Randomized Energy Aware Routing Algorithms in Mobile Ad Hoc Networks

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Randomized Energy Aware Routing Algorithms
in Mobile Ad Hoc Networks
Israat Tanzeena Haque
Computer Sci.and Soft.Eng.
Concordia University
Montr ´eal,Qu´ebec,Canada
it
haque@cse.concordia.ca
Chadi Assi
Concordia Institute for
Information Systems Eng.
Montr ´eal,Qu´ebec,Canada
assi@ciise.concordia.ca
J.William Atwood
Computer Sci.and Soft.Eng.
Concordia University
Montr ´eal,Qu´ebec,Canada
bill@cse.concordia.ca
ABSTRACT
We consider the problem of energy aware localized routing
in ad hoc networks.In localized routing algorithms,each
node forwards a message based on the position information
about itself,its neighbors and the destination.The objec-
tive of energy aware routing algorithms is to minimize the
total power for routing a message from source to destina-
tion or to maximize the total number of routing tasks that
a node can perform before its battery power depletes.In
this paper we extend our previous work on randomized lo-
calized routing algorithms that achieve high packet delivery
rates and show that they have good overall power consump-
tion.We present two different variants of energy aware ran-
domized routing,namely “greedy” and “compass”,and we
study their performance using different cost metrics (e.g.,
forwarding power,remaining node energy,or a combination
of both).We study their performance experimentally on
different topologies and compare it with other existing algo-
rithms.Our simulation results show that energy aware ran-
domized algorithms achieve superior packet delivery rates
and moderate energy consumption.
Categories and Subject Descriptors
C.2.1 [Computer-Communication Networks]:Network
Architecture and Design—Wireless communication;C.2.2
[Computer-Communication Networks]:Network Pro-
tocols—Routing protocols
General Terms
Performance,algorithms
Keywords
wireless networks,mobile ad hoc and sensor networks,rout-
ing,position based routing,energy aware routing
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage an d that copies
bear this notice and the full citation on the first page.To cop y otherwise,to
republish,to post on servers or to redistribute to lists,requires prior specific
permission and/or a fee.
MSWiM’05,October 10–13,2005,Montreal,Quebec,Canada.
Copyright 2005 ACM1-59593-188-0/05/0010...$5.00.
1.INTRODUCTION
We consider the problemof energy aware routing,in which
a message has to be sent efficiently from a source node to a
destination node,in a sensor or an ad hoc network.Amobile
ad hoc network (MANET) is a collection of autonomous mo-
bile devices that can communicate with each other without
having any fixed infrastructure.Each node in the network
can have an omni-directional antenna and communicate us-
ing wireless broadcasts with all nodes within its transmis-
sion range.A multi-hop routing protocol is needed to enable
communication between nodes that are not in transmission
range of each other.However,the absence of infrastruc-
ture in MANETs,the dynamic topology of these networks,
the autonomous heterogeneous nodes,and the resource con-
straints (battery power,bandwidth,computational power,
etc.) all contribute to make the problemof routing a tremen-
dous challenge.The constrained resources,especially the
battery power,make the routing problem a very challeng-
ing issue for MANETs.Moreover,mobile devices in an ad
hoc network need to forward packets for other nodes;this
extra activity consumes a significant amount of the energy
of mobile devices.Therefore,it is critical to design efficient
routing algorithms with the objectives of (1) minimizing the
overall energy usage in routing packets and (2) maximizing
the packet delivery rate.Applications of minimum energy
networks include soldiers deployed on a hostile terrain and
multisensor networks,where sensors communicate with each
other without having any central control or base station.
Energy is consumed at two levels during routing,namely
communication energy and the energy dissipated at the nodes.
The communication energy or the energy needed per routing
task can be optimized if nodes can adjust their transmission
power to efficiently select the next hop along the route.This
is equivalent to hop count if the transmission power is kept
constant [13].Routing algorithms that solely focus on the
communication energy are not eventually a good choice for
network lifetime.Some of the nodes (hot spots) in this ap-
proach will be chosen very often and this will ultimately
drain out the battery power of these nodes quickly,there-
fore partitioning the network abruptly.Routing algorithms
that minimize the energy required per routing task are called
power/energy aware algorithms [13];on the other hand,cost
aware routing algorithms ensure optimal use of node’s bat-
tery power and hence prolong network’s life time [13].
In this paper we focus on the energy aware localized rout-
ing algorithms,where each node forwards the routing pack-
ets based only on the position or geographic coordinates of
71

