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Using Central Nodes for Efficient Data Collection in Wireless Sensor
Networks
Vitaly Milyeykovskia, Michael Segala, Hanan Shpunginb, Vladimir Katza
aCommunication Systems Engineering Department, Ben-Gurion University of the Negev, Israel. Email:
{milyeyko,segal}@bgu.ac.il
bDepartment of Electrical Computer Engineering, University of Waterloo, Waterloo, Canada. Email:
hanan.shpungin@uwaterloo.ca
Abstract
We study the problem of data collection in Wireless Sensor Networks (WSN). A typical WSN is
composed of wireless sensor nodes that periodically sense data and forward it to the base station
in a multi-hop fashion. We are interested in designing an efficient data collection tree routing,
focusing on three optimization objectives: energy efficiency, transport capacity, and hop-diameter
(delay).
In this paper we develop single- and multi-hop data collection, which are based on the definition
of node centrality: centroid nodes. We provide theoretical performance analysis for our approach,
present its distributed implementation and discuss the different aspects of using it. Most of our
results are for two-dimensional WSNs, however we also show that the centroid-based approach is
asymptotically optimal in three-dimensional random node deployments. In addition, we present
new construction for arbitrary network deployment based on central nodes selection. We also
present extensive simulation results that support our theoretical findings.
1. Introduction
A wireless sensor network (WSN) consists of small autonomous low-cost low-power devices that
carry out monitoring tasks. Initially developed for military use, WSNs can nowadays be found in
many civil applications, such as environmental monitoring, biomedical research, seismic monitoring,
and precision agriculture [1]. The devices are called sensor nodes and the monitored data is typically
collected at a base station, following a specific collection pattern of activated wireless links.
As these networks have no hard-wired underlying topology, one of the most fundamental issues
when a WSN is deployed is the formation of an efficient communication backbone, or in other words,
answering the question which links to use in order to collect the data from the sensor nodes?
Efficiency can be defined in many ways, for example it can be maximizing the rate at which
data is collected ([23, 44, 46]) from the sensor nodes, prolonging the network lifetime by reducing
the energy consumption ([6, 9, 33, 35, 39]), minimizing the number of hops from the sensor nodes
to the collecting base station ([16, 22]), and other optimization objectives. It is apparent that the
topological structure of the communication backbone plays a vital role in its efficiency. However, it
is also important to note that a communication backbone which has good performance in some of
the criteria can have a bad one in others. For example, using the minimum spanning tree (MST)
as the backbone provides an optimal network lifetime performance for same initial battery charges
Preprint submitted to Elsevier June 6, 2015
*Article
Click here to view linked References
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[5], however it can have a very poor hop-diameter, which is critical for delay minimization.1Thus,
the network designer has to take special care when deciding which links to activate for the purpose
of data collection, as different optimization objectives may be have a negative effect on each other.
The problem of data collection can be divided into two major paradigms. Data collection with
aggregation ([25, 42]) allows each sensor node to accumulate the messages of its descendants and
then pass only one fixed-size message towards the base station. The second paradigm, is data
gathering without aggregation ([29, 30]) which requires that all messages initiated by the sensors
will eventually reach the base station.
Our main objective in this paper is to construct efficient communication backbones for single-
and multi-hop data collection with aggregation in WSNs for random sensor node deployments, while
measuring the efficiency based on the next three metrics.
•The transport capacity metric represents the sum of rate-distance products over all the active
links. It is measured in bit-meters and was first introduced by [20]. The idea behind this
measure is to capture both the notion of the overall rate and distance that the information
travels in a network.
•Hop-diameter is another important metric which reflects the depth of the data gathering tree,
i.e. the maximum number of hops from any of the sensor nodes to the base station.
•Total energy consumption is probably one of the most important parameters of a WSN as the
sensor nodes are often deployed in areas where battery replacement is infeasible [8]. Wireless
communication is a major contributer to the energy budget of a node. In this paper we focus
on minimize the total energy consumed by all nodes for communication purposes.
We propose a novel approach for the construction of communication backbones by identifying
central locations in the deployment area and routing all data through these regions. The general
idea is that these locations would serve as aggregation points both on a local and global scale. In
particular, we use an interesting geometrical notion of centroids, which is defined as the central
geometrical position of a collection of nodes, which are used as a guide for the construction of
hierarchical aggregation trees.
The rest of this paper is organized as follows. In Section 2 we present our system settings and
state our objective. Related works are surveyed in Section 3. Sections 4 and 5 are the technical
sections of the paper and show the construction of data collection communication backbones for
three scenarios, single-hop general network and multiple-hop random network. We present addi-
tional construction for arbitrary network deployment based on central nodes selection in Section
6. Simulation results for various types of networks are presented in Section 7 and compared to the
results of similar spirit obtained in [14]. Finally, we conclude and discuss future work in Section 8.
2. System settings
A WSN consists of nwireless sensor nodes, S={s1, . . . , sn}, distributed in some area A.
These nodes perform monitoring tasks and periodically report to a base station rwhich is located
somewhere within the area A(we consider different locations throughout the paper). During the
1Imagine nsensor nodes located on a straight horizontal segment, with the base station being to the right of the
right-most sensor. It is easy to show that the hop-diameter of MST in this case is n.
