Keywords battery · sensor · routing protocols · NP-complete 1 Introduction of multihop routing There is a common problem in energy efficiency consid-erations in wireless ad-hoc sensor ne
Trang 1DOI 10.1007/s11036-012-0403-1
The Energy-Aware Operational Time of Wireless Ad-Hoc
Sensor Networks
Nguyen Thanh Tung · Phan Cong Vinh
Published online: 26 August 2012
© Springer Science+Business Media, LLC 2012
Abstract Sensor networks are deployed in numerous
military and civil applications, such as remote target
detection, weather monitoring, weather forecast,
nat-ural resource exploration and disaster management
Despite having many potential applications, wireless
sensor networks still face a number of challenges due
to their particular characteristics that other wireless
networks, like cellular networks or mobile ad hoc
net-works do not have The most difficult challenge of the
design of wireless sensor networks is the limited energy
resource of the battery of the sensors This limited
resource restricts the operational time that wireless
sen-sor networks can function in their applications Routing
protocols play a major part in the energy efficiency
of wireless sensor networks because data
communica-tion dissipates most of the energy resource of the
net-works This paper studies the importance of
consider-ing neighborconsider-ing nodes in the energy efficiency routconsider-ing
problem After showing that the routing problem that
considers the remaining energy of all sensor nodes is
NP-complete, heuristics are proposed for the problem
Simulation results show that the routing algorithm that
N T Tung
International School, Vietnam National University in Hanoi,
144 Xuan Thuy St., Cau Giay District, Hanoi, Vietnam
e-mail: tungnt@isvnu.vn
P C Vinh ( )
IT Department, NTT University, 300A Nguyen Tat Thanh
St., Ward 13, District 4, HCM City, Vietnam
e-mail: pcvinh@ntt.edu.vn
considers the remaining energy of all sensor nodes improves the system lifetime significantly compared to that of minimum transmission energy algorithms Also, the energy dissipation of neighboring nodes accounts for a considerable amount of the total energy dissipa-tion Therefore, a method that reduces the energy dis-sipation by notifying the neighboring nodes to turn off their radio when not necessary is proposed By reducing the unnecessary energy dissipation of the neighbors, the lifetime is increased significantly
Keywords battery · sensor · routing protocols ·
NP-complete
1 Introduction of multihop routing
There is a common problem in energy efficiency consid-erations in wireless ad-hoc sensor networks (WASNs): maximizing the amount of data sent between any pair
of all sensor nodes until the first sensor node is out of battery As in sensor networks, sensors send data peri-odically during each fixed amount of time, the problem
is the same as maximizing network operation lifetime until the first sensor node run out of battery
There are many energy efficient routing methods proposed in wireless networks In [1, 2], a minimum total power routing (MTPR) was proposed In this protocol, the route with the minimum total power
con-sumption is selected from a set S containing all possible
paths The transmission energy and the reception en-ergy are used as a link cost metric
Trang 2Mobile Netw Appl (2013) 18:454–463 455
where Psent(i, j) is the transmission energy between
Node i and Node j Preceive( j) is the reception energy at
Node j The total energy for route l, P lcan be derived
from
P l=
D−1
i=0
for all node in the route, where i = 0 and j = D are
the source and the destination node, respectively The
desired route k can be obtained from:
P k= min
where A is the set containing all possible routes.
