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LIFL 2002-n
o
08 Juillet 2002
Publication LIFL 2002-n
o
08
LOCALIZED MINIMUM-ENERGY
BROADCASTING IN AD-HOC NETWORKS
Julien CARTIGNY, David SIMPLOT, and Ivan STOJMENOVIC
Juillet 2002
UNIVERSIT
´
E DES SCIENCES ET TECHNOLOGIES DE LILLE
Laboratoire d’Informatique Fondamentale de Lille, B
ˆ
at. M3 Cit
´
e Scientifique, 59655 VILLENEUVE D’ASCQ CEDEX
T
´
el. 03.20.43.47.24 Fax 03.20.43.65.66
TECH. REPORT LIFL 2002-8 1
Localized minimum-energy broadcasting in ad-hoc
networks
Julien Cartigny, David Simplot, and Ivan Stojmenovic
Abstract— In the minimum energy broadcasting prob-
lem, each node canadjust its transmission power in order to
minimize total energy consumption but still enable a mes-
sage originated from a source node to reach all the other
nodes in an ad-hoc wireless network. In all existing solu-
tions each node requires global network information (in-
cluding distances between any two neighboring nodes in the
network) in order to decide its own transmission radius. In
this paper, we describe a localized protocol where each node
requires only the knowledge of its distance to all neighbor-
ing nodes and distances between its neighboring nodes (or,
alternatively, geographic position of itself and its neighbor-
ing nodes). In addition to using only local information, our
protocol is shown experimentally to even provide more en-
ergy savings than the best known globalized BIP solution.
Our solutions are based on the use of relative neighborhood
graph which preserves connectivity and is defined in local-
ized manner.
Index Terms—Energy conservation, wireless ad-hoc net-
works, broadcasting, localized algorithms.
I. INTRODUCTION
I
N wireless ad-hoc networks, such as sensor networks,
all nodes cooperate to handle network facilities. These
networks are power constrained as nodes operate with re-
stricted battery power. We consider nodes that have the
capacity to modify the area of coverage with its transmis-
sion. Indeed, control of the emitted transmission power
allows to reduce significantly the energy consumption and
so to increase lifetime of the network. However, the ad-
justment of transmission signal strength generally implies
topology alterations like loss of the connectivity. Hence,
nodes have to manage their transmission area while main-
taining the connectivity of the network.
Julien Cartigny and David Simplot are at the Fundamental Com-
puter Science Laboratory of Lille (LIFL), France (http://www.lifl.fr).
E-mail: {cartigny,simplot}@lifl.fr.
Ivan Stojmenovic is with SITE at the University of Ottawa, Ontario,
K1N 6N5, Canada. E-mail: ivan@site.uottowa.ca.
This work was partially supported by a grant from Gemplus Re-
search Labs., an ACI Jeunes Chercheurs “Objets Mobiles Commu-
nicants” (1049CDR1) from the Ministry of Education and Scientific
Research, France, the CPER Nord-Pas-de-Calais TACT LOMC C21
and NSERC.
In the broadcasting task, a message originated from a
source node needs to be forwarded to all the other nodes
in the network. In this paper, we focus on the develop-
ment of protocols for energy-efficient broadcast commu-
nications. All existing solutions are globalized, meaning
that each node needs global network information. Mobil-
ity of nodes, or changes in their activitystatus (from active
to passiveand vice versa) may cause global changes in any
MST based structure. Therefore topology changes must
be propagated throughout the network for any globalized
solution. This may result in extreme and unacceptable
communication overhead for ad-hoc networks. Hence, be-
cause of the limited resources of mobile nodes, it is ideal
that each node can decide on its own behavior based only
on the information from all nodes within a constant hop
distance. Such distributed algorithms and protocols are
called localized [4], [7], [17], [18], [28]. Of particular
interest are protocols where nodes make decisions based
solely on the knowledge of its 1-hop or 2-hops neighbors,
and distances to them. In non-localized distributed, or
globalized, algorithms nodes require knowledge of whole
network topology to make decision.
Several different protocols have been proposed to man-
age energy consumption by adjusting transmitting pow-
ers. Among existing protocols, we can distinguish two
families of protocols: topology control oriented protocols
and broadcast oriented protocols.
The first family (topology control oriented protocols)
assigns the transmission power for each node such that
the network is connected independently of broadcast uti-
lization. That means that all nodes can be a source of
a broadcast and are able to reach all nodes of the net-
work using pre-assigned transmission radii at each node.
