DSpace at VNU: The energy-aware operational time of wireless Ad-Hoc sensor networks

Mobile Netw Appl (2013) 18:454–463 DOI 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, natural 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 networks 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 sensor networks can function in their applications Routing protocols play a major part in the energy efficiency of wireless sensor networks because data communication dissipates most of the energy resource of the networks This paper studies the importance of considering neighboring nodes in the energy efficiency routing 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 dissipation Therefore, a method that reduces the energy dissipation 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 Introduction of multihop routing There is a common problem in energy efficiency considerations 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 periodically 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 consumption is selected from a set S containing all possible paths The transmission energy and the reception energy are used as a link cost metric Ci, j = Psent (i, j) + Preceive ( j) (1) Mobile 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, Pl can be derived from D−1 Pl = Ci,i+1 (2) i=0 for all node in the route, where i = and j = D are the source and the destination node, respectively The desired route k can be obtained from: Pk = Pl l∈ A (3) cti (4) The battery cost for route l consisting of D nodes is D−1 Pl = fi (cti ) (5) i=0 The desired route k can be obtained from: Pk = Pl l∈ A (6) where A is the set containing all possible routes However, since only the sum of the battery costs is considered, a route containing nodes with low remaining battery may still be selected For example, if we have a route with nodes and in the route, Node has very low battery energy If Node and Node 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: Pl = max fi (cti ) i∈route_l (7) where fi (cti ) is given by Eq The desired route k can be obtained from: Pk = Pl l∈ A Rcj = cti i∈route_ j where A is the set containing all possible routes Minimum battery cost routing (MBCR) [1] was another 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 cti be the battery capacity of node ni at the time t The battery cost of node ni , fi (cti ) is fi (cti ) = But the disadvantage of the algorithm, like MBCR, is that they not try to minimize the energy consumption In order to achieve both low energy consumption and long network lifetime, a conditional max–min battery 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 Rcj be the battery capacity for route j at time t: (8) (9) Let A be the set containing all possible routes between any two nodes at time t satisfying the following equation: Rcj ≥ γ (10) for any route j ∈ A, where γ is a predefined energy capacity threshold If Eq 10 is 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 studies use the Dijkstra algorithm [3] or variants of this algorithm to calculate the shortest routes to a destination with different types of energy metrics Unfortunately, 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 algorithm 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 networks in that transmission from a source node to a destination node causes neighboring nodes to dissipate energy 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 shows that the transmission from the source to the destination will be listened by other seven neighboring nodes 456 Mobile Netw Appl (2013) 18:454–463 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: Maximize : min(e(i)), ∀i ∈ n (11) where e(i) is the remaining energy of Node i after the path is established Unfortunately, Problem 11 is NP-complete, therefore there is no polynomial time algorithm to find the energy efficient path We will prove the NPcompleteness of Eq using graph models and a well known NP-complete problems Firstly, we give some preliminary results Fig 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 considered the reception energy usage in the selection of energy efficient paths In [9], an analytical model for optimal transmission range for minimizing the total energy consumption was presented Unfortunately, all of these papers are designed for mobile ad hoc networks but not sensor networks Unlike sensor networks, battery 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 operation time of practical sensor networks under the battery constraint of the sensors Formulating routing problem Original routing problem Given a network of n sensors, in which any sensor node can connect to all other sensor nodes by adjusting its transmission power Each 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 j or its power to pi 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 Maximize : min(e(i)), ∀i ∈ V (12) where e(i) is the remaining energy of Node i after the path is established Problem 12 can be converted to a decision problem: Decision problem Eq 13 A sensor