Minimum battery cost routing (MBCR) [1] was
an-other way to approach In this protocol, the inverse of
battery energy of nodes is used as the metric The path
metric is the sum of the link metrics The path with the
smallest metric is selected from the possible routes Let
c t
i be the battery capacity of node n i at the time t The
battery cost of node n i , f i (c t
i ) is
f i (c t
i ) = 1
The battery cost for route l consisting of D nodes is
P l=
D−1
i=0
f i (c t
The desired route k can be obtained from:
P k= min
where A is the set containing all possible routes
How-ever, since only the sum of the battery costs is
con-sidered, a route containing nodes with low remaining
battery may still be selected For example, if we have a
route with 3 nodes and in the route, Node 3 has very low
battery energy If Node 1 and Node 2 have very high
battery energy, the total cost is still quite small and the
route is still selected
In order to improve the previous protocols, min–max
battery cost routing (MMBCR) was developed [1,2]
In this protocol, the node with smaller battery energy
will be avoided in selected paths to balance energy
consumption across the networks The battery cost for
route l is redefined as:
P l= max
i ∈route_l f i (c t
where f i (c t
i ) is given by Eq.4 The desired route k can
be obtained from:
P k= min
But the disadvantage of the algorithm, like MBCR,
is that they do not try to minimize the energy consump-tion In order to achieve both low energy consumption and long network lifetime, a conditional max–min bat-tery capacity routing (CMMBCR) was proposed [1] In the protocol, when all nodes on a path have remaining energy above a threshold value then MTPR is used When the remaining energy of any node is no longer higher than the threshold value, routing protocol is
switched to MMBCR Let R c
j be the battery capacity
for route j at time t:
R c j= min
i ∈route_ j c
t
Let A be the set containing all possible routes be-tween any two nodes at time t satisfying the following
equation:
for any route j ∈ A, where γ is a predefined energy
capacity threshold If Eq.10is satisfied, MTPR routing protocol is used Otherwise, MMBCR is used
There have been numerous studies on the energy efficiency of multi-hop routing in literature These stud-ies use the Dijkstra algorithm [3] or variants of this algorithm to calculate the shortest routes to a desti-nation with different types of energy metrics Unfortu-nately, only few studies mentioned about the remaining battery of sensor nodes, and it is difficult to apply the Dijkstra algorithm or its variants to the lifetime problem of wireless ad-hoc sensor networks (WASNs)
In other words, it is preferred to use the variants of the Dijkstra algorithm as routing methods because the al-gorithm is Linear Programming (LP) problem, but the algorithm cannot provide an optimal routing solution for the lifetime problem of WASNs
Wireless transmission is different to wire-line net-works in that transmission from a source node to a des-tination node causes neighboring nodes to dissipate en-ergy when they detect the transmission Unfortunately, the energy dissipation of neighboring nodes may be comparable to the energy dissipation of the nodes in the path and can degrade the performance of the routing methods It is shown that the reception energy of 802.11 products is at least 50 % that of the transmission energy [4,5] For example, Stemm and Katz measure the idle: receive: send energy consumption ratios of 1:1.05:1.4 [6] This data measurement emphasizes the importance
of considering reception energies by neighboring nodes
in energy-efficient routing models For example, Fig.1 shows that the transmission from the source to the destination will be listened by other seven neighboring nodes
Trang 3Fig 1 Transmission from a source to a destination drains the
energy of the source, the destination and neighboring nodes
There are many studies in the literature to work
out the best transmission power because the reduction
of the transmission range will lead to the reduction
of the energy consumption of neighboring nodes The
authors in [7] proposed a routing method considering
the reception energy of neighboring nodes to control
the transmission power In [8], the authors also
con-sidered the reception energy usage in the selection of
energy efficient paths In [9], an analytical model for
optimal transmission range for minimizing the total
en-ergy consumption was presented Unfortunately, all of
these papers are designed for mobile ad hoc networks
but not sensor networks Unlike sensor networks,
bat-tery constraint is not a major issue of mobile ad hoc
networks Therefore, only few papers in the literature
consider the limited energy storage of nodes, which is
the major challenge when designing sensor networks
For examples, [10,11] mentioned about the control of
sensor range to maximize the operation time of sensor
networks under battery limits This paper concentrates
on multi-hop routing methods that prolong the
opera-tion time of practical sensor networks under the battery
constraint of the sensors
2 Formulating routing problem
Original routing problem Given a network of n
sen-sors, in which any sensor node can connect to all other
sensor nodes by adjusting its transmission power Each
sensor node i has the energy storage of e (i) A random source node s wants to transmit data to a destination node d Obviously, there are many possible paths from
s to d Each path results in an energy reduction of
all nodes on the path (including the nodes are within the transmission range of the data transmission) The
routing problem is to find a path from s to d so that after
the data transmission, the minimum remaining energy storage of all sensor nodes is maximized:
where e (i) is the remaining energy of Node i after the
path is established
Unfortunately, Problem 11 is NP-complete, there-fore there is no polynomial time algorithm to find the energy efficient path We will prove the NP-completeness of Eq.1using graph models and a well known NP-complete problems Firstly, we give some preliminary results
Graph problem A sensor network is modelled as
G (V, E), where V is the set of nodes and E is the set of links between the nodes Node i sets its power to zero
or its power to p i j if Node i wants to transmit to Node
j, ∀i, j ∈ V Every node i has the remaining energy capacity of e (i) Given a source node s and a destination node d, find a path from s to d that maximize the minimum remaining energy of all nodes i ∈ V.