itself,its neighboring nodes,and the destination.We model
a MANET by a unit disk graph,where two nodes are con-
nected if and only if their Euclidean distance is at most
the transmission range [1].We classify the position based
algorithms as deterministic and randomized.In the first cat-
egory,the current node holding the packet selects the next
node deterministically out of its neighbors,whereas in the
second case the selection is random.
Extensive work has been done on energy efficient routing,
however in this paper we will mainly focus on position based
routing.We will start first by presenting the performance
analysis of some of the existing energy aware and non aware
position based routing algorithms.Then,we will present our
new randomized algorithms that control the communication
power with high packet delivery rates.Essentially,to decide
on the next node to which the packet should be forwarded,
our algorithms pick one neighbor of the current node from
above the line passing through the current node and the
destination,and another neighbor below this line.Then the
next node is chosen randomly from these two neighbors ac-
cording to some probability distribution.The exact choice
of the neighbors and the probability distribution determines
the algorithm.
In our simulation,we consider both the uniform and clus-
ter distributions of nodes in a given area.Our simulation
results show that on both data distributions our algorithms
have a substantially higher delivery rate than the determin-
istic algorithms in unit disk graphs.The rest of the paper is
organized as follows.The next section reviews some of the
enery aware routing related to our work.Section 3 gives a
brief description of the exsisting position based routing al-
gorithms.Section 4 describes the system model and other
relevant preliminaries.Randomized energy aware routing
algorithms are presented in Section 5.The simulation en-
vironment is given in Section 6.In Section 7 we present
the empirical results of our simulations and provide an in-
terpretation of the behavior of the algorithms.We conclude
with a discussion of the results and future directions of this
research in Section 8.
2.OVERVIEWOF ENERGY AWARE
ROUTINGALGORITHMS
In [12],the authors propose several power aware routing
metrics to increase the lifetime of the nodes and the net-
work.Conventional routing protocols in ad hoc networks
use delay or hop count to calculate the path to the destina-
tion.This approach might accelerate the battery drainage of
some specific nodes,which forward packets for many source-
destination pairs.The effect would be early node failure and
network partition.Following a longer path of a set of nodes
with plenty of energy would be a better choice [12].The
energy-aware metrics proposed by Singh et al.are as fol-
lows.
Minimum energy consumed per packet:This metric is
used to minimize the total communication energy of a packet
regardless of the available energy at the nodes.Assume a
packet j traverses a set of nodes n
1
,n
2
,........,n
k
,where n
1
is
the source and n
k
is the destination.Let P(n
i
,n
i+1
) be the
power needed to forward j from the node n
i
to n
i+1
.The
total energy consumed by packet j to reach the destination
is then the sum of the energy over the entire path.The
optimization of this metric,which is called power metric in
[13],is then subject to
min
∀j
e
j
,e
j
=
k−1

i=1
P(n
i
,n
i+1
) (1)
Minimum cost per packet:This metric tries to prolong
the lifetime of the nodes and networks through the careful
selection of next route node with plenty of energy.Let f
i
(x
i
)
be a function that denotes the cost or weight of node i,
where x
i
represents the total energy that node i already
expended.Hence,the total cost c
j
of sending a packet j
from the node n
1
to n
k
is sum of the cost of the entire
route.The optimization of this metric is then subject to,
min
∀j
c
j
,c
j
=
k−1

i=1
f
i
(x
i
) (2)
Another way of using the cost metric is normalizing f
i
to
reflect remaining battery lifetime of a node.Then,f
i
(x
i
)
can be modified as f
i
(z
i
) = 1/1 − g(z
i
),where z
i
is the
measured voltage and g(z
i
) is the normalized (between 0
and 1) remaining lifetime of node i.Singh et al.consider a
static random graph,where two nodes are connected with a
fixed probability p,and use nonlocalized Dijkstra’s shortest
path algorithm to evaluate the performance of their pro-
posed metrics compared to the hop count metric.
In [13],the authors proposed localized power,cost,and
power*cost algorithms based on the observation that if ad-
ditional intermediate nodes are placed at desired positions
between the source and destination that are at distance d
apart,then transmission power may be linearly dependent
on d,rather than d
α
(α ≥ 2).The authors first defined two
optimal algorithms in terms of communication energy and
battery lifetime.The SP −power algorithm that minimizes
the total communication energy of a packet is obtained by
applying Dijkstra’s shortest path algorithm with the weight
of the edge as ad
α
+ C,where a,α,and C are constants
that depend on the radio models.On the other hand,the
SP −cost algorithm,which maximizes the node’s lifetime,
can be obtained by applying Dijkstra’s shortest path algo-
rithm with an edge-weight of f(n
i
) = 1/g
i
.Stojmenovic et
al.use these optimal energy aware algorithms to compare
the performance of their proposed algorithms.
Let c,N
i
,and d be the current node holding the packet,
set of neighbors of c,and destination of the packet,respec-
tively.The distance between c and one of its neighbors is x
and the remaining distance to the destination is d−x,where
d is the total distance between c and d.The localized loop
free power and cost aware algorithms are next defined:In
the power algorithm,c selects the next neighbor such that
min
∀j∈N
i
e
j
,e
j
= E
c→x
i
+E
x
i
→d
(3)
Ec→x
i
= ax
α
+C (4)
E
x
i
→d
= C(d−x)[
a(α−1)
C
]
1/α
+C(d−x)[
a(α−1)
C
]
(1−α)/α
(5)
Here,E
c→x
i
and E
x
i
→d
are the energy needed for direct
communication.This process continues untill the packets
reach the destination or the next node is one from which c
received the packet,in which case algorithm fails to deliver
the packet.In the cost algorithm,c selects the next neighbor
72

