Analysis of Minimum-Energy Path-Preserving
Graphs for Ad-hoc Wireless Networks
Mahmuda Ahmed, Mehrab Shariar, Shobnom Zerin and Ashikur Rahman
Department of Computer Science and Engineering
Bangladesh University of Engineering and Technology
Dhaka, Bangladesh
Abstract— We consider ad-hoc wireless networks and the topology control problem defined as minimizing the amount of power
needed to maintain connectivity. The issue boils down to selecting
the optimum transmission power level at each node based on
the position information of reachable nodes. Local decisions
regarding the transmission power level induce a subgraph of
the maximum powered graph Gmax in which edges represent
direct reachability at maximum power. We propose an analysis
for constructing minimum-energy path-preserving subgraphs of
Gmax , i.e., ones minimizing the energy consumption between
node pairs. We also propose an algorithm for constructing
subgraph of Gmax based on 1-hop neighbor information. By
presenting experimental results we show the effectiveness of our
proposed algorithm.
keywords: Ad-hoc Network.
I. I NTRODUCTION
Minimizing energy consumption is a major design goal in
ad-hoc wireless network. If networks are designed preserving
minimum energy paths then significant reduction in energy
consumption can be achieved. We have to choose the minimum
power level for each node to minimize the energy consumption
for the whole network. We use the same model as [1] which
considers, an n-node, multi-hop, ad-hoc, wireless network
deployed on a two-dimensional plane. Suppose that each
node is capable of adjusting its transmission power up to a
maximum denoted by Pmax . Such a network can be modeled
as a graph G=(V,E), with the vertex set V representing the
nodes, and the edge set defined as follows:
E = {x, y|x, y ∈ V × V ∧ d(x, y) ≤ Rmax }
(1)
where d(x,y) is the distance between nodes x and y and
Rmax is the maximum distance reachable by a transmission
at the maximum power Pmax . The graph G defined this way
is called the maximum powered network. By setting a reduced
power level for each node, messages can be transmitted with
a minimum energy among all the paths in the network. This
will lessen average node degree and produce a subgraph of G,
while maintaining global connectivity.
We say that a graph G ⊆ G is a minimum-energy pathpreserving graph or, alternatively, that it has the minimum
energy property, if for any pair of nodes (u,v) that are connected in G, at least one of the (possibly multiple) minimum
energy paths between u and v in G also belongs to G .
Minimum-energy path-preserving graphs were first defined in
[2]. Typically, many minimum-energy path-preserving graphs
SPECTS 2008
can be formed from the original graph G. It has been shown
that the smallest of such subgraphs of G is the graph Gmin =
(V, Emin ), where (u, v) ∈ Emin iff there is no path of length
greater than 1 from u to v that costs less energy than the energy
required for a direct transmission between u and v. Let Gi =
(V,Ei ) be a subgraph of G = (V,E) such that (u, v) ∈ Ei iff
(u, v) ∈ E and there is no path of length i that requires less
energy than the direct one-hop transmission between u and v.
Then Gmin can be formally defined as follows:
(2)
G4 . . . Gn−1
Gmin = G2 G3
It is easy to see that any subgraph G of G has the minimumenergy property iff G ⊇ Gmin . Thereby, each of Gi ⊇ Gmin ,
for any i = 2, 3,. . . , n-1 is a minimum-energy path-preserving
graph.
G2 is the first of the series Gi , it is the most practically
useful derivative of the maximum powered graph G. An
algorithm for constructing one such graph, G2 , was presented
in [2]. It works reasonably well in dense networks but its
performance degrades considerably when the network density
drops to medium or low. Construction of G2 efficiently in
sparse and moderately dense networks with some assistance
of the Medium Access Control (MAC) layer was presented in
[1]. To construct G3 we need 2-hop information that means
information from the immediate neighbors and the neighbors
of those immediate neighbors. Number of links in G2 and G3
is considerably less compared to the number of links within
the maximum powered network. Interestingly, number of links
in G3 does not reduce significantly compared to G2 but G3
requires 1 hop more information than G2 . On the other hand, if
we can construct G3 and G2 in a distributed way, we can also
construct G2 ∩ G3 . G2 ∩ G3 is definitely more close to Gmin
than G2 and it has a great amount of savings in comparison
to G2 .