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report phase, the sensor nodes propagate a message to the base station through a data collection
tree,TS= (S∪{r}, ES), rooted at r. We consider data collection with aggregation, where every node
s∈Sforwards a single unit size report message to its parent. The message holds an accumulated
information collected from a subtree of TSrooted at s. An example of this scenario can be found
in temperature monitoring systems for fire prevention, intrusion detection, seismic readings, etc.
We assume the use of frame-based MAC protocols (see [10]) which divide the time into frames,
containing a fixed number of slots. The main difference from the classic TDMA is that instead of
having one access point which controls transmission slot assignments, there is a localized distributed
protocol mimicking the behavior of TDMA. The advantage of a frame-based (TDMA-like) approach
compared to the traditional IEEE 802.11 (CSMA/CA) protocol for a Wireless LAN is that collisions
do not occur, and that idle listening and overhearing can be drastically reduced. When scheduling
communication links, that is, specifying the sender-receiver pair per slot, nodes only need to listen to
those slots in which they are the intended receiver – eliminating all overhearing. When scheduling
senders only, nodes must listen in to all occupied slots, but can still avoid most overhearing by
shutting down the radio after the MAC (slot) header has been received. In both variants (link and
sender-based scheduling) idle listening can be reduced to a simple check if the slot is used or not.
Several MAC protocols were developed to adapt classical TDMA solutions which use access points
to ad-hoc settings that have no infrastructure; these protocols employ a distributed slot-selection
mechanism that self-organizes a multi-hop network into a conflict-free schedule (see [34, 45]).
Let d(u, v) be the Euclidean distance between two sensor nodes u, v ∈S. It is customary to
estimate that the energy required to transmit from uto vis proportional to d(u, v)α, where αis
the path-loss coefficient. In perfect conditions α= 2, however in more realistic settings (in presence
of obstructions or noisy environment) it can have a value between 2 and 4 (see [32]). In this paper
we assume α= 2 for simplicity. However, it is possible to extend our results for other values of α
which are greater than 2.
Let E(TS) be the energy requirement to execute a single report phase. Note that every
sensor performs a single transmission, during which it sends a single message to its parent in TS.
Thus, the energy requirement is proportional to the sum of squares of lengths of the edges E(TS).
The focus of this paper is to study the asymptotic performance of data collection trees, thus we
can express E(TS) as follows, E(TS) = (u,v)∈ESd(u, v)2.
Minimizing the energy requirement is one of the primary optimization objectives when
deploying a WSN due to the very low battery reserves at the sensor nodes and the high costs that
are associated with replacing these batteries (if at all possible).
Another critical aspect in the design of a WSN is the hop-diameter of TS. The data flows from
the leafs of the delivery tree to the base station, where each intermediate node waits to receive the
report messages from all its children, before sending its own report message to its parent. Therefore,
the hop-diameter of TS, denoted as H(TS), determines the delay of data collection.
The third measure that we are interested in is transport capacity,D(TS), of the data col-
lection tree TS. As mentioned earlier, the main idea which stands behind this metric is to capture
the spatial rate of the network, which is represented by the total rate over some distance. In our
scenario, the rate on all links is fixed as all the nodes transmit an aggregated, unit-size message, to
the parent in the collection tree and the schedule is conflict-free. Thus, to maximize the transport
capacity we need to minimize the total distance traveled by information, which is the sum of lengths
of all the links, D(TS) = (u,v)∈ESd(u, v).
Unfortunately it is impossible to achieve optimal performance in all three measures at the same
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time. For example, minimizing the hop-diameter results in all nodes transmitting to the base
station, which is disastrous in terms of energy consumption, whereas the best topology to minimize
energy consumption2results in a relatively high hop-diameter.
Our main objective in this paper is to construct data collection trees for several node distri-
bution scenarios which produce good performance in all three measures simultaneously.
3. Related work
This work is a continuation of works [14, 16] which take into account all of the above 3 per-
formance measures simultaneously. Below we discuss some of the related work on data collection,
energy efficiency, bounded-hop communication, and transport capacity.
In terms of total energy consumption measure, it was proved in [38] that using the minimum
spanning tree for data collection (gathering) with aggregation achieves optimality. A different
criterion used to measure energy efficiency is network lifetime, which is defined as the time the
first node depletes all its power reserves due to periodic data transmission. Segal [37] developed
an optimal maximum lifetime algorithm for data collection with aggregation. One of the possible
variants is to allow the use of different collection trees, which makes the maximum network lifetime
data gathering problem more challenging. Interestingly, if aggregation is allowed, the problem
is still polynomially solvable [25, 31], and is NP-complete otherwise [30]. Kalpakis et. al. [25]
developed an optimal data collection with aggregation algorithm in O(n15 log n) time. To counter
the slowness of the algorithm, Stanford and Tongngam [42] proposed a (1 −ε)-approximation in
O(n31
εlog1+εn) time based on Garg and K¨onemann [19]. For more details we refer the reader to
a recent survey by Ramanan et al. [27], which covers a diverse set of data gathering algorithms in
ad-hoc networks.
The notion of transport capacity was introduced by Gupta and Kumar in [20]. They showed
that for any layout of nwireless nodes in an area of size A, with each node being able to transmit W
bits per second to a fixed range, the overall transport capacity is at most (W√An) bit-meters per
second under both interference models (protocol and physical). In [24] the authors derive upper
bounds on the transport capacity as a function of the geographic location of the nodes. It has
also been shown that the scaling of transport capacity depends, among other factors, on channel
attenuation and path loss [47, 48, 49].