x
i
such that
min
∀j∈N
i
c
j
,c
j
= f(x
i
) +t(d −x)/r,(6)
Here,r is the transmission radius of the nodes and t is a
network dependent constant.For example,t = f(x
i
) means
the rest of the nodes have the same cost of the node x
i
,
which is not realistic.Also,the other values of t,(f
￿
(x
i
) and
1/g
￿
(x
i
)),where f
￿
(x
i
) and 1/g
￿
(x
i
) represent the average
cost and remaining battery power of x
i
and its neighbors,
respectively,and do not consider actual cost of the nodes.In
our algorithms (presented in Section 4),nodes periodically
broadcast their remaining battery power to their neighbors
to make routing decision based on accurate energy level of
nodes.
Finally,the authors combine power and cost metrics in
additive and multiplicative forms to obtain a single metric
that considers both the communication energy and the bat-
tery power of the nodes.When multiplication of two metrics
is used,the energy needed to forward a message from c to
next node x is f(x)E
c→x
.The additive form of the met-
rics is αE
c→x
+βf(x),where α and β may be fixed by the
source node s as f
￿
(s) and average of E
s→N
i
,respectively.
The corresponding optimal energy consumption algorithms
are called SP −power +cost and SP −power ∗cost.The lo-
calized version of these algorithms select the next node that
not only minimizes the communication energy but also se-
lects the next node with plenty of remaining battery power.
On the other hand,[3] proposed a distributed non-localized
shortest cost routing algorithm for sensor networks,where
the cost takes into account both the communication energy
consumption and the residual energy level at nodes.Their
objective is to maximize the network life by increasing the
number of routing tasks,before the first node is dead (does
not have enough energy to forward message).Their link cost
can be formulated as follows.
cost
ij
= (e
t
ij
)
x
1
E
−x
2
r
E
x
3
i
+(e
r
ij
)
x
1
E
−x
2
r
E
x
3
i
,(7)
Where E
i
and E
r
are initial and remaining energy at nodes,
respectively.x
1
,x
2
,and x
3
are some positive weighting
factors.For example,if x
1
= x
2
= x
3
= 0,then the resulting
path is the minimum hop path,however,if x
2
= x
3
= 0,
then the path is the minimum energy consumption path.
In [15] Xue et al.proposed a location aided or position
based energy aware routing algorithm,called LAPAR,under
node mobility and unidirectional links.In particular,each
node constructs relay regions based on the position of the
neighbors.The next node is then chosen such that its relay
region covers the destination.In case of multiple neighbor-
ing nodes that covers the destination,the greedy approach is
applied.This algorithm is loop free,but might fail if there
is no neighbor whose relay region covers the destination.In
this case LAPAR is combined with the perimeter routing.
A planar subgraph of the original topology is required in
perimeter routing,and the packet traverses those faces of
the subgraph that are intersected by the source and desti-
nation nodes.
Kuruvila et al.in [10] proposed another set of power
and cost aware routing algorithms that guarantee progress
In power progress algorithm the current node forwards the
packet to one of its neighboring nodes that is closer to the
destination than itself,such that it minimizes (d
α
cx
+C)/d
t

d
r
,where d
cx
,d
t
,and d
r
are the distances between the cur-
rent and neighboring node,current and destination node,
and neighboring node and destination,respectively.Simi-
larly,in projection power progress,the next node is the one
that minimizes (d
α
cx
+C)/cd.cx,where cd.cx is the dot prod-
uct of two vectors.
3.POSITION BASED ROUTING
ALGORITHMS
In position based routing algorithms,each node makes
a decision about which neighbor to forward the message
to based solely on the location of itself,its neighboring
nodes,and the destination.Although routing-table-based
solutions merely keep the best neighbor information on a
route toward the destination,the communication overhead
for maintenance of routing tables due to node mobility and
topology changes is quadratic in network size [6].On the
other hand,position-based algorithms do not require route
establishment and maintenance,hence these algorithms may
efficiently utilize the scarce memory resources and the rel-
atively low computational power available to the wireless
nodes.More importantly,given the numerous changes in
topology expected in ad-hoc networks,no reconfiguration of
the routing tables is needed and therefore we expect a signifi-
cant reduction in the route maintenance overhead.Position-
based routing algorithms are classified in different ways in
the literature,and we will describe some of those related to
our work.
In [6],the progress of a node x,with a given transmitting
node c and the final destination node d,is defined as the
projection of |cx| on the
cd line.A neighbor of c is in the
forward direction if the progress is positive;otherwise,it is
said to be in the backward direction.
The Most Forward within Radius (MFR) [14] routing al-
gorithm maximizes the progress toward the destination by
forwarding the packets to one of the neighboring nodes,
whose projection onto the line between the current node
and the destination is closest to the destination.However,
it also consumes maximum power to cover maximum dis-
tance,which in turn increases collisions with other nodes.
Hence,instead of forwarding packets to the farthest neigh-
boring node,the Nearest with Forward Progress (NFP) [8]
scheme sends the packet to the node closest to the sender.
Recently in [13],Stojmenovic et al.have introduced another
new routing method called Nearest Closer (NC),which is a
variation of NFP method.In this method,c selects one of
its closest neighbors that are closer to d than c.The trans-
mission can thus be accomplished with minimum power;
hence the interference with the other nodes is minimized,
while the probability of a successful transmission is maxi-
mized.In Greedy routing [5,6],a node forwards a packet
to the neighbor that is closest to the destination.Compass
or directional routing [9] moves the packet to a neighboring
node such that the angle formed between the current node,
the next node,and the destination is minimized.All of the
above mentioned algorithms (except NC) choose the next
node from among all the neighbors of the current node c,
and fail to deliver the packet if the chosen next node is the
one from which c receives the packet.Whereas,in NC,c
considers only the neighbors closer to the destination than
itself,and drops the packet if no such neighbor is available.
Let us consider the example given in Figure 1 to illustrate
the successful operation of each of the above mentioned al-
gorithms,where the source and the destination are s and
73

