IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 9, SEPTEMBER 2006 2563

Joint Optimization of Energy Consumption andAntenna Orientation for Multicasting in

Static Ad Hoc Wireless NetworksSong Guo, Member, IEEE, Oliver Yang, Senior Member, IEEE, and Victor Leung, Fellow, IEEE

Abstract— To explore the energy saving advantage offered bythe use of directional antennas, we consider the case of sourceinitiated multicast traffic in static ad hoc wireless networks thatuse switched antennas and have limited energy resources. Wepresent a constraint formulation for the joint optimization prob-lem MEM-AO (Minimum-Energy Multicast and Antenna Orien-tation) in terms of MILP (Mixed Integer Linear Programming).The optimal solution can be obtained using an MILP solver ina timely manner for moderately sized networks with switchedantennas. In addition to the theoretical effort, we also providetwo heuristic algorithms and a general post-process operationTR (Tree Reconstruction) for handling larger networks. Theexperimental results show that our TR operation significantlyimproves the performance of both heuristic algorithms.

Index Terms— Wireless communication network, ad hoc net-work, directional antenna, multicast, energy consumption opti-mization.

I. INTRODUCTION

W IRELESS ad hoc (and sensor) networks are expectedto be deployed in a wide variety of civil and military

applications. Energy conservation is a critical issue in buildingsuch networks when all nodes are equipped with a finite andnon-renewable amount of energy. One major metric for energyconservation is to route a communication session along theroutes which require the lowest total energy consumption.There have been many recent papers addressing the mini-mum energy routing problem for broadcast (e.g. [1]–[3]) ormulticast (e.g. [4]–[6]). Recent use of directional antennas inwireless communication has further enabled new approachesto energy saving for energy-constrained wireless networks [7],[8]. Indeed, use of directional antennas allows concentration ofthe beam toward the intended destination without wasting en-ergy in unwanted directions. In [7], Wieselthier et al. extendedthe results in [4] and made a major step in the systematicstudy of the multicast routing problem for energy-constrainedwireless ad hoc networks employing directional antennas.Although their D-MIP algorithm offers good performance insolving the minimum energy multicast routing problem, there

Manuscript received July 22, 2004; revised April 11, 2005; acceptedDecember 21, 2005. The associate editor coordinating the review of this paperand approving it for publication was S. Sarkar.

S. Guo and V. Leung are with the Department of Electrical and ComputerEngineering, University British Columbia, Vancouver, British Columbia, V6T1Z4, Canada (email: {sguo, vleung}@ece.ubc.ca).

O. Yang is with the School of Information Technology and Engineering,University of Ottawa, 161 Louis Pasteur, Ottawa, Ontario, K1N 6N5, Canada(email: yang@site.uottawa.ca).

Digital Object Identifier 10.1109/TWC.2006.04499.

is a very subtle detail in the algorithmic design of D-MIP thatmotivates us to further investigate this important problem.

In this paper, we consider the important problem of multi-cast routing with the objective of minimizing energy consump-tion for wireless ad hoc networks that use switched antennasand have limited energy resources. The contributions includeboth theoretical problem formulation and heuristic algorithmdesign for the multicast routing problem. From a theoreticalperspective, we formulate a joint optimization problem, MEM-AO (Minimum-Energy Multicast and Antenna Orientation), asa mixed integer linear programming problem. In particular,the antenna orientation is formulated as a continuous variableinstead of a finite number of possible values. The advantageis that the complexity of the MILP model is reduced and thusmight increase the speed to solve the problem. We also providetwo heuristic algorithms and a general post-process operationTR (Tree Reconstruction) for handling larger networks. Theexperimental results show that our TR operation significantlyimproves the performance of both heuristic algorithms.

II. SYSTEM MODEL

We have considered a wireless ad hoc network consisting ofa specified number of nodes located over a two-dimensionalplane. For wireless communication, each node is equippedwith a switched antenna, whose beamwidth is fixed and whoseorientation can be shifted to any desired direction. Based ona similar antenna propagation model as in [7], the RF powerpvu needed by node v to reach node u using a beamwidth θvis therefore

pvu =qu · rαvu · θv

2π(1)

where α is the path loss exponent, rvu is the distance betweennode v and node u, and qu represents the signal detectionpower threshold at node u. In such a network, each node’stransmitter has power control capability. That is, by adjustingthe transmission power level to not exceed some maximumvalue pmaxv , the sender v can reach destination nodes locatedat different distances. Furthermore, each node can also controlthe beam orientation of its directional antenna.