Construction of G3 involves exchanging many messages,
and thus causes obvious energy overheads. We have tried
not to increase the overhead but improve the performance
of G2 . Then we construct G3 with the information collected
from only the neighbors found within the region having radius
Rmax . The resulting graph is Local Minimum Energy
Communication Network (LMECN) is nothing but G2 G3 with
limited information. We
present here a distributed algorithm to
locally construct G2 G3 . This approach requires no further
114
ISBN: 1-56555-320-9
Authorized licensed use limited to: Bangladesh Univ of Engineering and Tech. Downloaded on August 28, 2009 at 03:48 from IEEE Xplore. Restrictions apply.
information than G2 but produces a reasonable amount of
savings.
tree. One problem with LMST is that the resulting graph does
not preserve the minimum-energy paths.
II. R ELATED W ORKS
III. A LGORITHM
Recent research has lead to exposure of mainly three
dimensions directed at power control algorithms for wireless
mobile networks. The first class of solutions looks at the issue
from the perspective of MAC layer. Monks et al. [3] propose a
modification to the IEEE 802.11 RTS-CTS handshake whereby
a node overhearing a CTS packet uses the received power to
estimate its distance to the sender. Also, Sing et al. [4] propose
techniques for powering-off the transceivers when they are not
active.
The significance of the second approach lies in its dealing
with network layer. Most of the schemes in this class use
distributed variants of the Bellman-Ford algorithm with various flavors of cost metrics derived from the notion of power.
Metrics taken as standard are as follows: energy consumed
per packet, time to network partition, variance in node power
levels, and(power) cost per packet.
The findings in the second approach can play a leading role
in the third approach that separates routing from topology
control. Ramanathan [5] describe two centralized heuristic
algorithms to minimize the maximum transmission power
while trying to maintain connectivity or bi-connectivity, called
Local Information No Topology (LINT) and Local Information
Link-State Topology (LILT).
COMPOW proposed by Narayanaswamy [6], chooses the
smallest common power level for each node emphasizing
mainly on 1) connectivity, 2)traffic carrying capacity maximization, 3)contention reduction in MAC layer, and 4) low
power requirement. It is pre-assumed that distribution of nodes
are homogeneous, which in the long run may appear as
a serious flaw. CLUSTERPOW was designed to overcome
the shortcomings of COMPOW by introducing node clusters.
However, it still suffers from a significant message overhead.
Cone-Based Topology Control (CBTC), generates a graph
structure similar to the one proposed by Yao [7]. Suitable
initial power level and the increment of power level at each
step are the driving force of CBTC, and plays a pioneer role
in determining the number of overhead messages.
Rodoplu [8] introduce the notion of relay region based on
a specific power model. Their algorithm was later modified
by Li [2] to trim some unnecessary edges. Our work closely
relates to these two studies.
N. Li [9] propose a distributed topology control algorithm
(called LMST) based on constructing minimum spanning
trees. Their algorithm achieves three goals: 1) connectivity,
2) bounded node degree (≤ 6) and 3) bi-directional links. In
LMST, each node u collects the position information of all
neighbors reachable at the maximum power. Based on this
information, u creates its own local minimum spanning tree
among the set of neighbors, where the weight of an edge
is the necessary transmission power between its two ends.