The optimization criterion is minimizing the total trans-
mission power assigned according to an energy consump-
tion model. This problem is known as min(-total) assign-
ment problem and was considered by Kiroustis et al. [11]
which established that this problem is NP-hard for tree-
dimensional space. Clementi et al. [8] showed that this
complexity result still occurs for two-dimensional space.
Approximate solutions [7], [15], [26] are based on min-
imum spanning trees or approximation of minimal span-
2 TECH. REPORT LIFL 2002-8
ning trees and are globalized.
The second family (broadcast oriented protocols)
achieves the same objectives but considers the broadcast
process from a given source node. For instance, Wieselth-
ier et al. [26] proposed greedy heuristics which are based
on Prim’s and Dijkstra’s algorithms. The more efficient
heuristic, called BIP for broadcasting incremental power,
constructs a tree starting from the source node and adds
new nodes one at a time according to a cost evaluation.
The constraints are not the same as for the first protocol
family since in this second case the subgraph induced by
the minimum-energy broadcast tree does not need to be
strongly connected: the only condition is that the source
can reach every node of the network. It has been proved in
[9], [13] that the minimum-energy broadcast tree problem
is NP-complete and [13] proposed an approximate glob-
alized algorithm which gives solutions with bounded ratio
against lower bound.
We can also distinguish several communication mod-
els: one-to-all model, one-to-one model and variable an-
gular range model. In one-to-all model, mobile nodes use
omnidirectional antennas and the communication zone of
a node is a disk centered at this node. All above cited
works (and all references except [6], [21], [27]) use this
model. In one-to-one model, nodes are equipped with di-
rectional antennas with small angles that can provide more
energy savings and interference reduction since the com-
munication zone of a node is a small beam from this node
to the targeted node [21]. With variable angular range
model, the nodes can choose direction and width of the
beam that allows to target several neighbor with one trans-
mission. Hardware solutions using directional antennas
(also called smart antennas) are more difficult to imple-
ment and we focus in this paper on one-to-all model. The
broadcast energy problem for other models are addressed
in [27] and our forthcoming paper [6].
In this paper we are mainly interested in broadcast
oriented protocols in one-to-all communication model in
wireless ad-hoc networks. The main contribution of this
paper is that we propose an algorithm that requires lo-
cal information while all existing solutions are globalized,
that is distributed where nodes require full knowledge of
network to make decision. The information needed in our
protocols are included in information needed by existing
protocols like BIP. In our localized protocols, each node
requires only the knowledgeof its distance to all neighbor-
ing nodes and distances between its neighboring nodes.
Distances can be measured by using signal strength, time
delay or more sophisticated techniques like microwave
distance [2]. If a positioning system (like GPS) is avail-
able, each node only needs position information from its
neighbor nodes.
The paper is organized as follows. In next section,
we present communication and energy models. In Sec-
tion III, we give a literature review of minimum energy
broadcast protocols. In Section IV, we describe how this
problem can be solved with localized algorithms. Sec-
tion V presents the results of our simulations where we
demonstrate the efficiency and superiority of our algo-
rithms. Finally, Section VI presents conclusion and future
directions.
II. PRELIMINARIES
A. Communication Model
We consider multi-hop wireless networks where all
nodes cooperate in order to fulfill a given communication
task. Such a network can be modeled as follows. A wire-
less network is represented by a graph G = (V, E) where
V is the set of nodes and E ⊆ V
2
the edge set which
gives the available communications: (u, v) belongs to E
means that u can send messages to v. In fact, elements of
E depend of node positions and communicating range of
nodes. Let us assume that maximum range of communi-
cation, denoted by R, is the same for all vertices and that
d(u, v) is the distance between nodes u and v.
For instance, the set E can be defined as follows:
E = {(u, v) ∈ V
2
| d(u, v) ≤ R}.
So defined graph is known as the unit graph, with R as its
transmission radius.
In given graph G = (V, E), we denote by n = |V |
the number of nodes in ad-hoc network. The neighbor set
N(u) of vertex u is defined as N (u) = {v | (u, v) ∈ E}.
The average degree of the network is the average number
of neighbors of its nodes.
