Antenna Orientation Optimization for Minimum-Energy Multicast Tree Construction in Wireless Ad Hoc Networks with Directional Antennas

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1 Antenna Orientation Optimization for Minimm-Energy Mlticast Tree Constrction in Wireless Ad Hoc Networks with Directional Antennas Song Go and Olier Yang School of Information Technology and Engineering Uniersity of Ottawa Ottawa, Ontario, Canada, KN 6N5 {sgo, ABSTRACT Energy conseration is a critical isse in wireless ad hoc networks since batteries are the only energy sorce to power the nodes. One major metric for energy conseration is to rote a commnication session along the rotes which reqire the lowest total energy consmption when all nodes are eqipped with a finite and nonrenewable amont of energy. To explore adantages offered by the se of directional antennas, we consider the case of sorce initiated mlticast traffic in wireless ad hoc networks that se switched antennas and hae limited energy resorces. In this paper, we present a constraint formlation in terms of mixed integer linear programming, which can be sed for an optimal soltion of the minimm-energy mlticast problem in wireless ad hoc networks with directional antennas. The optimal soltions can be sed to assess the performance of heristic algorithms for mobile networks by rnning them at discrete time instances. Categories and Sbject Descriptors C.. [Compter-Commnication Networks]: Network Architectre and Design Wireless Commnication; G.. [Discrete Mathematics]: Graph Theory trees; G..6 [Nmerical Analysis]: Optimization integer programming. General Terms Algorithms, Performance, Experimentation, Theory. Keywords Wireless Ad Hoc Networks, Directional Antenna, Minimm- Energy Roting, Mlticast Tree, Integer Programming.. INTRODUCTION Ad hoc wireless networks are expected to be deployed in a wide ariety of ciil and military applications. The commnicating nodes might be distribted randomly and are assmed to hae the Permission to make digital or hard copies of all or part of this work for personal or classroom se is granted withot fee proided that copies are not made or distribted for profit or commercial adantage and that copies bear this notice and the fll citation on the first page. To copy otherwise, or repblish, to post on serers or to redistribte to lists, reqires prior specific permission and/or a fee. MobiHOC 4, May 4-6, 4, Roppongi, Japan. Copyright 4 ACM /4/5 $5.. capacity of packet forwarding to commnicate with each other oer a shared and limited radio channel. Bilding sch networks poses a significant technical challenge becase of the constraints imposed by the characteristics of the ad hoc networks. One important constraint is the scarce power resorce if the nodes are operated by batteries. Ths, for increasing longeity of sch networks, it is imperatie that we find ways of either increasing battery power or alternatiely optimizing the se of the battery power ia energy-efficient algorithms and mechanisms. Obiosly, the first soltion is technology dependent, and we focs on the second one that is of mch interest in network research. Recently, the problem of minimizing the energy consmption of wireless ad hoc networks has been stdied comprehensiely. This problem is referred to as the Minimm-Energy Roting (MER) [] [], which can be classified into two categories: minimmenergy nicast roting [4, 5, 6,, 3, 4, 5] and minimmenergy broadcast or mlticast roting [3, 6, 7, 8, 9, ]. The minimm-energy nicast roting is essentially a shortest directed path problem based on arios power cost fnction. Howeer, for broadcast applications, and in general mlticast applications, the minimm-energy roting is far more challenging, which has been shown to be NP-complete [7]. Since the MER problem is hard, seeral heristic algorithms for bilding a sorce based energyefficient broadcast/mlticast tree hae been deeloped. A straight greedy approach is the se of broadcast trees that consist of the best nicast paths to each indiidal destination from the sorce node (broadcast session initiator). This heristic first applies the Dijkstra s algorithm to obtain a Shortest Path Tree (SPT), and then to orient it as a tree rooted at the sorce node. Similarly the Minimm Spanning Tree (MST) heristic first applies the Prim s algorithm to obtain a MST, and then to orient it as a tree rooted at the sorce node. In [3, ], another heristic called Mlticast Incremental Power (MIP) was presented. It exploits the wireless mlticast adantage property in the formation of the mlticast trees, and ths proides better performance than the greedy algorithms SPT and MST. The wireless mlticast adantage property, originally introdced in [3], means that all nodes within commnication range of a transmitting node can receie a mlticast message with only one transmission if they all se omni-directional antennas. 34