Once the tree has been constructed, u contributes to the final
topology those nodes that are its neighbors in the spanning
SPECTS 2008
A. Power Model
We are using the same power model as [1]. Here we assume
the well known, generic, two-way, channel path loss model,
where the minimum transmission power is a function of distance [10]. To send a packet from node x to node y, separated
by distance d(x,y), the minimum necessary transmission power
is approximated by
Ptrans (x, y) = t × dα (x, y),
(3)
where α ≥2 is the path loss factor and t is a constant. Signal
reception is assumed to cost a fixed amount of power denoted
by r. Thus, the total power required for one-hop transmission
between x and y becomes
Ptotal (x, y) = t × dα (x, y) + r,
(4)
The model assumes that each node is aware of its own position
with a reasonable accuracy, e.g., via a GPS device.
B. Previous approach to constructing G2
An algorithm was presented in [8] based on relay region.
The concept is like this:
Given a node u and another node v within u’s communication range (at Pmax ), the relay region of node v as perceived
by u (with respect to u), Ru→v , is the collection of points
such that relaying through v to any point in Ru→v takes less
energy than a direct transmission to that point (see 1)). Given
Fig. 1. (a) The relay region of v with respect to u (b) The enclosure of node
u
the definition of relay region, the algorithm for constructing
G2 becomes straightforward. Construction of G2 efficiently in
sparse and moderately dense networks with some assistance
of the Medium Access Control (MAC) layer was presented in
[1].
115
ISBN: 1-56555-320-9
Authorized licensed use limited to: Bangladesh Univ of Engineering and Tech. Downloaded on August 28, 2009 at 03:48 from IEEE Xplore. Restrictions apply.
LMECN Algorithm:
1.
/* s main node */
2.
/* ℵ is the set of discovered node so far */
3.
/* ℵs Neighbor set of node s */
4.
/* Suppose, s got a reply from node r */
5.
begin
6.
ℵs = ℵ s ∪ r
7.
for each n in ℵ{
8.
/*get list of common neighbors
9.
between n and r */
10.
D=getCommonNeighbor(n,r,ℵ)
11.
/*get list of common neighbors
12.
between s and r */
13.
G=getCommonNeighbor(s,r,ℵ)
14.
Segment 1:
15.
if r is in ℵs {
16.
/* case 1(a): */
17.
/* check for two length path
18.
between s and r relaying through n*/
19.
if (cost(s, r) ≥ cost(s, n) + cost(n, r)){
20.
ℵs = ℵs − {r}
21.
goto Segment 2
22.
}
23.
/* case 1(b): */
24.
/* check for three length path between s
25.
and r relaying n and d */
26.
for each d in D{
27.
if (cost(s, r) ≥ cost(s, n)
28.
+cost(n, d) + cost(d, r)){
29.
ℵs = ℵs − {r}
30.
goto Segment 2
31.
}
32.
}
33.
}
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
}
Segment 2:
if n is in ℵs {
/* case 2(a): */
/* check for two length path
between s and n relaying through r */
if (cost(s, n) ≥ cost(s, r) + cost(r, n)){
ℵs = ℵs − {n}
goto Segment 3
}
/* case 2(b): */
/* check for three length path between
s and n relaying r and d */
for each d in D{
if (cost(s, n) ≥
cost(s, r) + cost(r, d) + cost(d, n)){
ℵs = ℵs − {n}
goto Segment 3
}
}
/* case 2(c): */
/* check for three length path between s
and n relaying through g and r*/
for each g in G{
if (cost(s, n) ≥ cost(s, g)
+cost(g, r) + cost(r, n)){
ℵs = ℵs − {n}
goto Segment 3
}
}
Segment 3:
continue
}
ℵ=ℵ∪r
End
C. Algorithm
We propose an algorithm for constructing close approximation of G2 ∩ G3 with much less overhead which works
in a distributed way. We shall refer to our algorithm as the
Local Minimum Energy Communication Network algorithm
or LMECN algorithm in short through this paper. In LMECN
algorithm, function cost(x, y) takes two nodes x and y as
parameters and calculates the cost of direct transmission
between them according to the equation 4 assuming t, the
predetection threshold equal to 10−7 mW, additional receive
power, r = 20mW and path loss factor, α = 4. Function
getCommonNeighbor(x,y,ℵ) determines the common neighbor
set of x and y having information of set of nodes ℵ.
