square and sink node in the middle of the square. The node number 80 ≤ N ≤ 160
and the transmission range r ∈ [60, 130]. The typical values for these parameters are Eelec =
50nJ/bit, Emp = 0.0013pJ/bit/m4 EDA = 5 nJ/bit and α = 4. The contrl packet size is 200bit
and the data packet size is 800bit. Assume that the initial energy and maximum transmission
range of each node are random values in a certain range. For these groups of simulations, each
node vi randomly chooses an initial energy ei ∈ [emin , emax ], where emin = 1.6 J and emax = 2 J.
In this experiment, we have two system parameters, namely, the number of nodes and common
transmission range of nodes. For the same system parameter settings, we randomly create 30
connected graph instances, compute a CDS for each instance, and measure its size and network
lifetime. Given that the energy consumption of backbone nodes is very large, the use of topology
maintenance (i.e., re-execute topology control algorithms) can prolong network lifetime. We
evaluate the performance of our algorithm with and without topology maintenance.
5.1
Evaluation of network lifetime without topology maintenance
To evaluate the performance of the three proposed algorithms under diﬀerent numbers of nodes,
the number of nodes n is increased by 10 from 80 to 160. Each node vi randomly chooses the
transmission range ri ∈ [rmin , rmax ], where rmin = 80 m and rmax = 120 m.
In Fig. 2(a), network lifetime decreases as the number of nodes increases because the number of
nodes that need to be dominated is larger when we deploy more nodes. The energy consumption
increases as the number of nodes that need to be dominated increases, which leads to a reduction in
network lifetime. As Fig. 2(a) shows, network lifetime obtained from TCDS is the longest among
all three algorithms because TCDS considers node lifetime when constructing a CDS. CDS-BD-D
considers the residual energy of nodes when selecting a CDS node. The results indicate that
the node with the most residual energy is not necessarily the node with the longest lifetime in
HWSNs.
Fig. 2(b) shows the comparison of network lifetime obtained from TCDS, CDS-BD-D, and DDA
with diﬀerent transmission ranges. The number of nodes is ﬁxed at 100. As Fig. 2(b) shows,
network lifetime decreases when the transmission range increases. The larger the transmission
range, the more energy a node needs to consume. As expected, TCDS can prolong network
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lifetime by nearly 14% and 52% compared with CDS-BD-D and DDA, respectively. The DDA
algorithm returns the shortest network lifetime because it selects the largest degree node as the
backbone node when constructing a CDS.
2600
3000
TkCDS
CDS-BD-D
DDA
2400
2600
Network Lifetime(round)
Network Lifetime(round)
2000
1800
1600
1400
1200
2400
2200
2000
1800
1600
1400
1000
800
80
TkCDS
CDS-BD-D
DDA
2800
2200
1200
90
100
110
120
130
Number of Nodes
140
150
1000
60-100
160
(a)
70-110
80-120
Transmission Range(m)
90-130
(b)
Fig. 2: Network lifetime comparison with topology maintenance
5.2
Evaluation of network lifetime with topology maintenance
With topology maintenance, we regenerate a CDS when the remaining energy of any backbone
node decreases to 50%. TCDS participates in a role-switching mechanism, in which dominators
and dominatees exchange roles. The experiment parameter settings are similar to those in section
6.1.
As shown in Fig. 3(a), network lifetime obtained from TCDS is 86% more than that of DDA
and 5.4% more than that of CDS-BD-D. Fig. 3(a) shows that the gap of network lifetime between TCDS and DDA decreases. Given that their message complexity is the same, the energy
consumption of each round is the same. Fig. 3(b) shows that network lifetime obtained from
TCDS is 73% more than that of DDA and 5.1% more than that of CDS-BD-D.
3500
4000
TkCDS
CDS-BD-D
DDA
2500
2000
1500
1000
80
TkCDS
CDS-BD-D
DDA
3500
Network Lifetime(round)
Network Lifetime(round)
3000
3000
2500
2000
1500
90
100
110
120
130
Number of Nodes
(a)
140
150
160
1000
60-100
70-110
80-120
Transmission Range(m)
90-130
(b)
Fig. 3: Network lifetime comparison with topology maintenance
6
Conclusion
In this paper, we investigate the lifetime-extended CDS with a bounded CDS size problem in
HWSNs. We propose a novel node weight function and a distributed approximation algorithm
for CDS, which eﬀectively prolongs network lifetime and maintains a small CDS size. We prove
the upper bound of CDS size constructed by TCDS algorithm and analyze message complexities.
Simulations verify that TCDS algorithm can eﬀectively prolong network lifetime and outperform
X. Li et al. /Journal of Computational Information Systems 10: 16 (2014) 6973–6981
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other work. Some open problems remain for further study, such as constructing a lifetime extended
(k, m) CDS in HWSNs, which will be investigated in future studies.
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