Finally, communication backbones with bounded hop-distances between participating nodes
has also been studied. For the linear layout of nodes and an upper bound on hop-distance, Kirousis
et al. [26] developed an optimal power assignment algorithm for strong connectivity in O(n4)
time. In the Euclidean case, [12] obtains constant ratio algorithms for the bounded-hop vertex
connectivity for well spread instances. Beier et al. [4] proposed an optimal algorithm to find a
bounded-hop minimum energy path between pairs of nodes. In [7] the authors obtain bicreteria
approximation algorithms for connectivity and broadcast while minimizing the hop-diameter and
energy consumption. Funke and Laue [18] provide a PTAS for the h-broadcast algorithm in time
linear in n. Elkin et al. [16] proposed the solution for the broadcast tree construction (which is easily
deformable into the data collection tree) such that the total energy consumption and transport are
of factor ρfrom optimal bound (which is proportional to the weight of minimal spanning tree for
the set of nodes where the weight of edge is defined as the squared Euclidean distance between
2As described later in the paper, using the Euclidean minimum spanning tree minimizes the energy consumption.
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the nodes) and the hop-diameter is n/ρ + log ρ, for any chosen integer parameter ρ, 1≤ρ≤n.
Additional results for bounded range assignments can be found in [11, 13, 40].
The additional work considering all of three measures has been done by Crowcroft et al. [14].
They [14] have shown two different approaches. The first is based on so-called balance nodes,
where the main motivation is to build data collection routes based on centrally located nodes in
topologies which are already efficient in terms of some of the metrics. In the second approach, they
examined the addition of shortcut links to the currently constructed topology in order to allow the
required tradeoff between studied criteria. We show, in our simulation scenarios, the comparison
between our and the first approach taken in [14] when producing tree TSwith guaranteed hop-
diameter H(TS) = O(log n) and transport capacity D(TS) = O(log n·w(M ST )), where w(M ST )
is the weight of minimal spanning tree for our set of nodes, when the edge weight is defined as the
Euclidean distance between two nodes. The reason we choose to make this specific comparison lies
in the fact that both constructions have logarithmic hop diameter and our goal was to check out
the performance of other characteristics.
4. Single-hop collection
We start be defining the notion of geometric centroids and then analyze the performance bounds
of single-hop communication backbone which is centroid-based. In the end we discuss the possible
pitfalls of using a single-hop collection tree.
For npoints P={p1, p2, . . . , pn},n≥2, placed in the Euclidean plane, with coordinates (xi, yi),
i= 1, . . . , n, and assuming general position, the centroid c(P) is a point defined as c(P) = (x, y),
where x=n
i=1 xi/n and y=n
i=1 yi/n, which conceptually represents the center location of P.
Apparently the centroid of npoints has two very interesting properties as outlined in the
following theorems that provide an analysis of the sum of squares of distances, which was done in
[28], and sum of distances, which we develop here, between the points and the centroid.
Theorem 4.1 ([28]). For any set of points Pand an arbitrary point p′in the Euclidean plane,
p∈Pd(p, c(P))2≤p∈Pd(p, p′)2.
Theorem 4.2. For any set of points Pand an arbitrary point p′in the Euclidean plane, p∈Pd(p, c(P)) ≤
2p∈Pd(p, p′).
Proof. Let p∗be the geometric median3of points P. Clearly for every p∈P,d(p, c(P)) ≤
d(p, p∗) + d(p∗, c(P)), and thus p∈Pd(p, c(P)) ≤p∈Pd(p, p∗) + |P| · d(p∗, c(P)). From the
convexity of the Euclidean norm it follows that the norm of an average of a set of points is at most
the average of the norms of the points in the set, that is d(p∗, c(P)) ≤p∈Pd(p, p∗)/|P|. Therefore
for any p′∈R2,d(p∗, c(P)) ≤2p∈Pd(p, p∗)≤2p∈Pd(p, p′).
Clearly, the bounds shown above represent the total energy consumption and transport capacity
measure for a single-hop data collection tree if the base station is located at the centroid of the
sensors, which is reasonable to expect.
Unfortunately, if we consider a single-hop tree TSrooted at the centroid and spanning all the
nodes it may be inefficient in terms of energy consumption and transport capacity. Consider the
3The geometric median p∗of a point set Pis the point in the Euclidean plane that minimizes the sum of distances
between itself and the points in P, i.e. ∀p′:∑p∈Pd(p, p∗)≤∑p∈Pd(p, p′).
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linear layout of nodes as depicted at Figure 1. If we consider the minimum spanning tree (MST)
of the nodes, we obtain D(M ST ) = E(M S T ) = n−1. On the other hand, for TS, we have
D(TS) = Ω(n2) and E(TS) = Ω(n3), which is quite poor compared to the one obtained by the
MST.
In the next section we describe a hierarchical topological structure, the k-layer centroid network,
which is used for multi-hop data collection in random WSNs and achieves better performance than
the single-hop tree.
Figure 1: Worst case performance of single-hop centroid-based data collection tree.
5. Multi-hop collection for random deployments
Our first construction of multi-hop data collection is for randomly deployed sensor networks.
In this scenario we assume that nsensor nodes are randomly and independently placed in the
area Awith uniform distribution. We also assume that Ais a unit square. We show an efficient
communication backbone construction which is based on centroid networks, which are hierarchical
geometrical structures on top of a point set Pwhich represents the sensor nodes. As in the single-
hop scenario, we assume the base station is located at c(P).