We will assume that each node can change the power
of its transmissions for energy savings reasons (see next
subsection). In this case, the range of a node u ∈ V rep-
resents the maximal distance between u and a node which
can receive its transmission. The range of a node u ∈ V
is denoted by r(u) (with 0 ≤ r(u) ≤ R). The graph in-
duced by the range assignment function r is denoted by
G
r
= (V, E
r
) where the edge set E
r
is defined by:
E
r
= {(u, v) ∈ V
2
| d(u, v) ≤ r(u)}.
It is straightforward to see that the graph G
r
with mod-
ified ranges is not always undirectional.
A (directed) graph is strongly connected if for any two
vertices u and v, a path connecting u to v exists. In the
broadcasting task, a message needs to reach all nodes in
CARTIGNY ET AL.: LOCALIZED MINIMUM-ENERGY BROADCASTING IN AD-HOC NETWORKS 3
the network by transmitting from the source and retrans-
mitting by other network nodes with variable transmission
radii. Hence, in case of broadcast, the strong connectiv-
ity is not needed, we only need connectivity from source
node to all the other nodes in the network.
B. Energy Model
Commonly, the measurement of the energy consump-
tion of network interfaces when transmitting a unit mes-
sage depends on the range of the emitter u:
E(u) = r(u)
α
,
where α is a real constant greater than 2 and r(u) is the
range of the transmitting node. This model is used in [1],
[7], [9], [14], [15], [16], [24], [25], [26]. In reality, how-
ever, it has a constant to be added in order to take into
account the overhead due to signal processing, minimum
energy needed for successful reception and MAC control
messages [10]. The general energy consumption formula
is:
E(u) =
½
r(u)
α
+ c if r(u) 6= 0,
0 otherwise.
For instance, Rodoplu and Meng [19] consider the
model with E(u) = r(u)
4
+ 10
8
. This last model, also
used in [12], is more realistic as illustrated is Fig. 1: with
parameters α = 2 and c = 0, it is clear that the trans-
missions illustrated in subfigure (b) cost the same energy
as the one in subfigure (a) by using Pythagoras theorem.
By induction, all illustrated configurations are supposed
to have the same energy consumption and can be arbi-
trary extended. For medium access, signal processing and
reception power reason, it is not in accordance with real
world.
Another example are nodes placed on a line segment.
Assuming c = 0 and α ≥ 2, it follows that energy savings
are obtained when arbitrary number of nodes are placed
between source S and destination D, and these nodes are
used to retransmit the message. This will certainly con-
tradict basic signal processing requirement for minimal
reception power, and cause significant amount of colli-
sions in medium access layer if used by many simultane-
ous routing, multicasting and broadcasting tasks.
C. Minimum energy broadcasting
A transmission range assignment on the vertices in V is
a function r from V into an real interval [0, R] where R is
the maximal range of nodes. In some wireless networks,
the transmission range at each node has finite number of
possible values meaning that r is a function into a finite
subset of R. In accordance to reviewed literature, each
node can adjust its own power level, i.e. that can adjust
its transmission range. Each node has to reduce its trans-
mission range while maintaining the connectivity of the
graph. The measurement of total power consumption is
given by the following formula:
E =
X
u∈V
E(u).
III. LITERATURE REVIEW
We start with topology control protocols that aim to ad-
just transmission power while preserving strong connec-
tivity of the network. In [11], the Kirousis et al. address
the tree construction in wireless networks by using glob-
alized protocols. The authors showed that this problem is
NP-hard for three dimensional space and give an approx-
imation algorithm for constructing a spanning tree that
minimizes the total power consumption. Clementi et al.
showed that the minimum energy range assignment prob-
lem is still NP-hard in two-dimensional case.
Wieselthier et al. define in [26] a topology control al-
gorithm based on minimum-power spanning tree (MST in
short). Let V be a set of nodes and G = (V, E) the in-
duced graph with maximal range R. We assume that the
graph G is strongly connected. The weights of edges are
given by the selected energy model (but in fact, the MST
does not depend on particular choice of the metric because
of monotonicity). The construction of the MST is possi-
ble if we can determine distances between nodes. For in-
stance, Fig. 2 and 3 show a graph of 100 vertices and its
MST. Notice that in the unit graph on Fig. 2 the average
degree is 8 while in the MST on Fig. 3, the average degree
is less than 2.
Fig. 2. A graph with average degree 8.
It is well-known that the graph MST (G) = (V, E
mst
)
of the MST is symmetric (undirected). It is easy to see
that every node of V can be a root of a spanning tree by
using MST (G). It also well-known that M ST (G) is al-
ways strongly connected for a strongly connected graph
4 TECH. REPORT LIFL 2002-8
S D
(a)
S D
90°
(b)
S D
(c)
S D
(d)
Fig. 1. Configuration with same energy consumption for α = 2 and c = 0.