where e (i) is the remaining energy of Node i after the
path is established
Problem12can be converted to a decision problem:
Decision problem Eq. 13 A sensor network is
mod-elled as G (V, E) Node i sets its power to zero or its power to p i j if Node i wants to transmit to Node j, ∀i, j ∈
V Every node i has the remaining energy capacity of
e (i) Given a source node s and a destination node
d, find a path from s to d that e (i) ≥ c, ∀i ∈ V, c is a
constant
Let us consider a simple case of Problem13, in which all nodes transmit at the same power:
Constant power problem Eq.14 A sensor network is
modelled as G (V, E) All nodes can transmit at a con-stant i = P, ∀i ∈ V In other words, a Node i transmits with power P or does not transmit Every Node i has the remaining energy capacity e (i) Given a source node
s and a destination node d, find a simple path from s to
d that e (i) ≥ c for all nodes i ∈ V, c is a constant.
The above problem can be polynomially reduced
to the Path with Forbidden Pairs problem This is a well known NP-complete problem Details are given in
Trang 4Mobile Netw Appl (2013) 18:454–463 457
[12,13] Details of the reduction are given inAppendix
As a result, the simple constant power problem 14 is
NP-complete and therefore, the original problem11is
also NP-complete From the above results, there are no
polynomial algorithms to find a path to maximize the
minimum residual energy of sensor nodes and hence we
need to propose heuristic algorithms for the problem
3 Heuristic algorithms
Three heuristic energy-efficient routing methods are
implemented to extend the lifetime of WASNs A
round of data transmission is defined as the duration
of time a random source node transmits a unit of data
to a random destination node The lifetime of WASNs
is defined as the total number of rounds sending data
between sensor nodes until the first node run out of
energy The heuristic routing methods are summarized
as below The shortest path for these methods is
calcu-lated using the Dijkstra algorithm [3]
Shortest path of the energy dissipation (SP) Given a
source node s and destination node d, find a simple path
from s to d that minimizes the total energy
dissipa-tion by all nodes on the path This means the following
equation is minimized, where r is the reception energy
consumption of any sensor node (end of algorithm)
i∈−d
Figure2shows the SP method, where E ijdenotes to
energy consumption to send data from Node i to Node
j and E jdenotes to energy consumption to receive data
from at Node j (E j = 1 unit, ∀ j ∈ V).
Fig 2 The SP method
Algorithm 1 Implementation of the SP algorithm
Input: n: The number of sensor nodes indexed from
1to N r: A current round number of data transmission
f (l): The total energy consumption of all nodes with path l
e (n): the current energy of node n e: Minimum energy of all nodes in the network
1: Set r= 0
2: for each round of data transmission do
3: Pick up a random source node s and random destination node d
4: Using the Dijkstra algorithm to minimize f (l) from s to d
5: for n from 1 to N do
6: Update e (n)
7: end for
8: if e < 0 then
10: else
11: r = r + 1
12: end if
13: end for
14: Record r
The total energy consumption for path (1, 2, 3, 6)
(not including neighboring nodes) is: E (1,2,3,6) = E12+
E23+ E36 + E2 + E3 + E6=1 + 1 + 1 + 1 + 1 + 1=6; The total energy consumption for path (1, 4, 5, 6) is: E (1,4,5,6) = E14 + E45 + E56 + E4 + E5 + E6= 1 +
2+ 1 + 1 + 1 + 1 = 7;
Fig 3 Path calculation and selection from SP_N algorithm
Trang 5Table 1 Energy metrics of
Therefore, the SP algorithm will select the path (1, 2,
3, 6) Node 7 and Node 8 are not involved in the path
selection process The pseudo code for the algorithm is
implemented in Algorithm 1
Shortest path of the energy dissipation including
neigh-boring nodes (SP_N) Given a source node s and a
destination node d, find a simple path
from s to d
that minimizes the total energy dissipation by all nodes
participating in the data transmission.
i∈−d (p i+j ∈N(i) r j ) is minimized, where N(i) is
the set of neighboring nodes in the transmission range
of Node i (end of algorithm).