Let us model the network by a simple directed graphG(N,A) in which the node locations are fixed, and thechannel conditions unchanging. The directed graph G has afinite node set N with |N | = n nodes and an arc set Acorresponding to the unidirectional wireless communicationlinks. Any directed arc (v, u) ∈ A only if pvu ≤ pmaxv . Our

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2564 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 9, SEPTEMBER 2006

focus in this paper is on the “logical” problem of establishingminimum-energy trees, rather than on the development ofpractical protocols for full-fledged multicast implementation.Thus, we do not address medium-access issues or the con-straints of insufficient equipment or bandwidth resources. Toassess the complex trade-offs one at a time, we assume thatthere is no mobility, even though the handling of mobilityis one of the eventual goals of the approach outlined in thispaper.

III. MILP FORMULATION OF PROBLEM MEM-AO

A feasible multicast tree requires that any tree node couldgenerate and steer a single beam to reach all its children inthe tree. In order to formulate the MEM-AO problem as anMILP model, we shall provide several theorems in this sectionto systematically construct the linear constraints required for afeasible multicast tree with the objective to minimize the totalRF power expenditure. The proof details can be found in [8].

A. Multicast Tree Constraints

A multicast session is supported by a set of multicastmembers M , including the source node s and all destinationnodes. All the nodes involved in the multicast form a multicasttree Ts rooted at the source s. We want to provide a set oflinear constraints such that a sub-graph T ∗

s obtained fromthe formulation is a multicast tree with minimum energyconsumption. The following variables are used: (i) Zvu is abinary decision variable which is equal to one if the arc (v, u)is in the sub-graph T ∗

s of G, and zero otherwise, and (ii) Fvuis a continuous variable that only represents fictitious flowproduced by the multicast initiator s going through arc (v, u),and thus helps prevent loops.

The extracted sub-graph T ∗s is a multicast tree if and only

if the following properties are all satisfied: (a) every tree nodeexcept the source s has exactly one incoming arc, and nodes has no incoming arcs, (b) T ∗

s does not contain cycles, and(c) T ∗

s spans all the multicast members. Theorem 1 belowachieves that if (x)∗ is the optimal solution of variable xobtained from our MILP model, then the graph T ∗

s is theoptimal multicast tree associated with this solution. In thisgraph, A(T ∗

s ) = {(v, u)|Z∗vu = 1} is its arc set, and N(T ∗

s ) ={u|∃(v, u) ∈ A(T ∗

s )or(u, v) ∈ A(T ∗s )} is its node set.

Theorem 1: T ∗s is a multicast tree rooted at node s, if

Problem MEM-AO satisfies Constraints (2-7).∑v∈N

Zvs = 0; (2)

∑v∈N

Zvu = 1; ∀u ∈M \ {s} (3)

∑v∈N

Zvu ≤ 1; ∀u ∈ N \M (4)

∑v∈N

Zuv ≤ (n− 1)∑v∈N

Zvu; ∀u ∈ N \M (5)

∑v∈N

Fvu −∑v∈N

Fuv =∑v∈N

Zvu; ∀u ∈ N \ {s} (6)

Zvu ≤ Fvu ≤ (n− 1)Zvu; ∀u ∈ N \ {s}, v ∈ N (7)

B. Antenna Orientation Constraints

Let dv(0 ≤ dv < 2π) be the antenna orientation at nodev and αvu(0 ≤ αvu < 2π) be the angle measured counter-clockwise from the horizontal axis to the vector −→vu, whichis pointing from node v to node u. Thus it is clear that thewireless link (v, u) really exists if and only if the orientationdv is bounded by the two pointing directions αvu− θv/2 andαvu + θv/2. In order to formulate this set of constraints, wedefine {cvu : (v, u) ∈ A} to be binary variables such thatcvu = 1 if node u is covered by the antenna beam of node v,and 0 otherwise.