Fig. 3. Minimum spanning tree of graph in Fig. 2.
G. Hence, in [26] the authors define the range adjustment
as follows:
∀u ∈ V r(u) = max{d(u, v) | v ∈ V ∧(u, v) ∈ E
mst
}.
That means that each node chooses to reduce its range
by just covering its neighbors in MST. We denote by
MST
∗
(G) = G
r
the graph with modified ranges by us-
ing MST edges. It is clear that M ST (G) is included in
MST
∗
(G) (E
mst
⊆ E
r
) and then MST
∗
(G) is strongly
connected. This protocol is called MTCP (MST Topology
Control Protocol) in the remaining of this paper. It applies
Prim’s algorithm to construct a minimum spanning tree.
Wieselthier et al. have proposed in [26] two other glob-
alized greedy heuristics for the minimum-energy broad-
cast problem. They are called BLU and BIP belong to the
family of broadcast oriented protocols.
The BLU heuristic (Broadcast Least-Unicast-cost) ap-
plies the Dijkstra’s algorithm. It merges low-energy uni-
casts from the source node to all other nodes in a single
tree that is used instead of MST. In this case, power ef-
ficient routing protocols [12], [19] can be used to gen-
erate the basic structure. The BIP (Broadcast Incremen-
tal Power) is a modified version of the Prim’s algorithm’s
where we consider additional cost in order to cover new
nodes. The next node v in BIP is selected to minimize the
additional power (either by increasing transmission power
at one already transmitting node or by changing r(u) = 0
to r(u) = d(u, v) at one of MST neighbors). Although
the authors [26] use an energy model with constant c = 0,
BIP fits well with the general model with arbitrary con-
stant.
The authors [26] proposed also the “sweep” operation
for removing some unnecessary transmissions, which is
illustrated Fig. 4. A node u whose communication area is
covered by one of its neighbors (i.e. ∃v ∈ N(u) such that
d(u, v) + r(u) ≤ r(v)) may choose a null range.
u
v
r(v)r(u)
Fig. 4. Communication area of node u is covered by node v.
There are some improvements of BIP algorithm but al-
ways in globalized manner and with an energy model us-
ing constant c = 0 [7], [16], [25]. Wan et al. [25] gave
analytical performance of BIP and showed that the ap-
proximation ratio of MST is bounded by 12. Liang [13]
showed that BIP algorithm can have Ω(n) performance
ratio with respect to the optimum algorithm in the worst
case. They propose a sophisticated globalized solution
with better performance ratio, but did not evaluate its aver-
age case performance. Mark et al. [16] proposed a generic
search based globalized protocol for constructing the min-
imum power tree and claimed about 10% improvement
over BIP.
Other works lead to approximation algorithm for the
problem of minimizing the total power with a constant
performance guarantee. For instance Lloyd et al. [15]
propose a globalized algorithm which builds a 2-node-
connected graph and assume an arbitrary energy model.
CARTIGNY ET AL.: LOCALIZED MINIMUM-ENERGY BROADCASTING IN AD-HOC NETWORKS 5
Lindsey and Raghavendra [14] proposed an algorithm
which is not based on tree construction but still achieves
the broadcast with less than 25% more energy consump-
tion than the optimal solution. Their broadcasting proto-
col is the following. The source node simply sends a mes-
sage to a central node (that is closest to all other nodes)
by using power efficient routing protocol and the central
node transmits the message to all other nodes with a sin-
gle message. It is obvious that this protocol is not local-
ized for designation of the central node. Moreover, this
scheme has good results only for an energy consumption
using α = 2 (the authors use an energy model with c = 0)
and is not efficient for higher exponents.
In our localized approach, we use the relative neighbor-
hood graph (RNG) [23]. RNG was already applied for
solving problems in wireless networks. For instance, [20]
applied it to minimize the number of messages needed for
broadcasting in one-to-one unit graph model. Borbash
and Jennings [3] described the localized construction of
RNG in details and proposed to use it as connected topol-
ogy to minimize node degrees, hop-diameter, maximum
transmission radius and the number of biconnected com-
ponents. However, [3] do not describe the use of RNG in
solving any specific problem.