network is modelled as G(V, E) Node i sets its power to zero or its j power to pi 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 Problem 13, 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 constant 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 Mobile Netw Appl (2013) 18:454–463 457 [12, 13] Details of the reduction are given in Appendix As a result, the simple constant power problem 14 is NP-complete and therefore, the original problem 11 is 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 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 calculated 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 dissipation 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) ( pi + r) i∈ (13) −d Figure shows the SP method, where Eij denotes to energy consumption to send data from Node i to Node j and E j denotes to energy consumption to receive data from at Node j (E j = unit, ∀ j ∈ V) Fig The SP method Algorithm Implementation of the SP algorithm Input: n: The number of sensor nodes indexed from to 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 = 2: for each round of data transmission 3: Pick up a random source node s and random 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: destination node d Using the Dijkstra algorithm to minimize f (l) from s to d for n from to N Update e(n) end for if e < then Stop else r =r+1 end if end for 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 = + + + + + = 6; The total energy consumption for path (1, 4, 5, 6) is: E(1,4,5,6) = E14 + E45 + E56 + E4 + E5 + E6 = + + + + + = 7; Fig Path calculation and selection from SP_N algorithm 458 Table Energy metrics of Fig Mobile Netw Appl (2013) 18:454–463 Energy usage Unit E12 E23 E36 E14 E45 E56 1 1 Therefore, the SP algorithm will select the path (1, 2, 3, 6) Node and Node are not involved in the path selection process The pseudo code for the algorithm is implemented in Algorithm Shortest path of the energy dissipation including neighboring 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 ( pi + 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) Figure shows 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 = + + + + + + + + + = 10; The total energy consumption for path (1, 4, 5, 6) is: E(1,4,5,6) = E14 + E45 + E56 + E4 + E5 + E6 = + + + + + = 7; Energy metrics of Fig are the same the energy metrics of Fig shown in Table Table Energy metrics of Fig Remaining energy SP E1 E2 E3 E4 E5 E6 E7 E8 1 1 1 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 W(i) = j∈N(i) e( j) (14) 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 shows path calculation and selection from the SP_RE algorithm, where E j denotes to remaining energy at Node j (Table 2) The total energy consumption for path (1, 2, 3, 6) is: E(1,2,3,6) = E12 + E13 + E16 + E14 + E17 + E17 + E18 = 0.5 + + + + + + = 6.5; The total energy consumption for path (1, 4, 5, 6) is: E(1,4,5,6) = E14 + E15 + E16 + = + + = 3; As a result, the SP_RE algorithm will select the path (1, 4, 5, 6) Simulation and comparison A number of simulators in C++ is developed to simulate 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: Et = kEelec + n amp kd (15) The reception energy is calculated by the equation as follows: Er = kEelec Fig Path calculation and selection from SP_RE algorithm (16) 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 Mobile Netw Appl (2013) 18:454–463 459 Fig Average energy dissipation per round (mJ) over 100 random 100-node networks settings for the simulations in the section are given below 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 ∗ 10−12 J/bit/m2 and mp = 0.0013 ∗ 10−12 J/bit/m4 Broadcast ID message: 16 bits Broadcast energy message: 32 bits Fig The first set of simulations Fig Number of rounds over 100 random 100-node networks In the first set of simulations, the lifetime performance 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 lifetime 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 (Table 3) It is also of interest to evaluate the total energy consumption of the routing methods Figure shows 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 Results for Fig Table Results for Fig Number of rounds Energy per round (mJ) Protocol Mean Variance 90 % confidence interval for the sample means SP 268.9 12.5 (267, 271) SP_N 363.3 45.5 (356, 371) SP_RE 519.3 30.7 (514, 524) Protocol Mean Variance 90 % confidence interval for the sample means SP 11.72 0.34 (11.66, 11.78) SP_N 7.79 0.28 (7.75, 7.84) SP_RE 6.68 0.28 (6.64, 6.74) 460 Mobile Netw Appl (2013) 18:454–463 Fig 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 (Table 4) 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 sourcedestination 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 routing 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 lifetimes until 50 nodes (50 % of nodes) fail are recorded The lifetime is the average lifetime of these topologies Figure shows 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 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 