d1,respectively.In this example Greedy and MFR choose
b,NFP and NC select a,whereas Compass routing picks c
as the next node.
d2
s
d1
a
b
c
e
f
h
g
i
Figure 1:A sample network topology to illustrate
the operation of the algorithms.
In most cases,MFR and Greedy require the same number
of hops to reach the destination.However,Compass routing
needs a few extra hops compared with the Greedy routing,
while the delivery rate is similar.All these methods have
high delivery rates for dense graphs,but low delivery rate
for sparse graphs.However,the performance of MFR and
Greedy routing come close to matching the path length given
by the shortest path algorithm in case of successful delivery
[6].Hence,we might expect similar energy consumption for
MFR,Greedy,and Compass algorithms.NC has fewer el-
igible next nodes compared to the other algorithms,which
in turn reduces its delivery rate.However,when this algo-
rithm succeeds,the probability of achieving the best power
dilation is high.The energy consumption of NFP may have
the power dilation in between NC and the other algorithms.
Now we will present another example given in Figure 1,
where the above mentioned algorithms fail to deliver the
packet.In this example,the destination is d2.The next
node is b in MFR,Greedy,and Compass algorithms.Then,
b selects f which forwards the packet to the next node g.
At this point,g forwards back the packet to f since its the
only closest neighbor to the destination.According to the
definition,all these algorithms will drop the packet at g.In
the Greedy algorithm,if a node does not have any neighbors
closer to the destination than itself,the packet gets stuck at
the local maximum that reduces the delivery rates.NFP
and NC also face a similar problem,when the packet arrives
at f both the algorithms will drop the packet,where f does
not have any neighbors closer to the destination than itself.
All of these algorithms are routing loop free except Compass
algorithm.
4.SYSTEM MODEL AND PRELIMINAR-
IES
In our MANET model,which is adapted from [1],a set
of mobile hosts are spread out in an environment that is
modeled by the Euclidean plane:each mobile host with x-
and y-coordinates is represented by the point (x,y).All
distances are Euclidean distances in the plane.We use the
following hypotheses and notation:
1.Any mobile host knows the coordinates (x,y) of its
position.
2.The transmission range of each mobile host is r,that
is,two hosts can directly communicate with each other
if their distance is at most r.
3.Each mobile host has an omni-directional antenna,which
covers a circular area of radius r.
4.Communication links are bidirectional,that is,if a
mobile host u is able to receive signals from a mobile
host v,then v is also able to receive signals from u.
Based on the above hypotheses,we can represent a MANET
as a geometric undirected graph G = (S,E),where vertices
represent mobile hosts and edges represent a link through
which a pair of mobile hosts can communicate directly.The
set of vertices S is thus a set of points in the Euclidean plane.
Let d(u,v) be the distance between the points u and v in the
plane.The set of edges E {{u,v}:u,v ∈ S,d(u,v) ≤ r},
that is,E contains all the pairs of mobile hosts at a distance
of at most r [1].The resulting graph UDG(S) is called a unit
disk graph.For node u,we denote the set of its neighbors
by N(u).
Given a unit disk graph UDG(S) corresponding to a set
of points S,and a pair (s,d) where s,d ∈ S,the problem of
energy efficient position-based routing is to construct a path
in UDG(S) from s to d,where in each step,the decision of
which node to go to next is based only on the coordinates of
the current node c,N(c),and d.At the same time,the en-
ergy consumption both at the nodes and for communication
must be minimized to maximize the network lifetime.Here,
s is termed the source and d the destination.Frequently,we
will also refer to the line
cd passing through c and d.
An algorithm is deterministic if,when at c,the next node
is chosen deterministically from N(c),and is randomized if
the next step taken by a packet is chosen randomly from
N(c).
The routing algorithm may or may not succeed in find-
ing a path from s to d.We use the following notion of
a graph defeating an algorithm from [2].A deterministic
algorithm is defeated by a graph G = (S,E) if there is a
pair (s,d) ∈ S such that a packet using the algorithm never
reaches the destination d when beginning at the source s [2].
A randomized algorithm is defeated by a graph G = (S,E)
if there is a source/destination pair (s,d) ∈ S such that a
packet using the algorithm and originating at source s has
probability 0 of reaching destination d in any finite number
of steps.
We are interested in the following performance measures
for routing algorithms:the delivery rate,that is,the per-
centage of times that the algorithm succeeds and the power
dilation,the average ratio of the total communication power
consumption by the algorithm to the energy consumption of
the shortest path in the graph.
5.RANDOMIZED ENERGY AWARE
ROUTINGALGORITHMS
The deterministic algorithms that follow a path,constructed
based on a specific heuristic,might face the local maxi-
mum or routing loop,or lack of eligible neighbors during
the packet forwarding process.Therefore,this may reduce
their packet delivery rate,which is the primary objective of
a routing algorithm.The deterministic face and GFG algo-
rithms successfully overcome the above mentioned problems
74