We first assume 0 ≤ αvu < θv/2. Then according to thedefinition of cvu, the range [αvu−θv/2, αvu+θv/2] of dv mustbe mapped into [0, αvu+ θv/2]∪ [2π+αvu− θv/2, 2π], sincewe restrict variable dv between 0 and 2π . Therefore, cvu = 1if and only if dv ∈ [0, αvu+ θv/2]∪ [2π+αvu− θv/2, 2π) asexpressed in Constraint (8a). Similarly, we have Constraints(8b) and (8c) for dv and cvu corresponding to the value ofαvu within different ranges.

Case 1: 0 ≤ αvu <θv

2

cvu =

⎧⎪⎨⎪⎩

1 0 ≤ dv ≤ αvu + θv/2 or

2π + αvu − θv/2 ≤ dv < 2π0 otherwise

(8a)

Case 2: θv

2 ≤ αvu < 2π − θv

2

cvu =

{1 αvu − θv/2 ≤ dv ≤ αvu + θv/20 otherwise

(8b)

Case 3: 2π − θv

2 ≤ αvu < 2π

cvu =

⎧⎪⎨⎪⎩

1 αvu − θv/2 ≤ dv < 2π or

0 ≤ dv ≤ αvu + θv/2 − 2π0 otherwise

(8c)

The above Constraints (8a) - (8c) are obviously nonlinear. Inthe following, we only illustrate how the constraint in Case 1can be linearized. Other cases can be done in a similar mannerand the details are omitted. We observe that cvu describedby Equation (8a) can be decomposed into a summation oftwo new binary variables c1vu and c2vu , which are defined inEquations (9) and (10).

c1vu =

{1 0 ≤ dv ≤ αvu + θv/20 αvu + θv/2 < dv < 2π

(9)

c2vu =

{1 2π + αvu − θv/2 ≤ dv < 2π0 0 ≤ dv < 2π + αvu − θv/2

(10)

Equations (9) and (10) are depicted by thick lines in thedv − c1vu plane and the dv − c2vu plane as shown in Fig. 1(a)and 1(b) respectively. In the dv−c1vu plane, the points (dv, c1vu)that satisfy Equation (9) must be within the shaded areabetween line P1P2 and line P3P4, where P1 = (0, 1/2), P2 =(αvu+θv/2, 0), P3 = (2π, 0), and P4 = (αvu+θv/2, 1). Since0 ≤ dv < 2π and c1vu ∈ {0, 1}, Equation (9) and Constraints(12-13) define the same point set in the (dv, c1vu) plane.Similarly, Equation (10) can be rewritten in Constraints (14)and (15) with the help of Fig. 1(b). In summary, the nonlinear

GUO et al.: JOINT OPTIMIZATION OF ENERGY CONSUMPTION AND ANTENNA ORIENTATION FOR MULTICASTING 2565

(a) the dv − c1vu plane (b) the dv − c2vu plane

Fig. 1. Illustration of constraint linearization (0 ≤ αvu < θv/2).

Constraint (8a) is linearized using Constraints (11) to (15).Finally, the connection between multicast tree constraints andantenna orientation constraints is given in Theorem 2.

cvu = c1vu + c2vu (11)

c1vu > −dv/(2αvu + θv) + 1/2 (12)

c1vu ≤ −2dv/(4π − 2αvu − θv)+4π/(4π − 2αvu − θv) (13)

c2vu > 2dv/(θv − 2αvu)−(4π + 2αvu − θv)/(θv − 2αvu) (14)

c1vu ≤ 2dv/(4π + 2αvu − θv) (15)

Theorem 2: T ∗s satisfies orientation requirement, if Problem

MEM-AO includes Constraint (16).

Zvu ≤ cvu; ∀v, u ∈ N (16)

C. MILP Model

Let pv be a continuous variable that represents the trans-mission power of node v required by the multicast treeT ∗s . The objective is therefore to minimize

∑v∈N pv. Recall

the WMA (wireless multicast advantage) property that thepower required at node v is the maximum of the individualtransmission power to each neighbour from v. It allows us toobtain the following constraint.

pv = max(v,u)∈A(T∗

s ){pvu, 0} (17)

The “max” form in Constraint (17) appears nonlinear ob-viously. The following theorem illustrates how this constraintcan be linearized.