5.0.1. k-layer centroid networks
We start by providing the definitions and notation used in the context of k-layer centroid
networks and then proceed to presenting several useful properties and observations regarding these
networks.
The k-layer centroid network, k > 2, based on a point set P(in short, k-centroid network), is a
k-layer undirected tree TP= (V, E) , where Vand Eare the node and edge sets, respectively. The
leafs of the tree VP⊂Vrepresent the points P, and the internal nodes VC=V\VPrepresent the
centroids of subsets of P. Let rbe the root of the tree. For convenience we use the notion of node
and point interchangeably instead of saying node that represents a point.
The nodes Vare divided into klayers, V1, . . . , Vksuch that V1={r},VC=V1∪...∪Vk−1, and
Vk=VP. The edges Econnect between nodes in adjacent layers such that the parent of u∈Vi,
π(u), is in Vi−1and the children of v∈Vj,N(v), are in Vj+1, for any i, j, 1 < i ≤k, 1 ≤j < k. We
use Eito denote the set of edges between layers iand i+ 1, 1 ≤i≤k−1. In a k-centroid network
the following two conditions hold for any node v∈VC:
• |N(v)|>0.
•Let Tvbe a subtree of T, rooted at v∈VC, and let Pvbe the points represented by the leafs
of Tv. Then, vis the centroid of Pv.
For example, Figure 2 shows a 3-centroid network where the second layer nodes are centroids of
points sets P1, P2, P3, P4.
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Figure 2: An example of a k-centroid network.
Note that according to the second condition above, the root of the tree is the centroid of the
whole point set P. Next we provide several useful properties of k-centroid networks. We start with
an observation which follows directly from the definition above.
Observation 5.1. Let TPbe a k-centroid network, k > 2, and let vbe a non-leaf node in TPwith
a height of l. Then Tvis an (l+ 1)-centroid network based on Pvand if l > 1then for every
u∈N(v),Tuis an l-centroid network based on Pu.
Another interesting characteristic of k-centroid networks is that it is possible to easily add or
remove layers with only local changes to the edge set E. We refer to the process of adding layers as
extension and to the removal of layers as simplification. Let TPbe the original k-centroid network.
Asimplified network T(−i)
P= (V(−i), E(−i)) is obtained by removing the i-th layer, 1 < i < k
(the root and the leafs cannot be removed), and connecting the parent of every removed node to
its grandchildren in TP. Formally,
V(−i)=V\Vi
and
E(−i)= (E\Ei)∪ {(π(u), v) : u∈Vi, v ∈N(u)}.
Adding a layer to extend the network is essentially providing an additional level of grouping
the points into subsets. To add a layer below an existing i-th layer, 1 ≤i < k (it is not possible to
extend the network below the leafs layer), we need to remove the edges that connect layers Viand
Vi+1, and to add new edges which connect the new layer to the rest of the tree. Formally, for a
k-centroid network, the new (k+ 1)-centroid network, T(+i)= (V(+i), E (+i)), is defined as follows.
For each node uj∈Vi, 1 ≤j≤ |Vi|, we partition its children N(uj) into mj, 1 ≤mj<|N(uj)|
disjoint subsets Uj
1, . . . , U j
mj⊆N(uj). Then, the new nodes of the added layer, V[i]↔[i+1], are the
centroids of the union of the leafs in the trees rooted at the nodes of these subsets, that is
V[i]↔[i+1] ={uj
l: 1 ≤j≤ |Vi|,1≤l≤mj},
where
uj
l=c({p:p∈Pv, v ∈Uj
l}).
The edge set is modified by disconnecting Viand Vi+1 and connecting these layers to the new nodes,
E(+i)=(E\Ei)∪ {(uj, uj
l) : 1 ≤j≤ |Vi|,1≤l≤mj}
∪ {(uj
l, v) : 1 ≤j≤ |Vi|,1≤l≤mj, v ∈Uj
l}.
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It is easy to see that in both cases, the result of either simplification or extension is a proper
(k−1)- or (k+ 1)-centroid network, respectively. Also note that the simplification process is
deterministic for every removed layer, whereas in the case of extension there are multiple possible
outcomes.
The following theorem shows that by extending an existing k-centroid network we actually
reduce the sum of squares of distances in the network. For the ease of exposition we denote by
d(u, v) the distance between the points or centroids represented by the nodes u, v ∈V, and by
S2(TP) the sum of squares of distances in TP, i.e. S2(TP) = (u,v)∈Ed(u, v)2.
Theorem 5.2. Let TPbe a k-centroid network based on a point set P. For any l-centroid network,
T′
P, which can be obtained through a series of extensions from TP, it holds that S2(TP)≥S2(T′
P).
Proof. In order to prove the theorem, it is enough to show that for l=k+ 1 it holds
S2(TP)≥S2(T′
P). Let T′
Pbe obtained from TPby adding a layer below some existing i-th layer,
for some i, 1 ≤i < k. We are going to show, separately for each centroid uj∈Vi, 1 ≤j≤ |Vi|,
that Σ(xm,ym)∈N(uj)d(uj,(xm, ym))2≥Σmj
l=1d(uj, uj
l)2+ Σmj
l=1Σp∈Uj
l
d(uj
l, p)2. Subtract from the left
side of the inequality the right side. If the theorem is true, the result should be non-negative.