IV. LOCALIZED PROTOCOLS
A. RNG Topology Control Protocol (RTCP)
The main disadvantage of existing protocols is that al-
gorithms are not localized. Our proposal is to substitute
MST by the relative neighborhood graph (RNG) [23]. Let
V be a set of vertices and G = (V, E) the induced graph
with maximal range. The relative neighborhood graph of
G is denoted by RN G(G) = (V, E
rng
) and if defined by:
E
rng
= {(u, v) ∈ G |6 ∃w ∈ V (u, w), (w, v) ∈ G
∧d(u, w) ≤ d(u, v) ∧ d(v, w) ≤ d(u, v)}.
This condition is illustrated Fig. 5: an edge (u, v) be-
longs to the RNG if there does not exists a node w is gray
area.The gray area is the intersection of two circles cen-
tered at u and v and with radius d(u, v). Wecan see in Fig.
6 the RNG of graph given Fig. 2. In this example, and typ-
ically in general, the average degree of RNG is around 2.5
(against 2 for MST).
Analogously, the range adjustment can be defined in or-
der that each node can reach all its neighbors in RN G(G):
∀u ∈ V r(u) = max{d(u, v) | v ∈ V ∧(u, v) ∈ E
rng
}.
u
v
w
Fig. 5. The edge (u, v) is not in RNG because of w.
Fig. 6. Relative neighborhood graph for graph in Fig. 2.
The induced graph G
r
is denoted by RNG
∗
(G). It
is well-known that M ST (G) is included in RNG(G)
[4] and it is easy to see that RNG(G) is a subset of
RNG
∗
(G). For a strongly connected graph G, the con-
nectivity of RNG
∗
(G) is then guaranteed. We will refer
to this protocol as RTCP (RNG Topology Control Proto-
col).
The RNG can be deduced locally by each node just
by using the distance with its neighbors. With positing
system (like GPS), nodes need to send periodically an
“HELLO” message with coordinates. In this way, each
node maintains a neighborhood list with neighbor loca-
tions that allows to determine whether or not an edge is in
RNG. In this case, we need only 1-hop information.
We can observe that if nodes do not have positioning
system, nodes can achieve RNG edges determination if
they are able to determine mutual distances (for instance
by using signal strength or time delay information). Ev-
ery node sends in its HELLO message the list of its neigh-
bors with distances. Hence, RNG construction does not
require more information or different HELLO message as
required to construct MST. More information about RNG
construction can be found in [3], [20]. The information
required to make decision is 2-hop distance information.
In both cases, with GPS or with distance ability, the al-
gorithm for RNG edges determination is localized (with a
6 TECH. REPORT LIFL 2002-8
S
A
B
C
E
F
G
D
Fig. 7. Example of RNG graph for broadcast.
knowledge of 1 or 2 hops distance neighborhood).
The connectivity of RNG assures that all nodes receive
the message for any choice of the source node. We now
discuss the adaptation and some improvements of RTCP
to derive a broadcast oriented protocol.
B. RNG Broadcast Oriented Protocol (RBOP)
Let us consider the graph illustrated Fig. 7 where non-
RNG edges have been omitted. If the node S wants to
send a broadcast message, it transmits it with the minimal
range which allows to join its RNG-neighbors (namely
A, B and C). Then S emits its message with the range
d(S, A) and A, B and C receive the message. Hence S
forwards the message with the range d(A, S) (since A is
its further RNG-neighbor). It is quite obvious that A could
adjust its range to d(A, G) since S already has the mes-
sage. In similar way, B does not have to retransmit the
message since all its RNG-neighbors (S) have already re-
ceived the message. This “trick” is similar to neighbor
elimination scheme [17], [22] but only applied to neigh-
bors in RNG graph.
Let us continue the broadcast. The node C also receives
the message from S. According to preceding remark, C
resends the message with range d(C, D). It is received by
nodes D, E but also F even if it is not a RNG-neighbor.
Hence F receives the broadcast from a non-RNG edge.
In this case, it is better that F applies neighbor elimina-
tion but does not retransmit the message immediately. In
fact, most of the time nodes get the message from one of
its neighbors in RNG, hence by processing only neigh-
bor elimination for transmissions coming from non-RNG
edges the RNG-neighborhood will be smaller. In our ex-
ample, F eliminates E for this broadcast message. The
set of remaining neighbors for F contains only A. At the
same time E decides not to send the message since all its
RNG-neighbors are eliminated with message from C. It
S
A
B
C
E
F
G
D
Fig. 8. Broadcast from S with neighbor elimination.
is the same case for D . When A forwards the message, F
and G eliminate A from their respective neighborhood list
and terminate the protocol for this broadcast since their
lists are empty. The broadcast is accomplished by 3 trans-
missions: from S with radius d(S, A), from C with radius
d(C, D) and A with radius d(A, G) (see Fig. 8).