periodically and the interval between two subsequent data transmission in the networks is usually very long, while in mobile ad hoc networks, data is transmitted continuously 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 protocols 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 Elimination of reception energy by neighboring nodes As discussed in Section 1, 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 Fig Transmission from a source to a destination tells neighboring nodes to turn off their radio during the transmission Mobile Netw Appl (2013) 18:454–463 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 Section 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: A source node broadcasts a list of ID of nodes on the path to all sensors in the broadcast area 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 consumption 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 Figure 10 shows that new SP method performs much better than the original SP method (Table 5) 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 dissipation per round by the new SP_RE method (Table 6) This is because the original SP_RE already avoids the energy drain of neighboring nodes in the data transmission, so the broadcast process does not significantly help We not consider SP_N method as without the reception energy at neighboring nodes, SP_N is the same as SP method (Fig 13; Table 7) 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 461 Fig 10 Number of rounds over 100 random 100-node networks Table Results for Fig 10 Number of rounds Protocol Mean Variance 90 % confidence interval for the sample means SP 268.9 12.5 (267, 271) SP_100 557.8 39.0 (551, 564) SP_500 480.5 31.9 (475, 486) 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 462 Mobile Netw Appl (2013) 18:454–463 Table Results for Fig 12 Number of rounds Protocol Mean Variance 90 % confidence interval for the sample means SP 519.3 30.7 (514, 524) SP_RE100 680.3 55.1 (671, 690) SP_RE500 562.0 44.6 (555, 570) Conclusion Fig 13 Average energy dissipation per round (mJ) over 100 random 100-node networks Table Results for Fig 13 Energy per round (mJ) Protocol Mean Variance 90 % confidence interval for the sample means 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 SP 7.79 0.28 (7.75, 7.84) SP_RE100 5.73 0.24 (5.70, 5.78) SP_RE500 6.87 0.24 (6.84, 6.92) 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 heuristic solutions are required to achieve reasonable energy efficiency Three heuristic routing methods were implemented: (1) Dijsktra algorithm to minimize the total energy dissipation 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: Fig 14 Average number of rounds versus the number of surviving nodes Question: Consider a graph G(V, E), given a source node s and destination node d, and a collection C = {(a1 , b ), , (am, b m )} of pairs of vertices in V Find a simple path from s to d that contains at most one vertex from each pair in C Mobile 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 problem to the Path with Forbidden Pairs problem (PFP) Without loss of generality, we assume that for any node, a reception usage of one unit of energy (i.e., ri = 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 problem 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 V = V ∪ {v xy |(x, y) ∈ C} (17) E = E ∪ {(x, v xy), (y, v xy)|(x, y) ∈ C} (18) ei is the remaining energy set to: ei = c if i ∈ {v xy |(x, y) ∈ C&(x = t)||y = t} ei = c + if i ∈ {v xy|(x, y) ∈ C&(x = t)||y = t} ei = 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 ei = c + 1, or ei = c, if the destination 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 ei by at least Hence this path is also the path in G Conversely, given a solution path of the instance (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 Toh CK (2001) Maximum battery life routing to support ubiquitous mobile computing in wireless ad hoc networks IEEE Communications Magazine, June 2001 Safwat A et al (2002) Energy-aware routing in MANETs: analysis and enhancements In: 5th ACM international workshop on modeling analysis and simulation of wireless and mobile systems, pp 46–53 Dijkstra algorithm http://en.wikipedia.org/wiki/Dijkstra’s algorithm Accessed 23 Sept 2011 Ilyas M, Mahgoub I (2005) Mobile computing handbook CRC Press, Boca Raton, FL Feeney L, Nilsson M (2001) Investigating the energy consumption of a wireless network interface in an ad hoc networking environment In: IEEE INFOCOM Stemm M and 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The path metric is the sum of the link metrics The path with the smallest metric is selected from the possible routes Let cti be the battery capacity of node ni at the time t The battery cost of. .. maximize the operation time of sensor networks under battery limits This paper concentrates on multi-hop routing methods that prolong the operation time of practical sensor networks under the battery... 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