Σ(xm,ym)∈N(uj)d(uj,(xm, ym))2−Σmj
l=1d(uj, uj
l)2−Σmj
l=1Σp∈Uj
l
d(uj
l, p)2=Σmj
l=1|Uj
l|x2
uj−2Σmj
l=1|Uj
l|xujxuj
l
+
Σmj
l=1|Uj
l|xuj
l
2−Σmj
l=1x2
uj+2Σmj
l=1xujxuj
l−Σmj
l=1|Uj
l|xuj
l
2+Σmj
l=1|Uj
l|y2
uj−2Σmj
l=1|Uj
l|yujyuj
l
+Σmj
l=1|Uj
l|yuj
l
2−
Σmj
l=1y2
uj+ 2Σmj
l=1yujyuj
l−Σmj
l=1|Uj
l|yuj
l
2= Σmj
l=1(|Uj
l|−1)(xuj−xuj
l
)2+ Σmj
l=1(|Uj
l|−1)(yuj−yuj
l
)2=
Σmj
l=1(|Uj
l|−1)((xuj−xuj
l
)2+ (yuj−yuj
l
)2) = Σmj
l=1(|Uj
l−1)d(uj, uj
l)2.Clearly the last expression is
equal or larger than 0.
Figure 3: Extending a k-network does not always improve the sum of distances.
Unfortunately, we cannot make a similar claim for sum of distances (as demonstrated in Fig-
ure 3). Let us consider the set Pof k(2a+ 2), k, a ∈N, points on line, ordered in increasing order by
their coordinates xi,i= 1, ..., n. The distance between points xiand xi+1, for i= 1, ..., k (2a+ 2),
i̸=k,i̸=k(a+ 1), i̸=k(2a+ 1), is 1; for i=k(a+ 1) it is 2; and for i=k,i=k(2a+ 1), it is
d. The coordinate of the centroid Cof the points in P, is x=k(a+ 1) −1 + d. We partition P
into two sets: P1=x1, ..., xk(a+1) and P2- the remaining points. Coordinate of the centroid C1
of the points in P1, is x1=k(a+1)2+2ad−3a−1
2(a+1) . The sum of the distances between Cand the points
in P1is
k(a+1)
i=1
(x−xi) = k2(a+1)2+k(a−1)+2kd
2. The sum of the distances between C1and the points
in S1is
k(a+1)
i=1 |x1−xi|=k2a(a+1)−2ak+2akd
a+1 . When comparing these two sums one can see that in
the case of a > 1 and for any given kthere is such dfor which
k(a+1)
i=1 |x1−xi|>
k(a+1)
i=1
(x−xi).
The same happens with a symmetric case while considering the sum of distances between Cand
C2(the centroid of P2) and the points of P2. Thus, we obtain that the sum of distances from Cto
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the entire set is less than the sum of distances from C1to P1plus the sum of distances from C2to
P2.
5.1. Data collection using centroid networks
First we describe the k-centroid network T= (V , E) which we then use to produce the commu-
nication backbone.
Let Pbe the points that correspond to the location of sensor nodes S. To construct Twe
repeatedly divide the unit square area Ainto sub-areas. First we divide Ainto 4 equal square
sub-areas, then each of these sub-areas is further divided into 4 sub-areas, and so forth. In every
sub-area we pick one centroid of the points in that sub-area and add it to T. The connections
between these centroids are added according to the point hierarchy as described above, while the
root of Tis the centroid of all the points P. The iteration proceeds in steps, where at each
step we handle subdivision of sub-areas of the same size; it ends once it is not possible to continue
subdividing the areas into non-empty square regions. In the final phase, the centroids are connected
to the points in their respective areas. We now describe this process in detail.
1. Let j←1, A1← {A},V←V1← {c(P)},E← ∅.
2. While it is possible to divide all the areas in Ajinto 4 non-empty equal square sub-areas:
(a) Initialize Aj+1 ← ∅,Vj+1 ← ∅,Ej← ∅.
(b) For every A′∈ Aj:
i. Let P′be the points which are within area A′.
ii. Divide A′into 4 equal square sub-areas A′
1, A′
2, A′
3, A′
4. Let P′
1, P ′
2, P ′
3, P ′
4be the
points sets in these areas, respectively.
iii. Add the centroids c(P′
1), c(P′
1), c(P′
1), and c(P′
1) to Vj+1.
iv. Add the edges (c(P′), c(P′
1)), (c(P′), c(P′
2)), (c(P′), c(P′
3)), and (c(P′), c(P′
4)) to Ej.
v. Add the areas A′
1,A′
2,A′
3, and A′
4to Aj+1.
(c) Update V←V∪Vj+1,E←E∪Ej, and increase j←j+ 1.
3. Initialize Vj+1 ← ∅ and Ej← ∅.
4. For every A′∈Aj:
(a) Let P′be the points inside A′.
(b) Add all the edges {(c(P′), p) : p∈P′}to Ejand P′to Vj+1 .
5. Update V←V∪Vj+1 and E←E∪Ej.
For example of an execution, see Figure 4.
Let kbe the last value of jin the above scheme. Clearly, the obtained T= (V, E) is a k-centroid
network. We are now ready to define the data collection tree TS= (S∪{r}, ES). The general idea
is to match between the virtual nodes in Vand the sensor nodes S. For every centroid c∈V, let
Pcbe the points that produced c, and let Scbe the sensor nodes that correspond to these points.