The operation of the above algorithm is described from the
viewpoint of one node s. Here s broadcasts a single neighbor
discovery message (NDM) at the maximum power Pmax .
While s collects the replies of its neighbors, it learns their
identities and locations. It constructs set ℵ and ℵs . Initially,
all those sets are empty (s doesn’t even know what neighbors
there are). Here N is defined as the set of all nodes that s has
SPECTS 2008
discovered so far and ℵs is the set of its neighbors. Node s
has to maintain direct link or direct communication with each
node in ℵs , |ℵs | ≤ |ℵ|.
When node s gets a reply from node r it is added in
the neighbor set ℵs , then it considers 2 cases for each node
in ℵ. First it tries to find out a node n in ℵ such that
cost(s, r) >= cost(s, n)+cost(n, r), then r is discarded from
ℵs because node n can be used as a relay to transmit to r (see
figure 2 case 1(a)). If cost(s, r) < cost(s, n) + cost(n, r) then
there is no path of length two that can be used to transmit
r with less energy than direct transmission between s and
r. So, r will not be discarded from neighbor set ℵs . Thus
G2 is forming here. Now a three length path that costs less
energy than a direct transmission between s and r is searched
on. If d is a common neighbor node between n and r then
cost(s, r) >= cost(s, n) + cost(n, d) + cost(d, r) condition
is checked. If there is a node that fulfils the above condition
then r is discarded from ℵs because there is path of length
three which costs less than direct transmission (see Figure 3
116
ISBN: 1-56555-320-9
Authorized licensed use limited to: Bangladesh Univ of Engineering and Tech. Downloaded on August 28, 2009 at 03:48 from IEEE Xplore. Restrictions apply.
Fig. 2.
Fig. 3.
Case 1(a)
SPECTS 2008
Case 2(b)
Fig. 6.
Case 2(c)
Case 1(b)
case 1(b)).
Now our algorithm LMECN search for the nodes that should
be removed from neighbor set ℵs after including r. First it
discards all the nodes for which r can be used as relay in case
of two length path (see figure 4).
Thus G2 is formed here. Then search for a node d in
common neighbor set of n and r and for a node g in common
neighbor set of s and r is done to find out a three length path
that is less costly than direct transmission between s and n
(see figure 5 and figure 6).
This algorithm constructs G2 in a proper way and then
tries to construct G3 from available (partial) node information. As we know that construction of G3 requires 2 hop
information from neighbors which causes more messages to
be exchanged and increases energy overheads. So we tried to
improve performance of G2 here. We constructed G3 using
Fig. 4.
Fig. 5.
Case 2(a)
the local information found by exchanging 1 hop information
like G2 . We refer to the graph found by LMECN as GLMECN
throughout this paper.
D. Why GLMECN close to G2 G3
In this section we present scenario where local information
from immediate neighbors are not enough to construct G3 .
In Figure 7 the circle represents Rmax neighborhood of node
a. node b and node c is inside its neighbor region but node
d is outside of its sensing range. Suppose node d is situated
very close to Rmax i.e., near the boundary of node a. As
node a has only local information in LMECN, it will see that
there is no two or three length path that costs less than direct
communication between a and b, therefore it maintains link
ab. But in actual case it might happen that relaying through
node c and d may cost less than direct communication between
a and b. So, acdb is the three length path that is considered in
G3 but not in LMECN.
This is a case where LMECN produces
more link than G2 G3 .
Lemma 1: GLMECN ⊇ G2 G3 .
Proof: A link between two nodes u, v ∈ V is deleted in
LMECN only when there is another minimum energy path of
two or three length between the nodes, i.e. LMECN applies G2
and locally G3 , thus, it does not destroy any minimum energy
path having length two or three. Besides, LMECN produces
some extra links as described in previous subsection. So
the
resultant graph GLMECN is definitely a superset of G2 G3 .