Theorem 3: The tree energy is minimized, if ProblemMEM-AO includes Constraint (18).

pmaxv ≥ pv ≥ qu · rαvu · θv

2π· Zvu; ∀v, u ∈ N (18)

Our previous derivation on the linear constraints can nowhelp us to rewrite the problem formulation as a MILP modelgiven in Table I, in which coefficients Aivu, Bivu, Civu, Di

vu,and Evu(i = 1 or 2, (v, u) ∈ A) are listed in Table I. In thisformulation, the number of variables is approximately 4n2 +2n, and the number of constraints is of the order of O(n2). Wealso note that the coefficient matrix of constraints is sparse. It

has only O(n2) nonzero entries, far fewer than the total O(n4)entries. This property would increase the speed to solve theMILP problem.

Finally in order to prove the correctness of our formulation,it remains to show that every feasible multicast tree can beexpressed by the variables defined in Problem MEM-AO. Thisis achieved by the following theorem.

Theorem 4: The formulation in Table I solves the MEM-AO problem.

IV. HEURISTIC ALGORITHMS

Since the minimum-energy broadcast problem in networkswith omni-directional antenna is NP-complete [1], the moregeneral multicast problem in networks with directional anten-nas appears to be at least as difficult. The amount of time re-quired to solve Problem MEM-AO based on the MILP modelmay be excessive for significantly large networks. In orderto handle such networks, we first present two polynomial-time heuristic algorithms, BS-MST (Beam-Shifting MST) andBS-MIP (Beam-Shifting Multicast Incremental Power). TheBS-MST algorithm is based on the use of a revised MSTalgorithm in which the new node selected to be added into thetree must satisfy the antenna orientation requirement. Similarto D-MIP [7], our BS-MIP algorithm explores the wirelessmulticast advantage in the formation of the tree, in whichthe incremental cost of adding a new node involves simplyincreasing transmission range, shifting the antenna beam, ora combination of these two operations. As variant versionsof the standard Prim’s algorithm, both BS-MST and BS-MIPhave the same time complexity O(n3) using a straightforwardimplementation. For BS-MST, a more sophisticated imple-mentation using a Fibonacci heap yields O(n2) complexity.However, it is not yet clear whether the Fibonacci heaptechnique is applicable for BS-MIP because of the need toupdate the costs at each step of the algorithm.

We now present a general post-process operation calledTree Reconstruction (TR) to refine the results obtained by anyheuristic algorithm. It iteratively reconstructs the broadcasttree based on the maximal decremental power of the corre-sponding multicast tree by switching a tree arc with a non-tree arc until no more power saving can be obtained. Finally, itprunes the broadcast tree to be a multicast tree. One distinctivefeature of our TR is that the pruning operation is embedded

2566 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 9, SEPTEMBER 2006

TABLE I

MILP MODEL FOR PROBLEM MEM-AO

Object Function: Minimize�

v∈N pv (19)

Subject to:

Constraints (2)-(7), and (18);

civu > Ai

vu · dv + Bivu;∀v, u ∈ N, i = 1, 2 (20)

civu ≤ Ci

vu · dv + Divu;∀v, u ∈ N, i = 1, 2 (21)

Zvu ≤ c1vu + c2vu + Evu; ∀v, u ∈ N (22)

0 ≤ dv < 2π,∀u ∈ N (23)

Zvu ∈ {0, 1}; c1vu ∈ {0, 1}, c2vu ∈ {0, 1}, ∀v, u ∈ N (24)

TABLE II

VALUES OF COEFFICIENTS IN MILP FORMULATION OF PROBLEM MEM-AO

0 ≤ αvu < θv/2 θv/2 ≤ 2π − θv/2 2π − θv/2 ≤ αvu < 2π

A1vu −1/(2αvu + θv) 2/(4π − 2αvu + θv) −1/(2αvu + θv − 4π)