Then we choose sc∈Scto be the sensor node which is closest4to c, i.e. d(sc) = mins∈Scd(s, c).
Note that we might not have a tree yet, as it is possible that there are cycles and self-loops. These
are easily removed by running a breadth-first search in the obtained graph, starting with the root
(the node closest to the centroid of all the sensors). The resulting breadth-first tree is TS.
In order to estimate the efficiency of the constructed data collection tree TSwe will use the
following theorems.
4The distance between a sensor node and a point is the Euclidean distance between the location of the sensor
node and the coordinates of the point in the Euclidean plane.
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(a) The single centroid in V1. (b) The centroids V2. (c) The centroids V3.
(d) The 3-centroid network
(without links to the points).
Figure 4: An example of the algorithm execution, showing the construction of a 3-centroid network (the links to the
points in the last layer are omitted for clarity). The triangles represent the centroids in all three layers.
Theorem 5.3 ([43]). The sum of the of edges of an MST in two (three) dimensional unit size
square (cube) for random uniform points is Ω(n1/2)(Ω(n2/3)).
Theorem 5.4 ([36], Theorem 2.2). The sum of the squares of edges of an MST in two (three)
dimensional unit size square (cube) for random uniform points is Ω(1) (Ω(n1/3)).
We claim the following.
Theorem 5.5. For the data collection tree TSit holds that H(TS) = O(log n),E(TS)is O(log n)
times the optimal, and D(TS)is O(1) times the optimal.
Proof. Suppose we run the scheme of constructing Tuntil j=h+ 1, h+ 1 ≤k. In other
words we perform step 2 of the scheme h−1 times and then proceed to step 3. We obtain the
h+1-centroid network. Denote the sum of squares of lengths of edges in Ehby [2]
hand the sum of
squares of lengths of the edges in E\Ehby [2]
c. Denote the sum of lengths of edges in Ehby h
and the sum of squares of lengths of the edges in E\Ehby c. Obviously, E(TS) = [2]
h+[2]
c
and D(TS) = h+c. After h−1 times of Apartition, Ahhas 4h−1square sub-areas, each
of which has diagonal of length 2
|Ah|=2
4h−1. Since, each A′∈ Ahhas one centroid (and
the total amount of centroids in Ahis |Vh|=|Ah|= 4h−1), then [2]
h≤2
|Ah|2(n− |Vh|)≤
2
4h−1n−4h−1=On
4h−1, and t≤2
|Ah|(n− |Vh|)≤2
4h−1n−4h−1=On
√4h−1. Sup-
pose the sensors are located on the vertex points of unit size grid, with grid cells of size 1
√n−1×1
√n−1.
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By applying the scheme of construction Tfor this case of sensors arrangement, it is easy to see
that [2]
c=O(log |Ah|) = Olog 4h−1and c=O|Ah|=O√4h−1. Thus, for the case of
grid, E(TS) = [2]
h+[2]
c=On
4h−1+ log 4h−1and D(TS) = h+c=On
√4h−1+√4h−1.
Returning to the stochastic case we observe that the sums [2]
cand care maximal, if in each
A′∈ Ahthere is a sensor. This implies that each A′∈ Ai,i= 1, . . . , h −1, is not empty. Therefore,
[2]
cand care equal to those on the grid up to a constant, even if we choose the centroids
arbitrary within their corresponding sub-area. Thus, E(TS) = [2]
h+[2]
c=On
4h−1+ log 4h−1
and D(TS) = h+c=On
√4h−1+√4h−1. For h−1 = log4n,E(TS) = O(log n) and
D(TS) = O(√n). Note that for the random uniform distribution of sensors in A, the algorithm for
constructing Tstops when the area of square in Ahis at most Olog n
n[41]. Since |Ah| ≥ n
log n, it
follows that k=Olog4n
log n=O(log n). Thus, E(TS) = O(log n) and D(TS) = O(√n).
Following Theorems 5.3 and 5.4, we conclude that E(TS) is O(log n) times the optimal, and
D(TS) is O(1) times the optimal solution.
Suppose that the sensors are uniformly distributed within a unit cube. According to the
scheme, similar to the above, we can build the k-centroid network, k=O(log n). Following
the assumptions and arguments similar to the above, it can be shown, that [2]
h=On
3
√(8h−1)2,
h=On
3
√8h−1, and [2]
c=O3
√8h−1,c=O3
(8h−1)2. Thus, E(TS) = [2]
h+[2]
c=
On
3
√(8h−1)2+3
√8h−1and D(TS) = h+c=On
3
√8h−1+3
(8h−1)2. For h−1 = log8n,
E(TS) = O(3
√n) and D(TS) = O3
√n2. Using the results of Theorems 5.3 and 5.4, we obtain
that for three-dimensional case E(TS) and D(TS) are O(1) times the optimal solution.
One can wonder whether the idea of routing all data through central nodes would generate a
congestion points in the network backbone and also influence the on the lifetime of the network.
First, we observe that in our centroid-based construction the average squared edge’s length is O(1)
which compares well with the similar bound for MST, see Zhang and Hou [50]. It means that it
terms of lifetime, we would expect the good performance of our solution. Regarding the congestion
points, a good synchronization and scheduling mechanism, for example see algorithm Time-Slots for
tree proposed by [17], efficiently deals with the problem. In particular, each node in the tree keeps
the total number dof its siblings (having the same parent) and it’s sequential number between the
ordering of siblings. Then this node is allowed to send a message to its parent if the time slot is
equal to the sequential number of the node modulo the total number of siblings.
