Construction of Power Efficient Routing Tree for Ad Hoc Wireless Networks using Directional Antenna

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1 Construction of Power Efficient Routing Tree for A Hoc Wireless Networks using Directional Antenna Qing Dai an Jie Wu Department of Computer Science an Engineering Floria Atlantic University Boca Raton, Floria {qai, jie}@cse.fau.eu Abstract In a hoc wireless networks, noes are typically powere by batteries. Therefore saving energy has become a very important objective, an ifferent algorithms were propose to achieve power efficiency uring the routing process. Directional antenna was suggeste to further ecrease transmission energy as well as to reuce interference. In this paper, we iscuss three algorithms for routing tree construction that take avantage of irectional antenna, i.e, Simple-Linear (), Linear-Insertion (), an Reverse-Cone-Pairwise (). Their performances are compare through a simulation stuy. 1 Keywors: a hoc wireless network, irectional antenna, energy-efficient routing, transmission power I. INTRODUCTION: BACKGROUND AND RELATED WORK Energy efficiency is an important consieration for a hoc wireless networks, where noes are typically powere by batteries. It is irectly correlate with network longevity an connectivity, therefore affecting network throughput. Among all ifferent components of power consumption, transmission cost appears to ominate, compare to receiving cost an computation cost. It has been shown that the power threshol p for a source noe to reach its estination noe is positively correlate to the istance between them, an can usually be expresse as p = r α, where is the istance between the two noes, an α is between 2 an 4. In previous stuies, ifferent metrics have been use, some measure the overall energy consumption of the network, while some others try to exten lifespan of iniviual noes. The Broacast Incremental Power algorithm (BIP) is a centralize algorithm attempting to minimize the overall energy in route etermination [1]. It is similar to Prim s Minimum Spanning Tree algorithm, in that at any time all reache noes form a single-roote tree. Each step as the noe with the minimum incremental cost, calculate as the aitional energy for it to be reache by any noe within the tree, either by increasing power of a transmitting noe, or by making a non-transmitting noe transmit at a specific power level. It has alreay been prove that BIP has a constant approximation ratio of between 6 an 12, compare to the optimal solution [4]. On the other han, 1 The work was supporte in part by a grant from Motorola Inc. an NSF grants CCR , ANI , an EIA Algorithm I: Reverse-Cone-Pairwise () 1 unreachenoes {all noes except sourcenoe} 3 egesingraph 4 totalcost 6 for j to size[reachenoes] 1 7 o fin minnoe i in unreachenoes with minimum incrementalcost(i, j) 8 reachenoes reachenoes minnoe 9 unreachenoes unreachenoes minnoe 1 egesingraph egesingraph {ege(i, j)} 11 totalcost totalcost + incrementalcost(i, j) Fig. 1. Algorithm I: Reverse-Cone-Pairwise some power-aware routing approaches select routes that avoi noes with low remaining power, which can be either absolute or relative power level. Different metrics may well lea to ifferent algorithms, but they can also be combine to achieve a balance, thus optimizing overall energy consumption without epleting crucial noes [5][2]. Recently, the use of irectional antenna was propose in orer to reuce interference an to further increase power efficiency, because ieally, power consume in case of irectional antenna is only r α θ 2 π, where θ is the beam angle. In one of the session-base stuies, two algorithms were propose: RB-BIP (Reuce Beam BIP) an D-BIP (Directional BIP). The former simply as a beamreucing step to the original BIP algorithm, so that each noe can now transmit at its smallest possible angle. The latter incorporates the use of irectional antenna at each step of the tree construction, i.e, a noe in the tree coul also increase its current transmission beam angle or shift the existing beam to reach a new estination noe, in aition to increasing its transmission power, whichever gives the lowest incremental cost [2]. In our stuy, we try to fin ifferent centralize routing