and always guarantee packet delivery at the price of follow-
ing a long path.
Another simple and efficient way to avoid these problems
might be using randomization when choosing a neighbor.A
position-based routing algorithm is randomized if the next
node is chosen randomly out of the neighbors of the cur-
rent node.In [2],Bose and Morin proposed a randomized
algorithm called Random Compass in the context of trian-
gulations.In Random Compass the next node is chosen
uniformly at random from the two nodes that satisfy the di-
rectional heuristic going in the clockwise and the anticlock-
wise directions.The algorithm has a higher delivery rate
than the deterministic algorithms at the price of a longer
path length.In [4],the authors proposed a new set of ran-
domized algorithms,called AB algorithms,to increase the
packet delivery rate with the control over the path length.
In AB algorithms,the current node selects two neighboring
nodes (candidate nodes) from above and below the
cd line
based on either the greedy or compass heuristic.The next
node x is then chosen from these two nodes based on some
probability distribution.For example,in uniform distribu-
tion,the next node is chosen uniformly at randomout of the
candidate nodes.Let n
a
and n
b
be the candidate nodes from
above and below the
cd line,and dis
n
a
d
and dis
n
b
d
be their
distances to the destination d,respectively.Furthermore,
let θ
n
a
=

n
a
cd and θ
n
b
=

n
b
cd be the angles formed by
n
a
and n
b
with c and d,respectively.In case of distance
based biasing,the candidate nodes n
a
and n
b
have weights
of dis
n
b
d
/(dis
n
a
d
+dis
n
b
d
) and dis
n
a
d
/(dis
n
a
d
+dis
n
b
d
),re-
spectively.The next node is chosen such that the probability
to pick the candidate node closest to the destination is high.
However,in angle based biasing,the weights of n
a
and n
b
are θn
b
/(θn
a
+θn
b
) and θn
a
/(θn
a
+θn
b
),respectively.The
next node is the one that minimizes the angle to the direc-
tion of the destination with high probability.The simulation
results in [4] shows that AB algorithms not only have high
packet delivery rates but also have good stretch factor.
However,the above mentioned randomized algorithms do
not consider the energy constraints while routing packets,
and as a result might not be directly applicable in an en-
ergy constrained environment.In this paper we extend the
above schemes and propose new energy aware randomized
algorithms that enable higher packet delivery rates and effi-
cient utilization of energy in the network.Below,we present
variant of our algorithms:
Let n
a
and n
b
be the neighbor of c from above and below
the
cd line,respectively.Furthermore,P
ca
= dis
2
cn
a
+C and
P
cb
= dis
2
cn
b
+C are the power needed to forward one bit
information from c to n
a
and n
b
,respectively.Here,dis
cn
a
and dis
cn
b
are the Euclidean distance between c and n
a
and
n
b
.Finally,the cost at n
a
and n
b
are Cost
n
a
= 1/g(n
a
) and
Cost
n
b
= 1/g(n
b
),respectively.Our randomized algorithms
can then be defined as follows:
1.PowerGreedy:Let n
a
be the neighbor of c from
above the
cd line such that P
ca
∗ dis
n
a
d
is minimized
among such neighbors.Similarly,let n
b
be the neigh-
bor of c from below the
cd line such that P
cb
∗dis
n
b
d
is
minimized among such neighbors.The next node x is
chosen fromn
a
and n
b
with probability dis
n
b
d
/(dis
n
a
d
+
dis
n
b
d
) and dis
n
a
d
/(dis
n
a
d
+dis
n
b
d
),respectively.
2.CostGreedy:Let n
a
be the neighbor of c from above
the
cd line such that Cost
a
∗dis
n
a
d
is minimized among
such neighbors.Similarly,let n
b
be the neighbor of c
from below the
cd line such that Cost
b
∗dis
n
b
d
is min-
imized among such neighbors.The next node x is cho-
sen from n
a
and n
b
with probability dis
n
b
d
/(dis
n
a
d
+
dis
n
b
d
) and dis
n
a
d
/(dis
n
a
d
+dis
n
b
d
),respectively.
3.Power*CostGreedy:Let n
a
be the neighbor of c
from above the
cd line such that (P
ca
∗Cost
a
)dis
nad
is
minimized among such neighbors.Similarly,let n
b
be
the neighbor of c frombelow the
cd line such that (P
cb

Cost
b
)dis
n
b
d
is minimized among such neighbors.The
next node x is chosen from n
a
and n
b
with probability
dis
n
b
d
/(dis
n
a
d
+dis
n
b
d
) and dis
n
a
d
/(dis
n
a
d
+dis
n
b
d
),
respectively.
4.PowerCompass:Let n
a
be the neighbor of c from
above the
cd line such that P
ca
∗ θ
n
a
is minimized
among such neighbors.Similarly,let n
b
be the neigh-
bor of c from below the
cd line such that P
cb
∗ θ
n
b
is
minimized among such neighbors.The next node x is
chosen fromn
a
and n
b
with probability θ
n
b
/(θ
n
a

n
b
)
and θ
n
a
/(θ
n
a

n
b
),respectively.
5.CostCompass:Let n
a
be the neighbor of c from
above the
cd line such that Cost
a
∗ θ
n
a
is minimized
among such neighbors.Similarly,let n
b
be the neigh-
bor of c from below the
cd line such that Cost
b
∗θ
n
b
is
minimized among such neighbors.The next node x is
chosen fromn
a
and n
b
with probability θ
n
b
/(θ
n
a

n
b
)
and θn
a
/(θn
a
+θn
b
),respectively.
6.Power*CostCompass:Let n
a
be the neighbor of c
from above the
cd line such that (P
ca
∗ Cost
a

n
a
is
minimized among such neighbors.Similarly,let n
b
be
the neighbor of c frombelow the
cd line such that (P
cb