Figure3shows that unlike SP algorithm, the SP_N
algorithm considers nodes on a selected path and
all neighboring nodes involved in the transmission
The total energy consumption for path (1, 2, 3, 6)
is: E (1,2,3,6) = E12 + E23 + E36 + E2 + E3 + E6 + E4+
E7 + E8 + E7 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +
1= 10;
The total energy consumption for path (1, 4,
5, 6) is: E (1,4,5,6) = E14 + E45 + E56 + E4 + E5 + E6=
1+ 2 + 1 + 1 + 1 + 1 = 7;
Energy metrics of Fig 3 are the same the energy
metrics of Fig.2shown in Table1
Fig 4 Path calculation and selection from SP_RE algorithm
Table 2 Energy metrics of
Shortest path of the remaining energy (SP_RE) Let us
define a weight for a link on any path as the following
equation, where N (i) is the set of neighboring nodes in the transmission range of Node i.
j ∈N(i)
1
Given a source node s and a destination node d,
find a simple path
from s to d that minimizes
the total weight by all links participating in the data transmission.
i∈−d W (i) is minimized (end of algorithm).
Figure 4shows path calculation and selection from
the SP_RE algorithm, where E jdenotes to remaining
energy at Node j (Table2)
The total energy consumption for path (1, 2, 3,
6) is: E (1,2,3,6)= 1
E2 + 1
E3 + 1
E6+ 1
E4+ 1
E7 + 1
E7 + 1
E8 =
0.5 + 1 + 1 + 1 + 1 + 1 + 1 = 6.5;
The total energy consumption for path (1, 4, 5, 6) is:
E (1,4,5,6)= 1
E4 + 1
E5 + 1
E6+ = 1 + 1 + 1 = 3;
As a result, the SP_RE algorithm will select the path (1, 4, 5, 6)
4 Simulation and comparison
A number of simulators in C++ is developed to sim-ulate the performance of SP, SP_N and SP_RE The energy dissipation model used is given below The total transmission energy of a message is calculated by the following equation:
The reception energy is calculated by the equation as follows:
where Eelec is the energy dissipation of the electronic
circuitry to encode or decode a bit, k is message size,
amp is the amplifier constant and d is the distance
between the transmitter and the receiver The network
Trang 6Mobile Netw Appl (2013) 18:454–463 459
Fig 5 The first set of simulations
Fig 6 Number of rounds over 100 random 100-node networks
Table 3 Results for Fig.6
Number of rounds
90 % confidence interval (267, 271) (356, 371) (514, 524)
for the sample means
Fig 7 Average energy dissipation per round (mJ) over 100
random 100-node networks
settings for the simulations in the section are given be-low This model is the same with the model in [13,14] Network size (200 m× 200 m)
Base station (50 m, 275 m) Number of sensor nodes: 100 nodes Energy message: 20 bits
Position of sensor nodes: Uniform placed in the area
Energy model: Eelec = 50 ∗ 10−9 J, fs= 10 ∗
10−12J/bit/m2andmp= 0.0013 ∗ 10−12J/bit/m4 Broadcast ID message: 16 bits
Broadcast energy message: 32 bits
In the first set of simulations, the lifetime perfor-mance of the above routing methods is studied for the above 100 random 100-node sensor networks (Fig.5) Each node begins with 50 mJ of energy The operation
of each sensor network is divided into rounds In each round, a random source node transmits a unit of data
to a random destination node The process is repeated until the first sensor node is out of energy and the life-time for each routing method in each network topology
is recorded On average, SP, SP_N and SP_RE perform
268, 363, and 519 rounds respectively These lifetimes are the time until the first sensor fails The results are shown in Fig.6(Table3)
It is also of interest to evaluate the total energy consumption of the routing methods Figure 7shows the performance over the 100 topologies On average,
SP, SP_N and SP_RE dissipate 11.7, 6.7 and 7.8 mJ per round respectively As expected, SP_N provides the minimum energy dissipation per round among the three routing methods This is because SP_N selects
Table 4 Results for Fig.7
Energy per round (mJ)
90 % confidence interval (11.66, 11.78) (7.75, 7.84) (6.64, 6.74) for the sample means
Trang 7Fig 8 Average number of
rounds versus the number of
surviving nodes
a route to minimize the total energy dissipation of all
sensor nodes in the path including neighboring nodes
Although SP_RE provides the best lifetime, it spends
more energy per round than SP_N This is because
SP_RE needs to preserve the residual energy of all
sensor nodes so it does not always select the minimum
energy path (Table4)
So far, we consider the absolute lifetime of WASNs,
which is the time until the first sensor node dies In many
applications, sensor networks are very dense and there
are usually hundreds or thousands sensors As there