2 j Case 1 Case 3 j j Case 2 j Case 4 MinBeam (srcnoe, estnoe) 1 calculate estangle 2 insert estangle into anglelist 3 maxcone 4 minbeam (beamstart, beamen ) 5 for i to size[anglelist] 6 coneangle anglelist[i + 1] anglelist[i] 7 if coneangle > maxcone 8 maxcone coneangle 9 minbeam (anglelist[i + 1], anglelist[i]) 1 return minbeam Fig. 4. Reverse-Cone metho to fin the minimum beam. Fig. 2. Incremental transmission cost. power is etermine by subtracting the energy of the previous transmission beam from the new power, calculate as Fig. 3. θ 2 a b θ 1 c Illustration of Reverse-Cone metho. algorithms for broacasting with the use of irectional antenna. The paper is organize as follows: three algorithms are introuce in Section II. We then simulate their performance with ranomly generate networks in Section III, an iscuss their applications an potentials in Section IV. II. ALGORITHMS Our objective is to fin routing algorithms that are power efficient using irectional antenna. In this stuy, we assume one beam for each noe, an α = 2 to calculate the transmission energy. Here we propose three algorithms, assuming global knowlege of noe locations, ajustable transmission range an ajustable beam angle. The first algorithm is a refinement of D-BIP [2], where the noe with the minimum incremental cost is ae at each step, while taking the ajustable antenna beam with into consieration (see Figure 1). The actual beam angle that is use has to excee a minimum beam angle. All noes that have alreay been reache can act as transmitting noe, an all the non-tree noes are potential estination noes. Each possible transmitting-estination noe pair is examine to etermine the minimum power an beam angle neee to a a new estination noe. When a transmitting noe is aing a new estination noe, if the new noe oes not fall into its current transmission beam, it can either shift or expan its previous beam to cover the new noe. The incremental incrementalcost(i, j) = (p θ p θ)/(2 π), where p an θ represent the new transmission power an beam angle, respectively, while p an θ are the previous power an beam angle. This process is emonstrate in Figure 2, where i an j are the source an estination noes, respectively, an the cone represents the current transmission beam. In case 1, the estination noe coul be enclose within the current beam, incluing through a beam shift. It will be covere without any incremental cost. Otherwise, either the transmission beam has to be expane (case 2) or transmission power nees to be increase (case 3), or both (case 4) for to be reache. It coul therefore be very costly. During this process, it is esirable to use the minimum transmission beam angle whenever possible. A simple heuristic metho to calculate the new beam is to keep the start an en point of the previous beam, an to expan either en that leas to a smaller increase of the overall beam span. We will see that this heuristic oes not always provie the optimal beam angle. An example is illustrate in Figure 3. At first, the source noe a is transmitting to two ownstream neighbors b an c with a beam angle θ 1. When a new estination noe joins, an when θ 1 is close to π, neither expaning beam to b-c- nor to c-b- provies the minimum beam. On the other han, b--c is the optimal solution in this case with a beam angle θ 2. The Reverse-Cone metho (Figure 4) is evelope to calculate the minimum beam span. Instea of merely keeping the start an en point of the previous beam, each noe keeps angle positions, calculate as the raius of a estination noe from itself, of all its estination noes in a sorte list anglelist. The 2π circle aroun a transmitting noe is then ivie into several cones, each efine by the two ajacent neighbors in the anglelist. Whenever aing a new estination noe, the transmitting noe will first insert the angle location of the new noe into its anglelist. The anglelist is traverse to fin the largest cone. The minimum new beam to cover all the noes in anglelist woul be the reverse of this largest cone. For the previous example in Figure 3, noe a first as noes b an c in its sorte anglelist. Of the two cones, c b counterclockwise is larger than b c, therefore the reverse of cone c b, i.e, cone b c, or θ 1, is selecte as the minimum

3 Algorithm II: Simple Linear () 1 unreachenoes {all noes except sourcenoe} 3 listhea sourcenoe 4 totalcost 6 for i 1 to size[unreachenoes] 1 7 o fin minnoe with minimum cost mincost 8 reachenoes reachenoes minnoe 9 listhea minnoe 1 unreachenoes unreachenoes minnoe 11 totalcost totalcost + mincost Fig. 5. Algorithm II: Simple Linear beam. When noe a as a new estination, is then inserte into a s angle list, which now becomes b c. After traversal of the new anglelist, b c is foun to be the largest cone, therefore the reverse of it, c b, or θ 2 will be use as the new transmission beam. The algorithm generally provies goo performance, except in the situations when the previous transmission beam was small, aing a new noe at a istant angle position can be very expensive. Two other heuristic algorithms are therefore evelope: Simple-Linear () an Linear-Insertion (). In both of them, each transmitting noe always uses the minimum angle to reach exactly one ownstream noe, an there is only one ownstream estination noe for each transmitting noe. As a result, a linear chain will be forme step by step, starting from the source noe. Initially, the reachen oes set inclues sourcen oe only, an the unreachenoes set inclues all other noes. In (see Figure 5), initially the source noe is the listhea, then at each step, a minnoe is etermine as an unreache noe closest to the current listhea, an is ae to the reachenoes set to become the new listhea. This listhea is the only possible transmitting noe to reach the next new noe, until the unreachen oes set is empty. In (Figure 6), an aitional backtrack step is inclue when each noe is ae to the reachen oes set. During the backtrack, after the minn oe is etermine, it is first teste for possible insertions into each position between any two ajacent noes within the existing linear chain. The insertioncost, the incremental cost for inserting the new noe, is calculate to check whether any insertion causes a saving of energy, compare to irectly aing the minnoe as the new listhea. If so, an insertion will take place where the insertioncost is the minimum. In this case, the previous listhea remains unchange. If not, minnoe woul be attache as the new listhea. In the example in Figure 7, s represents the sourcenoe, 1 through 6 are the estination noes to be reache. The Linear- Insertion algorithm woul a 6 between s an 1 in the backtrack process, which leas to a lower overall energy cost. While seems to be a better solution than, unfortunately, it oes not always outperforms ue to its heuristic nature (examples not shown). Algorithm III: Linear-Insertion () 1 unreachenoes {all noes except sourcenoe} 3 listhea sourcenoe 4 totalcost 6 for i to size[unreachenoes] 1 7 o fin minnoe with minimun cost mincost 8 for j to size[reachenoes] 1 9 o insert minnoe after reachenoe[j] 1 fin minimum InsertCost 11 if InsertCost < mincost 12 o mincost mininsertcost 13 insert minnoe 14 else 15 o listhea minnoe 16 reachenoes reachenoes minnoe 17 unreachenoes unreachenoes minnoe 18 totalcost totalcost + mincost 6 Fig. 6. Algorithm III: Linear Insertion Simple-Linear 2 s s 1 2 Linear-Insertion Fig Illustration of Linear-Insertion. 5 5 The complexity of the three algorithms are as follows: SI an have the similar complexity level of θ(n 2 ). has a higher complexity of θ(n 4 ), with the Reverse-Cone metho. To evaluate the relative performance of the three algorithms, simulations are performe on ranom networks. The result is shown in Section III. III. SIMULATION RESULT AND CONCLUSION We generate ranom network instances, an the three algorithms are use to obtain routing graphs, respectively. Their transmission costs are calculate an compare with each other. Our results show that in most occasions, prouces routing solutions with goo energy-efficiency. Between an