Cost
b

n
b
is minimized among such neighbors.The
next node x is chosen from n
a
and n
b
with probability
θ
n
b
/(θ
n
a

n
b
) and θ
n
a
/(θ
n
a

n
b
),respectively.
This process continues until the packet reaches the desti-
nation or traverses a number of hops equal to three fourths
of the total number of nodes in the network.In the later
case,we drop the packet where randomized algorithms fail.
Also,in cost aware routing,current node c needs to know
the exact remaining power available at its neighbors.Hence,
in our work,nodes periodically broadcast the battery power
information to their neighbors.It is obvious that the smaller
the broadcast period,the more accurate the battery power
information.However,if the period is too small,it will in-
crease the communication overhead.On the other hand,a
larger period might lead the nodes to use inaccurate battery
power level.Hence,we choose the broadcast period in be-
tween two extremes.When a node forwards 3 consecutive
routing packets,it broadcast its remaining battery lifetime
to the neighbors.
Our algorithms first select two candidate nodes according
to the definition to control both the energy requirement and
distance or direction (ensure progress) to the destination.
The difference between our algorithms and the AB algo-
rithms is that in AB algorithms the communication and/or
battery power is not taken into account to select the candi-
date neighbors.In its new version we expect a performance
that is similar to ABalgorithms in terms of the packet deliv-
ery rates;however,since we control the energy consumption
both at communication and node levels,we expect a better
75

overall energy consumption when routing packets between
source and destination.
Let us again consider the second example in Figure 1,
where the deterministic algorithms fail to deliver the packet.
The PowerGreedy will follow c −b −f,whereas PowerCom-
pass will follow a − b − f with higher probability.At this
point the algorithms may pick g and forward the packet to
it.However,at some point h will be chosen as the next node,
and the packet may reach the destination through the node
i.Hence,it is clear that our randomized algorithms still
offer higher packet delivery rates.In addition,we consider
energy consumption of routing packets and nodes to pick the
next node,this in turn ensures moderate power dilation.
An immediate application of this class of algorithms is as
follows:Mobile nodes may not always be distributed uni-
formly.For example,two sessions of a conference on wire-
less communications are going on in two different rooms of a
building.People from these two rooms might need to com-
municate with each other through some of the intermediate
mobile devices placed in between them.The resulting topol-
ogy forms two distinct clusters of mobile devices.In this
topology,only a few mobile devices might be placed between
the clusters,we call this sparse part of the topology as a hole.
Another such topology may appear in sensor networks.In
sensor networks there are thousands of tiny sensors with
constrained resources that are placed in an inaccessible en-
vironment.These sensors monitor the target area and send
back the gathered information to one or more sinks (devices
with more energy and computational power),which are ac-
cessed by the end users.Although the sinks are not usually
energy constrained,direct communication between sinks and
sensors might dissipate their energy abruptly.In [13] the au-
thors mentioned that multihop routing might achieve better
performance than direct communication to save the energy
of sensors.However,in multihop routing,nodes close to the
sink become hot spots and may lose their energy quickly be-
cause they are continuously forwarding messages for other
sensors.This will create a hole around the sink.Hence,
energy aware randomized algorithms are well suited for this
particular situation.
6.SIMULATION ENVIRONMENT
In the simulation experiments,a set S of n points (where
n ∈ {75,100,125,150}) is randomly generated on a square
of 100m by 100m.For the transmission range of nodes,we
use 15m (experiments showed that with lower transmission
radii,the graph was too often disconnected,and with higher
transmission radii,the generated graphs were so dense that
the delivery rate of all algorithms approached 100%).Each
node has initial energy level between 3M and 4M,where
M = 10
6
,which is assigned randomly.After generating
a fully connected UDG(S),a set of 100 source-destination
pairs is randomly chosen.A fixed size data packet of length
16 bytes is used in addition to a 6 byte control packet that
contains the IDand current battery level of nodes.This con-
trol packet is periodically broadcast by a node to its neigh-
bors to advertise the current energy levels (nodes broadcast
their energy level after forwarding every three consecutive
routing packets).All the routing algorithms are then applied
in turn on the chosen source-destination pairs.Clearly,an
algorithm succeeds if a path to the destination is discovered.
The deterministic algorithms are deemed to fail if they enter
a loop,while the randomized algorithms are considered to
fail when the number of hops in the path computed so far
exceeds three fourths of the number of nodes in the graph.
To compute the average packet delivery rate,this process
is repeated with 100 random graphs and the percentage of
successful deliveries determined.Additionally,the average
power dilation is computed.
There are different radio models for energy aware routing.
For example,in [11],Rodoplu et al.proposed a power con-
sumption radio model based on the observation that direct
transmission is more power consuming than relaying mes-
sages through the intermediate nodes.Also,transmission
power is related to the path loss as 1/d
n
,where n is the
path loss exponent.Finally,transmit and receive circuitries
are subject to the energy consumption.Hence,the energy
required to transmit a message from node A to node B at
distance d is d
n
+c/t,where t and c are the required energy
at the transmitter and receiver,respectively.In their simu-
lation,they adopt the values of n,t,and c as 4,10
−7
mW,
and 20 mW.The power needed to transmit one bit of infor-
mation is then d
4
+2∗10
8
.In [13] Stojmevovic et al.call this
model as RM model.Another radio model is,called HCB
model in [13],proposed by Heinzelman et al.in [7].In their
model,the energy needed to transmit one bit of information
between two nodes is ￿
amp
d
n
+ 2E
elec
,where n = 2 is the
path loss exponent.The radio dissipates at transmitter and
receiver circuitry is E
elec
= 50nJ/bit and transmitter ampli-
fier is ￿
amp
= 100pJ/bit/m
2
.We can further normalize the
energy requirement by setting E = E
elec
/￿
amp
.Hence,the
final expression becomes (d
2
+1000) to transmit one bit of a
message.We use HCB model to evaluate the performance
of the algorithms.
In addition to the uniform distribution,we also consider
cluster distribution.In such distribution,60 nodes are dis-
tributed randomly on the above mentioned two dimensional
plane,and the remaining 27 nodes form three clusters A,B,
and C.They are centered at (15,15),(25,25),and (75,75)
each with 9 nodes.Uniformly distributed nodes are used
to ensure connectivity among clusters.The first two clus-
ters are overlapped and C is disjoint.We randomly pick the
source from A and the destination from C.We mainly con-
sider 100 source-destination pairs from 100 differnt topolo-
gies and compute the average packet delivery rates and power
dilation of all the algorithms.
7.DISCUSSION OF RESULTS
Detailed simulation results for all the routing algorithms
are given in Tables 1 and 2 for the case when the trans-
mission radius is 15m.In particular,we are interested in
the performance of our proposed randomized routing algo-
rithms compared to the previously published routing algo-
rithms Greedy,Compass,PowerProgress,and Projec-
tionPowerProgress.
The randomization helps us to avoid the local maximum
or routing loop and offers high packet delivery rate.How-
ever,this also means that the extra paths found can be
long,and this contributes to the higher power dilations of
the randomized algorithms.Our simulation results give us
the exact expected results.It is immediately evident from
the results given in Tables 1 to 2 that all the deterministic
algorithms have the worst delivery rates but the best power
dilations.All the randomized algorithms improve on the de-
livery rates of the four deterministic algorithms.Among all
the algorithms,PowerProgress has the best power dila-
76