are many possible paths between a particular
source-destination pair, the data communication of the pair still
operates normally even though a small portion of the
sensors runs out of energy Therefore, it is also of interest
to investigate the lifetime performed by the above
rout-ing algorithms until a few percent of sensor nodes dies
In the next simulations, SP, SP_N and SP_RE are run
again over the above network topologies and the
life-times until 50 nodes (50 % of nodes) fail are recorded
The lifetime is the average lifetime of these topologies
Figure8shows that SP_RE only performs the best for
the system lifetime of above 90 % nodes surviving
However, if the system lifetime is considered as the
time until 50 % of nodes surviving, then SP_N becomes
the best solution The results come from the facts that
SP_RE balances the energy load among all sensor
nodes so every sensor node tends to die at the same
time (just after the first node dies) SP_N, however,
minimizes the total energy consumption of all sensors
in each round so after a few of nodes are off; it still can
operate for a significant number of rounds
5 Elimination of reception energy by neighboring
nodes
As discussed in Section1, when a source node transmits
data to a destination, neighbors that are inside the
transmission range of the source node also receive the
data As the reception energy at each neighbor is the
same to that of the destination and is more than half of
the transmission energy, the total energy consumption becomes much higher than the actual energy needed for the data transmission Therefore, in order to achieve a better lifetime for the sensor networks, it is desirable to reduce the unnecessary energy spent at the neighbors Data transmission in sensor networks requires much lower bandwidth (on the order of 1–100 kb/s) than that of mobile ad hoc networks (on the order of 1–
100 Mbps) [4] In sensor networks, nodes send data pe-riodically and the interval between two subsequent data transmission in the networks is usually very long, while
in mobile ad hoc networks, data is transmitted continu-ously and randomly so mobile device has to wake up to listen to the channel Therefore, unlike mobile ad hoc networks, the MAC design for sensor networks can use time-division multiple access (TDMA) based proto-cols that conserve more energy than contention-based protocols like carrier sense multiple access (CSMA) (e.g., IEEE 802.11) TDMA protocol allows sensors to turn off their radio when not necessary As a result, a
Fig 9 Transmission from a source to a destination tells
neigh-boring nodes to turn off their radio during the transmission
Trang 8Mobile Netw Appl (2013) 18:454–463 461
pre-broadcast scheme is proposed to reduce the energy
dissipation of neighboring nodes
In more detail, when a source node wants to send
data to a destination, it will calculate a routing path
using any routing method presented in Section3 The
source node then sets its transmission power to reach
the destination node and broadcasts a message that
contains the list of ID of nodes that are involved in the
data transmission Any node in the broadcast area will
receive the message If a node’s ID matches the ID of
any node in the path, it will turn on its radio during
the data transmission Otherwise, the node turns off its
radio until the end of this data transmission All nodes
wake up at the beginning of the next round for new
data transmission The pre-broadcast scheme requires
two overhead energies:
1 A source node broadcasts a list of ID of nodes on
the path to all sensors in the broadcast area
2 All sensor nodes in the area receive the message
from the source node
However, as the actual data message size is bigger
than the broadcast message, it is expected that the
pre-broadcast scheme will reduce the total energy
con-sumption significantly (Fig.9)
In the next set of simulations, SP method was run
with the pre-broadcast message of 100 bits (SP_100)
and 500 bits (SP_500) respectively The lifetime (the
number of rounds) for each case is recorded Figure10
shows that new SP method performs much better than
the original SP method (Table5) On average, SP with
the broadcast message of 100 bits achieves the lifetime
of about two times longer than the original SP This is
because the new SP method only spends energy of 60 %
of the original SP method as shown in Fig.11
The same simulations were repeated for SP_RE with
the pre-broad cast message of 100 bits (SP_RE_100)
and 500 bits (SP_RE_500) Unlike the new SP method,
the new SP_RE method only performs slightly better