A2vu 2(θv − 2αvu) −2(2αvu + θv) 2(4π + θv − 2αvu)

B1vu 1/2 −(2αvu − θv)/(4π − 2αvu + θv) 1/2

B2vu −(4π + 2αvu − θv)/(θv − 2αvu) 1/2 −(2αvu − θv)/(4π + θv − 2αvu)

C1vu −2/(4π − 2αvu − θv) 2/(2αvu − θv) −2/(8π − 2αvu − θv)

C2vu 2/(4π + 2αvu − θv) −2/(4π − 2αvu − θv) 2/(2αvu − θv)

D1vu 4π/(4π − 2αvu − θv) 0 4π/(8π − 2αvu − θv)

D2vu 0 4π/(4π − 2αvu − θv) 0

Evu 0 −1 0

in the iterative broadcast tree reconstruction (in order tocalculate the tree power of the corresponding multicast tree)instead of only being performed at the last step. In this way,the refinement may proceed along the direction of possiblymaximizing the total power saving using global topologyinformation. The time complexity of the TR operation can beexplained as follows. In each iteration of tree updates, there areat most O(n2) number of possible arc switches, resulting inat most O(n2) number of pruning operations to be performed,and each pruning operation has complexity O(n). Thereforethe complexity of the TR operation is O(Kn3), where K is thetotal iteration number. In our experiment in the next section, Knever exceeded a small number even for a large-scale network.

V. PERFORMANCE EVALUATION

We have evaluated the performance of a set of algorithmsI = {BS-MST, BS-MST-TR, BS-MIP, BS-MIP-TR} for manynetwork examples, where X-TR indicates the algorithm thatperforms X followed by a TR operation. Networks with aspecified number of nodes are generated within a square re-gion (1000 meters×1000 meters). The maximum transmissionpower is restricted by the maximum radio propagation rangeof 300 meters. One of the nodes is randomly chosen to bethe source. Multicast groups of a specified size m = |M |are chosen randomly from the overall set of nodes. Eachantenna can point to any desired direction with a specifiedantenna beamwidth. We have only considered propagationloss exponents of α = 2 and the signal detection powerthreshold of qv = 1 for any node v. In all cases, (i.e., fora specified network size, a multicast group size, an antenna

beamwidth, and an algorithm i ∈ I), our results are based onthe performance of 100 randomly generated networks.

Our performance metric is the total power of the multicasttree. To facilitate the comparison of our algorithms over a widerange of network examples, we use the notion of normalizedtree power [4] for each network example, defined as the ratioof actual total power required using one heuristic algorithmto the best solutions. For our small (20-node network) exam-ples, the best solution is the optimal solution, which can beefficiently obtained based on our MILP model in Table I. Forlarge (100-node network) examples, the best solution is theminimum total power obtained from all heuristic algorithms.

Table III summarizes performance of the algorithms for 20-node networks with various multicast group sizes and variousbeam widths. We list mean and variance of the normalized treepower as (mean, variance) for each algorithm in the table. Asnoted above, the normalization is taken with respect to the op-timal solution. We solved the MILP model using the CPLEX[10] software package on an MS WIN2000 workstation witha PIII 800-MHz processor and 128 MB memory, and the usertime is several seconds on each 20-node network example.Some key observations are summarized as follows. (1) For allcases, BS-MIP performs better than BS-MST, both in terms ofmean and variance. (2) The TR operation can provide muchbetter performance for both algorithms. (3) BS-MIP-TR hasthe best performance over all other algorithms. In particular,a larger multicast group can averagely achieve a much betterperformance when the antenna beam-width is relatively small( θv < 180◦ ), but a slightly worse performance when theantenna beam-width is relatively large ( θv > 180◦ ). (4) BS-

GUO et al.: JOINT OPTIMIZATION OF ENERGY CONSUMPTION AND ANTENNA ORIENTATION FOR MULTICASTING 2567

TABLE III

PERFORMANCE OF 20-NODE NETWORKS

|M| BS-MST BS-MST-TR BS-MIP BS-MIP-TR BS-MST BS-MST-TR BS-MIP BS-MIP-TR

θv = 45◦ θv = 90◦

5 (2.19, 2.27) (1.66, 0.86) (1.63, 0.72) (1.54, 0.68) (1.61, 0.19) (1.28, 0.12) (1.30, 0.19) (1.25, 0.16)