Algorithms
n = 75
n = 100
n = 125
n = 150
Greedy
61.17
72.33
84.52
92.44
Compass
63.08
73.24
85.81
93.97
PowerProgress
52.82
64.21
78.68
88.67
ProjecPProgress
57.98
69.79
84.12
91.53
PowerGreedy
76.59
86.06
94.82
98.81
CostGreedy
70.99
85.51
95.54
99.08
Power*CostGreedy
69.18
82.93
94.29
99.04
PowerCompass
79.19
88.46
96.59
99.13
CostCompass
78.46
88.11
96.14
99.05
Power*CostCompass
78.99
87.32
96.29
99.17
Table 1:Average packet delivery rate in terms of
percentages on UDG,for transmission radius r =
15m.
Algorithms
n = 75
n = 100
n = 125
n = 150
Greedy
1.02
1.03
1.03
1.03
Compass
1.05
1.07
1.08
1.09
PowerProgress
1.01
1.02
1.02
1.02
ProjecPProgress
1.07
1.09
1.10
1.12
PowerGreedy
2.24
2.19
1.97
1.68
CostGreedy
2.38
2.39
2.08
1.69
Power*CostGreedy
2.46
2.51
2.27
1.87
PowerCompass
1.74
1.64
1.54
1.34
CostCompass
1.75
1.66
1.55
1.37
Power*CostCompass
1.75
1.70
1.58
1.38
Table 2:Average power dilation on UDG,for trans-
mission radius r = 15m.
tion and the worst packet delivery rate.In this algorithm,
candidates nodes are closer to the destination than the cur-
rent node.Hence,if there is no such candidate node to
forward the packets,the algorithm fails,however,when it
succeeds the resultant path minimizes the total energy con-
sumption of the packet.The greedy algorithm has power
dilation close to the PowerProgress algorithm with much
better delivery rate.The other two deterministic algorithms
also have similar performance.
We can divide our randomized algorithms in two groups
based on their biasing strategy,namely greedy and com-
pass based algorithms.The compass based algorithms dom-
inate the other group both in terms of packet delivery rate
and power dilation.PowerCompass,CostCompass,and
Power*CostCompass algorithms select candidate nodes
that are close to the direction of the destination in addition
to minimizing the power and/or cost.After that the next
node is chosen with higher probability such that it again
minimizes the angle formed between that node,the cur-
rent node,and the destination.In greedy based algorithms,
PowerGreedy,CostGreedy,and Power*CostGreedy
algorithms select the candidate nodes in a slightly differ-
ent way.The candidate nodes minimize both the distance
to the destination and energy requirement.The next node
is picked with higher probability such that it reduces the
distance to the destination.Our simulation results show
that the first group has better performance in terms of both
packet delivery and power dilation compared to the other
one.Following the direction helps the packet to reach the
destination early.On the other hand,in greedy based al-
gorithms,candidate nodes are chosen such that these nodes
minimize the distance to the destination and the energy re-
quirement.However,choosing farthest neighbor of the cur-
rent node is more energy consuming than picking a node
that minimizes the angle formed between that node,the
current node,and the destination.This contributes higher
power dilation in our randomized greedy based algorithm.
Also,in these algorithms,the packets may deviate from the
direction of the destination,and this may offer low packet
delivery rates.
We also examine the behavior of the variants of our algo-
rithms.In these variants,c picks the candidate nodes just
as the above mentioned algorithms,however,the next node
is chosen out of two candidate nodes that minimize power,
cost,or power*cost to reach the destination.These algo-
rithms also have delivery rates similar to our above men-
tioned algorithms.However,they have very high (almost
three times larger than the deterministic algorithms) power
dilations.This might be happening due to the fact that
their corresponding biasing does not provide any progress
(either following the direction or reducing the distance to
the destination).
The three compass based algorithms have similar perfor-
mance.This is expected because three of the algorithms use
the same biasing method to reach the destination.Hence,
they have similar delivery rates.The power dilation is also
similar with PowerCompass having slightly better results
compared to the other two algorithms.We also expect it
since PowerCompass always tries to minimize the power
requirement during routing while following the direction to
the destination.However,the CostCompass algorithmmight
follow a slightly longer path than PowerCompass,which
mainly considers remaining lifetime of a node to pick the
candidate nodes.Finally,Power*CostCompass algorithm
just combines both power and cost to pick next nodes and
has performance close to both methods.The greedy based
algorithms also show similar behavior with PowerGreedy
having the highest packet delivery rate as well as lowest
power dilation.The other two algorithms have similar per-
formance.
Algorithms
Delivery Rate
Power Dilation
Greedy
52.17
1.04
Compass
51.01
1.10
PowerProgress
37.20
1.02
ProjecPProgress
45.31
1.19
PowerGreedy
68.03
2.31
CostGreedy
66.73
2.78
Power*CostGreedy
66.74
2.86
PowerCompass
77.03
1.95
CostCompass
75.73
2.09
Power*CostCompass
74.74
2.10
Table 3:Average packet delivery rate and power
dilation on UDG of 87 nodes under cluster distribu-
tion,for transmission radius r = 15m.
The results of our simulation on cluster distribution are
given in Table 3.We investigate the performance of all
the algorithms on cluster node distribution.The random-
ize algorithms still dominate the deterministic algorithms
in terms of packet delivery rate though the performance on
power dilation is just opposite.Furthermore,compass based
algorithms again have better performance than greedy based
algorithms.
77