than the original SP_RE Figure 12 also shows that
there is not significant improvement of the energy
dissi-pation per round by the new SP_RE method (Table6)
This is because the original SP_RE already avoids the
energy drain of neighboring nodes in the data
trans-mission, so the broadcast process does not significantly
help We do not consider SP_N method as without the
reception energy at neighboring nodes, SP_N is the
same as SP method (Fig13; Table7)
It is also of interest to evaluate the best routing
method for the lifetime problem of WASNs until a
specified proportion of nodes fails Although the new
broadcast scheme improves the lifetime of the SP
method significantly, Fig 14 shows that the SP_RE
Fig 10 Number of rounds over 100 random 100-node networks
Table 5 Results for Fig.10
Number of rounds
90 % confidence interval (267, 271) (551, 564) (475, 486) for the sample means
Fig 11 Average energy dissipation per round (mJ) over 100
random 100-node networks
Fig 12 Number of rounds over 100 random 100-node networks
Trang 9Table 6 Results for Fig.12
Number of rounds
90 % confidence interval (514, 524) (671, 690) (555, 570)
for the sample means
Fig 13 Average energy dissipation per round (mJ) over 100
random 100-node networks
Table 7 Results for Fig.13
Energy per round (mJ)
90 % confidence interval (7.75, 7.84) (5.70, 5.78) (6.84, 6.92)
for the sample means
Fig 14 Average number of rounds versus the number of
surviv-ing nodes
method is still the best routing method for the lifetime Therefore, it is recommended to use SP_RE to prolong the network lifetime of WASNs
6 Conclusion
It is shown that the problem of locating a simple path that maximizes the minimum remaining energy of all sensor nodes is NP-complete Therefore, there is no polynomial time algorithm for the problem and heuris-tic solutions are required to achieve reasonable energy efficiency
Three heuristic routing methods were implemented: (1) Dijsktra algorithm to minimize the total energy dis-sipation of nodes on a selected path (SP), (2) Dijsktra algorithm to minimize the total energy dissipation of nodes on a selected path including neighboring nodes (SP_N), and (3) the Dijskstra algorithm considering the remaining energy of all sensor nodes on a selected path (SP_RE) Simulation results show that SP_RE can double the lifetime of SP on average, while SP_N can minimize the total energy dissipation to half of SP
As discussed in the paper, the energy dissipation of neighboring nodes accounts for a considerable amount
of the total energy dissipation Therefore, a method that reduces the energy dissipation by notifying the neighboring nodes to turn off their radio when not necessary was proposed The method operates using broadcast messages When a source node computes a path to a destination node, the node can broadcast a message to all sensor nodes in the transmission range
to the destination This message contains the IDs of forwarding nodes in the path The sensor nodes receive the message and determine if they belong to the path
If not, these nodes turn off their radio during the data transmission
By reducing the unnecessary energy dissipation of the neighbors, the lifetime is increased significantly
Appendix
Proof of the NP-completeness of problem (14) [11,12]
Path with forbidden pairs problem (PFP)
Instance: Consider a graph G (V, E), given a source
node s and destination node d, and a col-lection C = {(a1 , b1), , (a m , b m )} of pairs
of vertices in V.
Question: Find a simple path from s to d that contains
at most one vertex from each pair in C.
Trang 10Mobile Netw Appl (2013) 18:454–463 463
The PFP problem is known to be well-known graph
theory NP-complete
Path with remaining energy problem (RE) A sensor
network is modelled as G (V, E) All nodes can transmit
at a constant i = P, ∀i ∈ V In other words, a node i
does not transmit or transmit with power P Every node
i has the remaining energy capacity e (i) Given a source
node s and a destination node d, find a simple path from
s to d that e (i) ≥ c for all nodes i ∈ V.
We now give a polynomial reduction from this
prob-lem to the Path with Forbidden Pairs probprob-lem (PFP)
Without loss of generality, we assume that for any
node, a reception usage of one unit of energy (i.e.,
r i = 1 for any node i) We first transform an instance
(G(V, E), s, d, C) of the PFP problem in an instance
(G(V, E), s, d, p, c) of the Remaining Energy
prob-lem by formally definition as follows, where s and d are
unchanged, c is the minimum tolerable capacity at any
node i and is set to an arbitrary positive value.