The distributed implementation of the k-centroid network is quite straightforward once we
established connectivity between the nodes and chose the leader (the root of the tree). In order to
establish connectivity we can use 2 different approaches. The first, described in Dolev et al. [15]
forms a temporary underlying topology in O(n) time using O(n3) message. The second (better)
approach is given by Halld´orsson and Mitra [21] that show how to do this in O(poly(log β , log n)),
where βis the ratio between the longest and shortest distances among nodes. After the topology is
established, we can use leader-election algorithm by Awerbuch [3] that shows how to find a leader in
a distributed fashion in a network with nnodes in O(n) time using O(nlog n) messages. Next, using
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the location of every node, the root of k-centroid network can be determined and the construction
of the k-centroid network is started in recursive fashion that takes O(k) time and O(n) messages
assuming omnidirectional communication.
6. Multi-centers for arbitrary deployments
The construction of our data collection tree is based on finding a Hamiltonian circuit in the
network. The existence of Hamiltonian circuit is evident since graph Grepresents a full graph
(all the nodes potentially capable transmitting to each other). The Hamiltonian circuit is built
on top of a minimal spanning tree of the graph, based on an algorithm presented by Andreae
and Bandelt [2]. Following the algorithm, we build in linear time a Hamiltonian circuit with the
following properties:
Theorem 6.1 ([2]). Let hbe a Hamiltonian circuit of graph G, built over the minimal spanning
tree of G,TM ST . The following applies:
1. W(h)=O(W(TMST )).
2. w(e*(h))=O(w(e*(TMST ))).
where W(g) is the weight of some graph g, calculated as the energy consumption of all the edges
in the graph, and e*(g) is the longest edge in graph g.
For full algorithm description and further information see [2] and [16]. Let k, d be natu-
ral numbers such that n=kd. We assign indexes to the nodes of circuit h={u1, u2, ..., un},
which consist of vectors vi∈V:S→Rdsuch that each vector consists of values in range
vi= (a1, ..., ad),0≤aj≤k−1,1≤j≤d. Subsequent nodes in the Hamiltonian circuit will
receive subsequent vector indexes (e.g. for k= 2, d = 2 {u1, .., u4}={00,01,10,11}). In fact,
parameter ddefines the number of centers that we will be using in the construction of our data
collection tree.
Figure 5: [k=3,d=3] Index Labeling based on Hamiltonian Circuit Order.
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Next, starting at empty graph Kn, we add an edge between every pair of nodes with indexes
subject to (a1, ..., aj,0, ..., 0) and (a1, ..., aj−1,0, ..., 0) for some j∈[1, d], resulting in graph Tk,d.
Figure 6: Tree construction example for k= 3, d= 3.
Theorem 6.2. Tk,d is a spanning tree of G.
Proof. Since by definition, every node is connected to the node with the last non-zero member of
index vector decremented (except node v0= (0, ..., 0) which is connected as well), every node in G
has at least one edge. Hence, Tk,d is a spanner. Next, we count the number of edges, created by the
change of each aimember of index vector. a1affects k−1 edges: (0,0, ..., 0) −(k−1,0, ..., 0). a2
affects k·(k−1) edges: (a1,0,0, ..., 0) −(a1, k −1,0, ..., 0), while a1∈[0, k]. Consequently, the total
number of edges created as a result of using the algorithm: d−1
i=0 (k−1) ·ki= (k−1)kd−1
k−1=n−1.
Since Tk,d spans G,Tk,d is a spanning tree.
Theorem 6.3. Let Tk,d be a tree constructed by the upper mentioned method. The following bounds
apply for Tk,d:
1. Hop-Diameter: H(Tk,d) = 2(k−1) ·d.
2. Energy Consumption: W(Tk,d) = O(n
k·W(TMST )).
3. Transport Capacity: D(Tk,d) = O(d·D(TM ST )).
Proof.
1. Diameter bound. In order to traverse from some node (a1, .., ad) to node v0we need to traverse
from each node with (a1, ..., aj−1, aj,0, ..., 0) to (a1, ..., aj−1,0,0, ..., 0) which is up to k−1
hops. Since 1 ≤j≤d, the maximal number of hops to node v0is d(k−1), hence diameter is
2d(k−1).
2. Energy Consumption bound. We assume that the distance between every two adjacent nodes
in the Hamiltonian circuit is bounded by l=O(e∗(TMST )), and thus require O(l2) energy to
communicate. We sum up all of the edges’ energy consumption, by iteratively summing up
the edges in each dimension 1≤j≤d:W(Tk,d) = e∈E(Tk,d )d(e) = d
j=1 e∈E(dim j)d(e) .
Each dimension jcontains (k−1)kj−1edges, each O(kd−j·l) long and consumes O((kd−j)2·l2)
energy. Hence,
W(Tk,d) =
d
j=1
(k−1) ·kj−1(kd−j)2·l2=l2(k−1) ·k2d−1
d
j=1
(1
k)j=
=l2(k−1) ·k2d−1(
d
j=0
(1
k)j−1) = l2(k−1) ·k2d−1((1
k)d+1 −1
1/k −1−1) =
=l2·k2d(k−1−k−(d+1)) = l2·O(k2d−1) = O(l2·n2
k) =
=O(n
k·W(TMST ))
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3. Transport Capacity bound. Transport capacity is measured by the total distance traveled by
information when transmitted by all nodes simultaneously. Thus, the metric depends on the
total sum of edges’ lengths, which we sum up by grouping them into dimensions, similarly to
before.