The improved protocol is then the following one:
1) the source node u of a broadcast emits its message
with determined range r(u) from RTCP,
2) when receiving a new broadcast message:
a) if the emitter is a RNG-neighbor: the node cal-
culates the furthest of its RNG-neighbors that
did not receive this message. The node re-
sends the message according to this range or
ignores the message if all its RNG-neighbors
have received the message,
b) otherwise, the node generates, for this broad-
cast, the list of RNG-neighbors that have not
received this message. After a given timeout,
if the neighbor list is not empty (neighbors can
be removedby action 3b), the node retransmits
the message with a range allowing to reach
furthest neighbor in the associated list,
3) when receiving an already received message:
a) the node ignores the message if it has already
forwarded it,
b) the node removes nodes that received this
message from the associated neighborhood
list,
c) the message is ignored if the associated list is
empty,
d) otherwise, if the message arrives on a RNG-
edge, send the message with range allowing
to reach furthest neighbor in the list of non-
CARTIGNY ET AL.: LOCALIZED MINIMUM-ENERGY BROADCASTING IN AD-HOC NETWORKS 7
0
10
20
30
40
50
6
8 10 12 14 16 18 20 22
EER
Degree
MTCP
RTCP
BIP
RBOP
(a) α = 2 c = 0
0
10
20
30
6
8 10 12 14 16 18 20 22
EER
Degree
MTCP
RTCP
BIP
RBOP
(b) α = 4 c = 10
8
Fig. 9. Expended energy ratio comparison.
eliminated RNG neighbors.
In next section, we give simulation results for presented
protocol, which is referred as RBOP (RNG Broadcast Ori-
ented Protocol), and other protocols described in this and
previous section.
V. PERFORMANCE EVALUATION
In our simulations, we compare four protocols. Two
of these protocols are globalized: MST Topology Control
Protocol (MTCP) and the Broadcast Incremental Power
(BIP) from [26] (enhanced with the sweep operation).
The two other protocols are the two localized algorithms
we propose: RNG Topology Control Protocol (RTCP)
and RNG Broadcast Oriented Protocol (RBOP). In or-
der to permit comparison with works in the literature,
we use two different energy models: α = 2, c = 0 and
α = 4, c = 10
8
.
The parameters of our simulations are the following.
The number of nodes n is always 100 and nodes are static.
The maximum communication radius R is fixed to 250
meters. The MAC layer is assumed to be ideal. Nodes are
randomly placed in a square area whose size is computed
to obtain a given density (from 6 nodes per communica-
tion zone to 30). Only connected sets are retained. For
each measure, 5000 broadcasts have been run.
Because of ideal MAC layer and nature of protocols,
we are sure that all nodes receive broadcasted messages.
Hence, the “reachability” is always 100%. The observed
parameter is the energy consumption (according to the
two energy models). For each broadcast, we calculate the
total energy consumption :
E
total
=
X
u∈V
E(u),
where E(u) depends of the transmission radius as ex-
plained in Section II. This total energy consumption E is
compared with total energy consumption needed for sim-
ple flooding protocol:
E
flooding
= n × (R
α
+ c).
For the four considered protocols, we computed the aver-
age expended energy ratio (EER) that is defined by:
EER =
E
total
E
flooding
× 100.
In Fig. 9 and Tables I and II, we show the comparison
of saved energy for the four protocols and the two energy
models. The average degree varies with density (in nodes
per communication nodes) but is not exactly the same be-
cause of border effect.
EER
density degree MTCP RTCP BIP RBOP
6 5.197 41.784 46.675 12.575 25.448
8 6.856 33.700 39.965 24.776 23.988
10 8.394 28.260 34.896 26.366 21.234
12 9.972 24.176 30.538 24.307 18.307
14 11.483 21.074 26.977 21.962 15.865
16 12.945 18.630 24.027 19.860 13.997
18 14.317 16.672 21.573 18.111 12.470
20 15.685 15.103 19.606 16.632 11.251
22 17.170 13.759 17.854 15.375 10.156
24 18.369 12.635 16.414 14.367 9.236
26 19.790 11.665 15.166 13.468 8.509
28 20.988 10.842 14.093 12.692 7.890
30 22.312 10.136 13.199 11.986 7.383
TABLE I
EXPENDED ENERGY RATIO FOR α = 2, c = 0.