E= E ∪ {(x, v xy ), (y, v xy )|(x, y) ∈ C} (18)
e iis the remaining energy set to:
1 e i = c if i ∈ {v xy |(x, y) ∈ C&(x = t)||y = t}
2 e i = c + 1 if i ∈ {v xy |(x, y) ∈ C&(x = t)||y = t}
3 e i = c + |V|, otherwise.
By the definition, G contains all the vertices of G
and m new vertices that represent a forbidden pair Let
us define F as the set of the m vertices Each vertex of
F is only connected to its two respective “forbidden”
vertices and is assigned e i = c + 1, or e i = c, if the
des-tination is part of the forbidden pair
We now prove that a solution of this instance of the
RE problem if and only if it is a solution for the original
instance of the PFP problem
It is easy to see that a solution from s to d for the RE
problem in(G(V, E), s, d, p, c) does not include any
of the vertices in F, as any vertex i of the path (except
the destination d) requires to decrement e iby at least 2
Hence this path is also the path in G.
Conversely, given a solution path
of the in-stance (G(V, E), s, d, C), we can verify that the path
is a feasible solution path for the RE problem in
(G(V, E), s, d, p, c) As G is a sub graph of G, a path
in G is also a path in G Hence we need to verify that
the path in G satisfies the remaining energy constraints.
As all nodes except nodes in F respect the remaining
energy constraints, only nodes in F may violate the
feasibility of the solution This can be seen that none
of nodes in F belong to the path as F ∈ G A vertex in
F can be a be neighbor of maximum a vertex of
As
each vertex of F can only be a neighbor of its forbidden
pair, the vertex cannot be a neighbor of two nodes in the path
As the proof follows a polynomial reduction to the Path with Forbidden Pairs (PFP) problem, the RE problem is NP-complete
References
1 Toh CK (2001) Maximum battery life routing to support ubiquitous mobile computing in wireless ad hoc networks IEEE Communications Magazine, June 2001
2 Safwat A et al (2002) Energy-aware routing in MANETs: analysis and enhancements In: 5th ACM international work-shop on modeling analysis and simulation of wireless and mobile systems, pp 46–53
3 Dijkstra algorithm http://en.wikipedia.org/wiki/Dijkstra’s al-gorithm Accessed 23 Sept 2011
4 Ilyas M, Mahgoub I (2005) Mobile computing handbook CRC Press, Boca Raton, FL
5 Feeney L, Nilsson M (2001) Investigating the energy con-sumption of a wireless network interface in an ad hoc net-working environment In: IEEE INFOCOM
6 Stemm M and Katz RH (1997) Measuring and reducing energy consumption of network interfaces in handheld de-vices IEICE Trans Fundam Electron Commun Comput Sci 80(8):1125–1131 (special issue on Mob Comput)
7 Liu BH et al (2004) An energy efficient select optimal neigh-bor protocol for wireless ad hoc networks In: Proceedings
of the 29th annual IEEE international conference on lo-cal computer networks (LCN’04) IEEE Computer Society, Washington, DC, USA, pp 626–633
8 Shrestha N, Mans B (2005) Reception-aware power control
in ad hoc mobile networks In: The third international con-ference on innovative applications of information technology for developing world (asian applied computing conference (AACC 2005)), Kathmandu, Nepal, 10–12 December 2005
9 Chen Y et al (2003) On selection of optimal transmission power for ad hoc networks In: 36th annual Hawaii interna-tional conference on system sciences (HICSS’03) - track 9, Washington, DC, USA
10 Tung NT (2011) The operational time of wireless ad-hoc sen-sor networks In: Proceeding in the 5th international confer-ence on software, knowledge information, industrial manage-ment and applications, Benevento, Italia
11 Tung NT (2012) Heuristic energy-efficient routing solutions
to extend the lifetime of wireless ad-hoc sensor networks In: 4th asian conference on intelligent information and data-base systems Lecture notes in computer science, vol 7197 Kaohsiung, Taiwan, pp 487–492
12 Garey M, Johnson DS (1979) Computers and intractability.
A guide to the theory of NP-completeness Freeman, San Francisco, CA
13 Tung NT (2009) Energy-efficient routing algorithms in wireless sensor networks PhD thesis, Monash University, Australia
14 Heinzelman WB, Chandrakasan AP (2002) An application specific protocol architecture for wireless microsensor net-works IEEE Trans Wirel Commun 1(4):660–670