10 (1.65, 0.21) (1.38, 0.12) (1.49, 0.13) (1.35, 0.12) (1.52, 0.12) (1.22, 0.08) (1.26, 0.08) (1.18, 0.05)

15 (1.56, 0.10) (1.23, 0.09) (1.24, 0.08) (1.20, 0.06) (1.64, 0.09) (1.13, 0.07) (1.19, 0.05) (1.12, 0.03)

20 (1.64, 0.06) (1.17, 0.03) (1.16, 0.03) (1.13, 0.01) (1.42, 0.06) (1.10, 0.03) (1.12, 0.02) (1.09, 0.01)

θv = 180◦ θv = 360◦

5 (1.18, 0.08) (1.07, 0.02) (1.13, 0.05) (1.04, 0.01) (1.15, 0.07) (1.05, 0.02) (1.12, 0.05) (1.01, 0.02)

10 (1.19, 0.05) (1.07, 0.02) (1.14, 0.03) (1.06, 0.01) (1.17, 0.06) (1.05, 0.02) (1.15, 0.03) (1.05, 0.02)

15 (1.17, 0.03) (1.08, 0.02) (1.14, 0.02) (1.07, 0.01) (1.18, 0.02) (1.06, 0.02) (1.16, 0.03) (1.07, 0.01)

20 (1.17, 0.03) (1.09, 0.01) (1.15, 0.01) (1.07, 0.01) (1.17, 0.02) (1.06, 0.01) (1.15, 0.02) (1.09, 0.01)

TABLE IV

PERFORMANCE OF 100-NODE NETWORKS

|M| BS-MST BS-MST-TR BS-MIP BS-MIP-TR BS-MST BS-MST-TR BS-MIP BS-MIP-TR

θv = 45◦ θv = 90◦

5 (1.46, 0.046) (1.07, 0.004) (1.38, 0.050) (1.04, 0.014) (1.19, 0.010) (1.03, 0.001) (1.15, 0.012) (1.01, 0.001)

25 (1.53, 0.027) (1.07, 0.002) (1.22, 0.009) (1.03, 0.003) (1.17, 0.003) (1.03, 0.001) (1.13, 0.003) (1.02, 0.001)

50 (1.64, 0.034) (1.04, 0.002) (1.12, 0.003) (1.02, 0.002) (1.14, 0.003) (1.02, 0.000) (1.09, 0.001) (1.01, 0.000)

100 (1.69, 0.027) (1.04, 0.002) (1.07, 0.001) (1.01, 0.001) (1.14, 0.003) (1.02, 0.001) (1.05, 0.001) (1.01, 0.000)

θv = 180◦ θv = 360◦

5 (1.21, 0.010) (1.01, 0.002) (1.17, 0.008) (1.00, 0.000) (1.21, 0.008) (1.01, 0.001) (1.18, 0.008) (1.00, 0.000)

25 (1.19, 0.008) (1.00, 0.000) (1.15, 0.005) (1.00, 0.000) (1.16, 0.005) (1.01, 0.000) (1.14, 0.004) (1.00, 0.000)

50 (1.15, 0.005) (1.00, 0.000) (1.11, 0.002) (1.00, 0.000) (1.14, 0.004) (1.00, 0.000) (1.11, 0.002) (1.00, 0.000)

100 (1.16, 0.002) (1.00, 0.000) (1.09, 0.001) (1.00, 0.000) (1.15, 0.001) (1.00, 0.000) (1.11, 0.001) (1.00, 0.000)

MIP-TR performs very closely to the optimal solution for asmall multicast group and a large antenna beam-width, e.g.within 1% close to the optimal solution when m = 5 andθv = 360◦.