117
ISBN: 1-56555-320-9
Authorized licensed use limited to: Bangladesh Univ of Engineering and Tech. Downloaded on August 28, 2009 at 03:48 from IEEE Xplore. Restrictions apply.
700
original
node
600
500
400
Fig. 7.
300
An Extraneous Edge
200
100
Lemma 2: LMECN preserves connectivity.
Proof: We presented Gmin as,
Gmin = G2
G3 G4 . . . Gn−1
0
0
Fig. 8.
According to [2] Gmin is the smallest subgraph with the
minimum energy property. Gmin has no redundant edges and
also preserves connectivity. Thereby, each of Gi ⊇ Gmin , for
any i = 2, 3,. . . , n-1 is a minimum-energy path preserving
graph and also preserves connectivity. G2 ∩ G3 is a superset
of Gmin according to Lemma 2 GLMECN ⊇ G2 ∩ G3 and
thus, preserves connectivity.
100
200
300
400
500
600
700
Original Graph (uniform distribution) no. of nodes 75
700
G2
node
600
500
IV. E XPERIMENTAL R ESULTS
400
A. Topological Analysis
300
By simulation we evaluate the performance of our algorithm. Initially we deployed 25-300 nodes over a flat square
area of 670m×670m according to uniform distribution. Figure
8 shows a typical deployment of 75 nodes, each having a
maximum communication range of 250m. This is the starting
graph G for our algorithm. The graph found by G2 is figure
9 which shows a great amount of savings than original graph
G. The next graph at figure 10 is found by G3 , this also has
a large amount of savings with respect to original graph G at
figure 8. So, both G2 and G3 shows good results. Now the
graph that is closer to Gmin than G2 and also G3 is G2 ∩ G3
is shown in figure 11. This graph has the least number of
links among the graphs mentioned above. Figure 12 shows the
graph we found by LMECN. Here the graph has just one more
link than that of G2 ∩ G3 . This implies that the performance
of LMECN is much closer to G2 ∩ G3 but incurs much less
overhead than constructing G3 (2 hop information) to calculate
G2 ∩ G3 .
We perform more simulation on our algorithm deploying 25100 nodes according to zipf distribution, figure 13 shows the
starting graph G of our algorithm. It is a typical deployment
of 75 nodes, each having a maximum communication range of
250m. The next two fig shows the graph found by G2 (figure
14) and G3 (figure 15) where both shows good savings. In
uniform distribution G2 ∩G3 (figure 16) and GLMECN (figure
17) both shows same no of links. We have got the same result
here without increasing the overhead, it is definitely a good
achievement.
SPECTS 2008
118
200
100
0
0
Fig. 9.
100
200
300
400
500
600
700
G2 (uniform distribution) no. of nodes 75
700
G3
node
600
500
400
300
200
100
0
0
Fig. 10.
100
200
300
400
500
600
700
G3 (uniform distribution) no. of nodes 75
ISBN: 1-56555-320-9
Authorized licensed use limited to: Bangladesh Univ of Engineering and Tech. Downloaded on August 28, 2009 at 03:48 from IEEE Xplore. Restrictions apply.
700
700
G2IntG3
node
600
G2
node
600
500
500
400
400
300
300
200
200
100
100
0
0
0
Fig. 11.
100
200
300
400
500
600
700
0
G2 ∩ G3 (uniform distribution) no. of nodes 75
100
Fig. 14.
700
200
300
400
500
600
700
G2 (Zipf distribution) no. of nodes 75
700
LMECN
node
600
G3
node
600
500
500
400
400
300
300
200
200
100
100
0
0
0
Fig. 12.
100
200
300
400
500
600
700
0
GLM ECN (uniform distribution) no. of nodes 75
100
Fig. 15.
700
200
300
400
500
600
700
G3 (Zipf distribution) no. of nodes 75
700
original
node
600
G2IntG3
node
600
500
500
400
400
300
300
200
200
100
100
0
0
0
Fig. 13.
100
200
300
400
500
600
700
0
Original Graph (Zipf distribution) no. of nodes 75
SPECTS 2008
Fig. 16.