8 TECH. REPORT LIFL 2002-8
EER
density degree MTCP RTCP BIP RBOP
6 5.188 26.182 30.739 10.115 16.363
8 6.869 19.041 24.315 20.374 14.165
10 8.385 14.848 19.860 21.590 11.783
12 9.948 11.935 16.267 19.994 9.590
14 11.459 9.843 13.516 18.079 7.972
16 12.936 8.328 11.336 16.370 6.655
18 14.312 7.271 9.767 15.019 5.731
20 15.709 6.442 8.530 13.799 5.069
22 17.151 5.806 7.570 12.863 4.513
24 18.388 5.282 6.766 12.007 4.095
26 19.784 4.891 6.173 11.334 3.750
28 20.990 4.570 5.677 10.675 3.446
30 22.269 4.313 5.283 10.136 3.236
TABLE II
EXPENDED ENERGY RATIO FOR α = 4, c = 10
8
.
We can observe that localized RTCP topology control
protocol has very close performance to the performance
of globalized MTCP protocol. The difference does not
exceed 6%. This fact illustrates that localized algorithms
can be very competitive with globalized one. It can be also
observed that expended energy ratios (EER) for both pro-
tocols decrease with increased density, or increased values
for α and c.
It can be observed that globalized BIP protocol has
roughly the same performance in both energy models. On
the other hand, EER of RBOP decreased with increased
α and c. Most important finding is that localized RBOP
protocol surprisingly outperforms globalized BIP proto-
col for all densities except for lowest considered density
(about 6). The reasons why BIP is better for small degree
graphs are not clear yet. It could be related to the con-
sideration of only connected random unit graphs which
are difficult to generate for low degree. Most of randomly
generated sparse graphs are disconnected and therefore re-
jected. Hence retained graphs may not reflect the general
case.
The differences in EER between RBOP and BIP in fact
appear to be significant. For α = 2, c = 0 model, BIP
requires about 50% more energy for average degrees over
12. In case of model α = 4, c = 10
8
, the differences are
even larger. They start at about 50% for average degree
8 and increase to over three times more energy at densest
networks. An explanation for better overall performance
of localized protocol over globalized one is given in the
coming conclusion section.
VI. CONCLUSION
In this paper we presented a localized RNG based mini-
mum energy broadcast RBOP protocol that outperformed
globalized BIP protocol [26]. This surprising achieve-
ments can be explained by observing that the nature of
broadcasting task differs from the nature of routing task.
While MST structure closely resembles energy require-
ments of a routing task, it does not necessarily captures
the structural properties in case of broadcasting. Increased
transmission radius beyond the value of furthest uncov-
ered neighbor in any MST like or RNG structure does not
necessarily increase the overall energy consumption. It is
quite possible that a small increase beyond longest RNG
edge will reach several new neighboring nodes, and there-
fore the energy needed per one reached node may actu-
ally decrease (in one-to-all communication model). This
explanation for better performance of RBOP protocol ac-
tually gives direction for further improvements in its per-
formance, which is currently investigated by our group.
The value r(u) in RBOP is actually the minimum possible
transmission radius which is required to maintain connec-
tivity of the broadcast process. We can sort all neighbors
(not already eliminated) by their distance to u, and con-
sider ratios E(u)/M(u), where E(u) is the transmission
power from the energy model, and M(u) is the number
of non-eliminated neighbors reached by transmitting with
transmission radius equal to the distance d (note that E(u)
and M(u) depend on the distance of selected neighbor to
u). The optimal ratio, constrained by distances ≥ r(u),
will then be selected. We are also exploring other direc-
tions for further improvements.
Networks where nodes can only choose between ac-
tive (range set to maximum) or inactive state (range set
to zero) are a special cases which have been addressed
by several works. Dominating sets protocols [28], [22]
can be seen as a solution for the min assignment problem
for this case. MPR (Multipoint relaying) broadcast [18]
and stochastic flooding [5] can be seen as energy-efficient
broadcast protocols for active-inactive power assignment
networks. Some ideas of these protocols, or some combi-
nations between RBOP and these protocols may allow to
improve our present results.
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