4 2 Simulation result: noes = = 1/12 PI Simulation result: noes = Simulation result: noes = (c) Fig. 8. Simulation results with from π/9 to π/4. Number of noes is 1, 3 an 1 (c), respectively, in a fixe area., has a better performance. In simulation with ifferent network ensity, we observe that when noe istribution is sparse, outperforms when the is relatively small. In the example shown in Figure 8, we set the number of noes from 1 to 1 in the same size of area. As the increases graually, we can see that energy costs of an routing increase with the linearly. This is because in SI an algorithms, the route construction process is not affecte by the, an the same routing graph will always be prouce just as omniirectional antenna is provie. The final cost is simply calculate by multiplying the original overall cost by a factor of /(2π). The result of is close to SI when the is small, = 1/6 PI = 1/4 PI (c) = 1/2 PI () Fig. 9. Simulation results with from 1 to 1 in a fixe area. is π/12, π/6, π/4 (c), an π/2 (), respectively.

5 Simulation result: noes = = 1/9 PI D-BIP 6 D-BIP Fig. 1. Simulation results of,, an compare with D-BIP. from π/18 to π/4, with 3 noes in a fixe area, an number of noes from 1 to 1 in a fixe area, with π/9. which makes sense since in this case woul most likely generate the same routing graph as. When increases to a certain point, will outperform, an eventually outperform, when shifting or expaning current beam shows avantage. Simulations are also performe to measure the overall cost over the. In a fixe area, the costs of an increment along with the increase of noes, but the cost of ecreases, except when the noe number an are very small (Figure 9). It seems that in most cases, has the best approximation ratio among the three. To compare the performance of our three algorithms, especially, with that of D-BIP, a simplifie version of D- BIP is implemente. Same as before, we consier only the transmission cost, which is a function of the istance. As expecte, shows better energy efficiency than D-BIP (Figure 1). Nevertheless, all three algorithms escribe above are greey algorithms an heuristic in nature, with none being optimal. In fact, we can fin special network instances between SI, or, where one outperforms the other two(examples not shown). From the simulation result, it seems that when noes are relatively sparse an is small, is the best choice. Not only oes it give the lowest routing cost, but its complexity is lower than also. When the increases to a certain extent or when network is ense, expaning current beam to inclue aitional estination noes woul be more cost efficient, an at this point, provies the best performance. Currently, we are further analyzing their performance statistically. In the future, it will be interesting to know the approximation ratio of the three algorithms. REFERENCES [1] J. E. Wieselthier, G. D. Nguyen, A. Ephremies, On the construction of energy-efficient broacast an multicast trees in wireless networks, Proceeings of IEEE INFOCOM 2, p [2] J. E. Wieselthier, G. D. Nguyen, A. Ephremies, Energy-limite wireless networking with irectional antennas: the case of session-base multicasting, IEEE INFOCOM 22, p [3] L. Li, J. Y. Halpern, P. Bahl, Y. -M. Wang, R. Wattenhofer, Analysis of a cone-base istribute topology-control algorithm for wireless multi-hop networks,proceeings of ACM Principles of Distruibute Computing, p , 21. [4] P. -J. Wan, G. Calinescu, S. -Y. Li, O. Frieer, Minimum-energy broacasting in static A Hoc wireless networks, Wireless Networks, Vol. 8, p67-617, 22. [5] J. Wu, F. Dai, M. Gao, I. Stojmenovic, On calculating power-aware connecte ominating set for efficient routing in a hoc wireless networks, IEEE/KICS Journal of Communications an Networks, Vol. 4, p59-7, March 22.