119
100
200
300
400
500
600
700
G2 ∩ G3 (Zipf distribution) no. of nodes 75
ISBN: 1-56555-320-9
Authorized licensed use limited to: Bangladesh Univ of Engineering and Tech. Downloaded on August 28, 2009 at 03:48 from IEEE Xplore. Restrictions apply.
700
300
LMECN
node
600
G2
LMECN
250
no. of links
500
400
300
200
200
150
100
100
50
0
20
0
no. of links
Fig. 17.
100
200
300
400
500
600
30
700
40
50
60
70
80
90 100
no. of nodes
GLM ECN (Zipf distribution) no. of nodes 75
Fig. 19.
comparison between G2 and GLM ECN in Zipf distribution
TABLE I
1000
900
800
700
600
500
400
300
200
100
0
G2
LMECN
C OMPARISON B ETWEEN G2 AND GLM ECN IN U NIFORM DISTRIBUTION
No of
Nodes
25
35
50
75
100
150
200
300
0
50
100
150
200
250
Edges
in G2
61.6
85.2
133.6
202.8
278.0
441.6
596.0
948.8
Edges
in GLM ECN
60.0
81.0
124.8
190.8
262.0
412.4
562.8
902.8
300
no. of nodes
Fig. 18.
Comparison Between G2 and GLM ECN in Uniform distribution
the
number of reply messages is equal to, NodeDegree(s)+
n∈neighbor(s) NodeDegree(n).
B. Comparison between G2 and GLMECN
Here we deployed 25-300 nodes over a flat square area of
670m × 670m according to uniform distribution and simulate
the network both for G2 and LMECN. For illustration, Table
I shows a comparison between the number of edges in G2
and in LMECN. For a specific number of nodes there are five
different scenarios. We have taken average number of edges or
links for each set. We found here a saving of 4.85% without
increasing overhead( see figure 18). Again, we simulate on a
network of 25-100 nodes deployed over a flat square area of
670m × 670m according to Zipf distribution. Table II shows
a comparison between the number of edges in G2 and in
LMECN. Up to 6.18% of links is saved here (see figure
19). In both distribution the savings is quite satisfactory and
considerably higher for denser networks.
Now, we took the number of reply messages as overhead.
To analysis on overhead cost by G3 and GLMECN (or G2 )
we simulate on networks of 25-300 nodes deployed over a flat
square area of 670m×670m spread under uniform distribution
( 20) and 25-100 nodes deployed over a flat square area of
670m × 670m according to Zipf distribution ( 21). For a
specific number of nodes there are five different scenarios.
We have taken average number of overhead messages for each
set. We found that construction of G3 causes huge amount of
overhead with respect to GLMECN in both the cases.
C. Overhead Analysis
A node s broadcasts NDM at first to know about its
neighbors. In case of constructing G2 , the number of reply
messages for any node s is equal to NodeDegree(s). Nodedegree is defined as the number of neighbors it has within
the region having radius Rmax . For constructing G2 ∩ G3 ,
SPECTS 2008
120
TABLE II
C OMPARISON B ETWEEN G2 AND GLM ECN IN Z IPF DISTRIBUTION
No of
Nodes
25
30
35
40
45
50
75
100
Edges
in G2
57.6
75.2
91.2
105.2
118.0
131.6
207.6
284.8
Edges
in GLM ECN
55.4
70.8
85.2
99.6
111.0
125.2
198.2
267.2
ISBN: 1-56555-320-9
Authorized licensed use limited to: Bangladesh Univ of Engineering and Tech. Downloaded on August 28, 2009 at 03:48 from IEEE Xplore. Restrictions apply.
3e+006
V. C ONCLUSION
G3
LMECN
The algorithm we have presented is a distributed algorithm
that ensures minimum energy path preserving graphs in adhoc wireless networks. Our construction of LMECN is assisted
by the rudimentary protocol of constructing G2 . Its need of
information for an additional hop is served with no additional
overhead and only through the previous information gained
through G2 . Our studies assisted by some simulation results,
have demonstrated the superiority and robustness of the new
algorithm over the previous solutions for networks with moderate and low density of nodes.