7.1 Effect of Node Density
As the number of nodes grows,the delivery rate of all
the algorithms increases.However,we notice the significant
change in delivery rates in the case of deterministic algo-
rithms that performwell in dense networks.In contrary,our
randomized algorithms show the same behavior in sparse to
dense networks.This gives another explanation of the ap-
plicability of our algorithms in a sparse network with the
presence of local maximum.
In terms of power dilation,however,the results are dif-
ferent for the deterministic and randomized algorithms.For
the deterministic algorithms,the power dilations increase
very slightly as the number of nodes increases.Whereas,
our randomized algorithms show the opposite trend.The
power dilations in the UDG decrease as the number of nodes
increases.For larger values of n,the number of possible
paths available to the randomized algorithms increase,so
the power dilation can be expected to decrease.For in-
stance,in a denser graph,the algorithm can recover from a
bad path earlier,which leads to a lower power dilation for
the algorithm.The deterministic algorithms on the other
hand might need to traverse some extra length in dense net-
works,which leads to slightly worse dilations.
8.CONCLUSIONS
In this paper,we extended our previous work [4] and pro-
posed a set of new randomized energy aware routing algo-
rithms for mobile ad hoc and sensor networks.In our algo-
rithms,the current node holding the packet always forwards
the packet to the next node based on the position of itself,
its neighbors,and the destination.The compass based al-
gorithms are called PowerCompass,CostCompass,and
Power*CostCompass algorithms.In particular,to deter-
mine the next node at any point,these algorithms pick one
candidate above and one below the line between the current
node to the destination,by using the heuristic that mini-
mizes both the power and/or cost and the angle (formed
between the current node,candidate node,and the destina-
tion).The next node is then chosen with higher probability
to be closer to the direction of the destination.On the other
hand,the greedy based algorithms PowerGreedy,Cost-
Greedy,and Power*CostGreedy first minimize the en-
ergy requirement and the distance to the destination for the
chosen candidate notes.Then,the next node is the one that
is closest to the destination,which is picked with higher
probability.
Our simulation results demonstrate that our randomized
energy aware algorithms yield a definite improvement over
all deterministic algorithms studied in terms of the delivery
rate.The best power dilations are achieved by the determin-
istic algorithms,however,by using weighted randomization
based on the angles created by the candidate neighbors and
the
cd line,we can maintain the improved delivery rates
while greatly reducing the power dilations of the random-
ized algorithms.Our algorithms retain their performance
even under cluster node distribution while deterministic al-
gorithms lose their performance.
9.ACKNOWLEDGMENTS
The anonymous referee’s comments are gratefully acknowl-
edged.This research is supported in part by the Natural
Sciences and Engineering Research Council,Canada.
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