D(Tk,d) =
d
j=1
(k−1) ·kj−1kd−j·l= (k−1)l·
d
j=1
kd−1=
= (k−1) ·l·kd−1·d=k−1
k·l·n·d=O(l·n·d) =
=O(d·D(TMST ))
Corollary For k=n
log nand d=log n−log log n
log n=O(1) we get the following bounds:
1. Hop-Diameter: H(Tk,d) = O(n
log n).
2. Energy Consumption: W(Tk,d) = O(log n·W(TM ST )).
3. Transport Capacity: D(Tk,d) = O(D(TM ST )).
Notice, that these results beat the results of Elkin et al. [16] that for logarithmic hop-diameter
data collection tree (having the same energy performance bounds) achieved logarithmic factor
approximation for transport measure, opposite to our constant factor approximation.
Comparing the obtained results with the cenroids-based approach for the uniform random
deployment, we can conclude that for similar transport and energy performance for both arbitrary
and unform cases, we can build better hop-diameter O(log n) data collection tree for uniform case
than for arbitrary case (O(n/ log n)).
7. Simulation results
In this section we show some simulation results of the k-centroid network constructed for the
multi-hop random scenario as described in Section 5. As we show, the simulation results fully
support and even slightly outperform our theoretical analysis. In what follows we compare the k-
centroid network topology with the optimal one, in terms of both energy consumption, hop-diameter
and total link distance, which is achieved by using the minimum spanning tree and balance nodes
partition based hierarchical tree as the delivery trees. The optimality of MST is straightforward in
the case of total distances, whereas for energy consumption it was shown to be the best possible in
[38].
In our experiments we have randomly and uniformly distributed nsensor nodes in a unit square,
with the network size nranging from 100 to 5000 in steps of 100. We have computed the total
distance of the communication links (Figure 7), the energy consumption (Figure 8), and the hop-
diameter (Figure 9) and the weight of the maximal edge of all three topologies. While transport,
energy and hop-diameter parameters are natural for evaluation, we choose to evaluate the weight
of the maximal edge since the network lifetime (in case of equal initial capacities of all sensors) is
heavily depends on this measure. The results below are an average of 20 tries for every network
size n.
In terms of total distance (Figure 7), our solution is consistently within a factor less than 2
from the best possible (MST), which matches the theoretical result of the O(1) approximation
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Uniform Balanced tree Uniform centroid Uniform MST
Total distance units
Figure 7: Total link distances, D(·).
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Uniform Energy
Uniform Balanced tree Uniform centroid Uniform MST
Total energy units
Figure 8: Total energy consumption (sum of square of distances), E(·).
ratio. Moreover, our algorithm performance beats the performance of the balanced tree solution
proposed in [14].
Interestingly, the energy consumption of our scheme (Figure 8) slightly outperforms the pro-
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jected theoretical bound of O(log n), with the ratio rising from 3 −4 for smaller networks and up
to 6 for larger ones with n≥3000. As for previous metric, our algorithm again outperforms the
solution given in [14]. The practical meaning of the obtained results is that the theoretical bound
obtained by us might not optimal and be even decreased by more delicate analysis.
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Uniform Hop diameter
Uniform Balanced tree Uniform centroid Uniform MST
Hop distance to the root
Figure 9: Max hop-distance from every sensor to the root, H(·).
The hop-diameter of our scheme is very close to the optimum (which is obviously 1), being only
8 for a 4000-node WSN. We have used a logarithmic scale in (Figure 9) to compare it to the one
produced by MST, which is 10−15 times greater than ours for small networks (nfrom 100 to 700),
and as high as 30 times greater for larger ones (n≥700). The balanced tree approach explained
in [14] also produces good results but still produces worse values than ours.
The length of maximal edge is obviously optimal for MST. As we can learn from the Figure 10,
our solution again outperforms balanced tree approach in [14] and lies within the small constant
factor for different size networks. This compares well with our analysis for the lifetime of the
networks where we show that the average length of the edge in the centroids-based construction
is similar to the average length of the edge’s length in MST. Moreover, we also have checked the
values of maximal edge for the arbitrary (opposite to uniform) deployment of sensor nodes. The
results are shown in Figure 11.
From Figure 11 we can see the similar performance as was obtained for the uniform distribution
of nodes. It means that the deployment (almost) does not influence the performance of our proposed
construction.
8. Conclusions
In this paper we developed various data collection topologies that were based on the location
theory notion of centroids and central nodes. We have shown that a centroids based hierarchy
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Figure 10: The length of maximal edge for uniform deployment.
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Figure 11: The length of maximal edge for arbitrary deployment.
provides good approximation factor solutions for energy, transport capacity, and hop-diameter
measures, in 2D, and performs asymptotically optimal in 3D for random sensors locations. Our
simulation results verify our theoretical findings and, in fact, suggest that a possible tighter analysis
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for two dimensional space may exist. We also presented a construction for arbitrary deployment
based on central nodes of the network that performs well for our desired measures. It would also
be interesting to investigate the construction, where one of the objectives is an average hop-count
between the nodes in the obtained network.
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