Performance results for 100-node networks are shown inTable IV. For any specified antenna beamwidth, BS-MST-TRand BS-MIP-TR significantly improve performance over BS-MST and BS-MIP respectively. We attribute this improvedperformance to the distinctive features of our TR operationusing the global topology information to reconstruct multicasttree dynamically instead of in an “expansion step” manner,e.g. the tree formation in [4], and minimizing the total energyconsumption of the final multicast tree instead of a broadcasttree. Finally, in order to investigate the time-complexity of TR,we also examine the parameter K for each set of experiments.We observed that it never exceeded 4 for all 20-node networkexamples and 6 for all 100-node network examples.

From the observations of experimental results for both smalland large networks, we have discovered that BS-MST-TRand BS-MIP-TR have very similar and good performance.Considering the implementation of distributed version andcomplexity, we believe that BS-MST-TR would be a betterchoice since the initial broadcast tree construction would usethe distributed algorithm in [9] with minor changes: only thefragment including the source is allowed to enlarge its size byabsorbing new node that must satisfy the antenna orientationrequirement. For a fully distributed algorithm, we are nowexploring the distributed implementation of the TR operation.

VI. CONCLUSION

In this paper we have presented an MILP formulation forthe minimum-energy multicast and antenna orientation opti-mization problem in wireless ad hoc networks. We have alsopresented a group of heuristic algorithms for handling largernetworks. Our initial results give rise to a number of morechallenging issues for further research. First, a major challengeis to extend our analytical model to networks with highlydirectional antennas [8] like adaptive antennas. Moreover, itis worth exploring the utility of our MILP formulation. Anear optimal solution may be obtained efficiently using convexoptimization. Second, applying a more realistic antenna model(not the perfect “pie-slice” directional antennas with no sidelobes) would induce nonlinear constraints in our formulationand we are examining the feasibility of using nonlinear inte-ger programming solvers for the MEM-AO problem. Third,building an efficient MAC layer is very important for takingadvantage of the energy savings of both WMA property anddirectional antennas. Finally, more localized algorithm designshould be explored since it is more essential for applications inmobile networks. A possible direction is to extend the designphilosophy of our tree refinement operation in a localizedfashion.

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[10] CPLEX, http://www.cplex.com

Song Guo is an NSERC (Natural Sciences andEngineering Research Council of Canada) Post-doc Fellow in the Department of Electrical andComputer Engineering, the University of BritishColumbia, Canada. He received his Ph.D. degree inComputer Science from the University of Ottawa,Ont., Canada in 2005. His recent research focuses onthe modeling, analysis and performance evaluationof low-power wireless networking systems.

Oliver Yang is a Professor in the School of Infor-mation Technology and Engineering at Universityof Ottawa, Ontario, Canada. Dr. Yang received hisPh.D. degree in Electrical Engineering from the Uni-versity of Waterloo, Ont., Canada in 1988. He hasworked for Northern Telecom Canada Ltd. and hasdone various consulting. His research interests arein modeling, analysis and performance evaluationof computer communication networks, their proto-cols, services and interconnection architectures. TheCCNR Lab under his leadership has been working

on various projects in the traffic control, traffic characterization, switcharchitecture and traffic engineering issues in both wireless and photonicnetworks. This has been published in more than 150 technical papers. Dr. Yangis also interested in queuing theory, simulations, computational algorithms andtheir applications such as reliability and traffic analysis. Dr. Yang is currentlythe editor of IEEE Communication Magazine.

Victor C. M. Leung received the B.A.Sc. and Ph.Ddegrees in electrical engineering from the Universityof British Columbia (UBC) in 1977 and 1981, re-spectively, where he was awarded the APEBC GoldMedal and the NSERC Postgraduate Scholarship.After working in MPR Teltech Ltd. and the ChineseUniversity of Hong Kong, he returned to UBC asa faculty member in 1989, where he is currentlya Professor, and holder of the TELUS MobilityResearch Chair in the Department of Electrical andComputer Engineering. His research interests are

in the areas of architectural and protocol design and performance analysisfor computer and telecommunication networks, with applications in satellite,mobile, personal communications, and high-speed networks. Dr. Leung is aFellow of IEEE, a voting member of ACM, and an Editor of the IEEE Trans-actions on Wireless Communications and IEEE Transactions on VehicularTechnology.