No. of overhead messages
2.5e+006
2e+006
1.5e+006
1e+006
500000
0
0
50
100
150
No. of nodes
200
250
R EFERENCES
300
Fig. 20. Overhead comparison between G3 and GLM ECN in Uniform
distribution
250000
G3
LMECN
No. of overhead messages
200000
150000
100000
50000
0
20
30
40
50
60
70
80
90
100
No. of nodes
Fig. 21.
Overhead comparison between G3 and GLM ECN in Zipf
distribution
TABLE III
OVERHEAD COMPARISON BETWEEN G3 AND GLM ECN IN U NIFORM
DISTRIBUTION
No of
Nodes
25
35
50
75
100
150
200
300
Overhead msg.
in GLM ECN
160.8
334.8
810.4
1637.2
3082.0
6978.8
12421.6
27777.6
[1] A. Rahman and P. Gburzynski, “Mac-assisted topology control for
ad-hoc wireless networks.” International Journal of Communication
Systems, vol. 19, no. 9, pp. 955–976, 2006.
[2] L. Li and J. Halpern, “Minimum energy mobile wireless networks
revisited,” Proc. IEEE International Conference on Communications
(ICC), june, 2001.
[3] J. P. Monks, V. Bhargavan, and W. M. Hwu, “A power controlled MAC
protocol for wireless packet networks,” in Proceedings of INFOCOM,
2001, pp. 219–228.
[4] S. Singh and C. Raghavendra, “Power efficient MAC protocol for
multihop radio networks,” in Proceedings of PIMRC, 1998, pp. 153–
157.
[5] R. Ramanathan and R. Rosales-Hain, “Topology control of multihop
wireless networks using transmit power adjustment,” in In Proc. of IEEE
INFOCOM 2000, Tel Aviv, Israel, March 2000, pp. 404–413.
[6] S. Narayanaswamy, V. Kawadia, R. S. Sreenivas, and P. R. Kumar,
“Power control in ad-hoc networks: Theory, architecture, algorithm
and implementation of the COMPOW protocol,” in Proceedings of the
European Wireless Conference - Next Generation Wireless Networks:
Technologies, Protocols, Services and Applications, Florence, Italy,
February 2002, pp. 156–162.
[7] Y.-S. C. h. Sze-Yao Ni, Yu-Chee Tseng and J.-P. Sheu, “The broadcast
storm problem in a mobile ad hoc network,” in Mobicom, 1999.
[8] V. Rodoplu and T. Meng, “Minimum energy mobile wireless networks,”
IEEE Journal of Selected Areas in Communications, vol. 17, pp. 1333–
1344, 1999.
[9] N. Li, J. C. Hou, and L. Sha, “Design and analysis of an MST-based
topology control algorithm,” in Proceedings of INFOCOM, 2003.
[10] T. Rappaport, “Wireless communications: principles and practice,” pp.
219–228, 1996.
Overhead msg.
in G3
1342.4
3910.8
15613.6
41130.8
106442.8
361218.0
852938.4
2821319.2
TABLE IV
OVERHEAD COMPARISON BETWEEN G3 AND GLM ECN IN Z IPF
DISTRIBUTION
No of
Nodes
25
30
35
40
45
50
75
100
SPECTS 2008
Overhead msg.
in GLM ECN
308.8
357.6
515.2
675.2
930.8
1146.4
2592.4
4377.2
Overhead msg.
in G3
4799.6
5274.4
9506.4
13656.4
23031.2
24828.4
104965.6
221260.8
121
ISBN: 1-56555-320-9
Authorized licensed use limited to: Bangladesh Univ of Engineering and Tech. Downloaded on August 28, 2009 at 03:48 from IEEE Xplore. Restrictions apply.