2 It has been shown earlier that the se of directional antenna in the context of wireless ad hoc networks can largely redce the radio interference, thereby improing the tilization of wireless medim and conseqently the network performance. Some papers [, ] sggest the se of mltiple directional antennas per node (or mltiple beam antennas) in order to increase the throghpt of 8. media access control protocol [3]. In [4], the athor explores the se of beam forming antennas in order to improe both throghpt and delay in ad-hoc networks. Another paper [5] has sggested the se of mltiple directional antennas to redce the roting oerhead of on-demand roting protocols for ad-hoc networks like DSR [6] and AODV [7]. Oer the past few years, energy efficient commnication in wireless ad hoc networks with directional antennas has receied more and more attention. An energy-efficient roting and schedling algorithm [] was sed to coordinate transmissions in ad hoc networks where each node has a single directional antenna. A recent paper [] extended the reslts in [3, ] and indced two protocols Redced-Beamwidth MIP (RB-MIP) and Directional-MIP (D-MIP) that exploit the se of directional antennas for mlticasting in wireless networks. In or earlier stdy [8], we introdced a new concept called irtal relay that allows the constrction of a minimm-energy mlticast tree to be eqialently mapped to the constrction of a minimm-energy irtal relay tree. This reslts in the minimmenergy mlticast (MEM) problem modeled as a mixed integer linear programming (MILP) problem. The optimal soltion wold therefore be obtained sing an MILP soler aailable in the pblic domain. This paper extends the work of [8], not only by presenting a more general stdy from broadcast to mlticast case, bt also by improing or analytical model with the se of directional antennas. We formlate a generalized antenna orientation optimization and minimm-energy mlticast problem that incldes the problems addressed in [8] as special cases. The remaining of this paper is organized as follows. In Section, we analyze the challenges for minimm-energy broadcast and mlticast roting in a wireless enironment. Section 3 gies a definition of minimm energy mlticast tree in the context of directional antenna applications as the basis of the formlation. From Section 4 to Section 8, we constrct linear constraints for Problem MEM systematically, complete formlation of the problem in a form of Integer Linear Programming, and proe that it prodces the optimal soltions. Finally in Section 9, we smmarize or reslts and point ot seeral ftre research problems. For the conenience of the readers, the notations sed in this paper are listed in the Appendix.. ANTENNA MODEL boresight Figre. Directional antenna propagation model. An ad-hoc wireless network consists of a fixed nmber of nodes, which are randomly distribted oer a two-dimensional plane. Each node is eqipped with a directional antenna, which permits energy saings by concentrating transmission energy where it is needed. We se an idealized directional antenna propagation model as shown in Fig, where the antenna orientation ( < π) of node is defined as the angle measred conter-clockwise from the horizontal axis to the antenna boresight, and the antenna directionality is specified as the angle of beamwidth ( < π). Based on this model, antenna can be loosely classified into omni-directional, modestly directional (switched antenna), and highly directional (adaptie antenna), which are listed in Table. There hae been a nmber of antenna prodcts in each of the category. Table. Antenna classification Antenna Directionality Antenna Orientation Omnidirectional fixed beamwidth Modestly directional fixed beamwidth Highly directional ariable beamwidth nsteerable steerable steerable When considering omni-directional antennas and niform propagation condition, we obsere that all nodes within the commnication range of a transmitting node can receie its transmission, and the receied signal power aries as r, where r (r > ) is the distance to the sender, and is a parameter that typically takes on a ale between and 4, depending on the characteristics of the commnication medim. Based on this model, the transmitted power reqired to spport a link between two nodes separated by range r is proportional to r. Withot loss of generality, all receiers hae the same power threshold for signal detection, which are typically normalized to one, reslting in p = r, where r is the distance between node and node, and p represents the power needed for link between node and node. For or directional antenna propagation model, we frther assme that for any node, all of the transmitted energy is concentrated niformly in a beamwidth, ignoring the possibility of sidelobe interference. Then, the transmission power needed by node to transmit to node in its antenna beam sing beamwidth is p = r π () Conseqently, the se of narrow beams allows energy saing for a gien commnication range or extends the antenna range for a gien transmission power leel when compared to the se of omnidirectional antennas. On the other hand, only the nodes located within the transmitting node s antenna beam can receie the signal, ths possibly diminishing the effect of the wireless mlticast adantage. We only focs on the modestly directional antenna in this paper. All the assmptions throgh the whole paper are smmarized below. () Beamwidth of each antenna cannot be adjsted, i.e., is fixed for any node. 35

3 () Orientation of each antenna can be shifted to any desired direction to proide connectiity to a sbset of the nodes that are within commnication range. (3) A single antenna beam is proided for each session in which a node participates. (4) Each node knows the precise locations of its potential neighbors. 3. MINIMUM ENERGY MULTICAST TREE Let s model the network by a simple directed graph G(N, A, p), where N is a finite node set, N = n, and A is an arc set corresponding to the nidirectional wireless commnication links. The arc weight fnction p: A R assigns power to each arc, where R denotes the positie real nmber set. That is, for each arc (, ), p is the power needed for the link from node to node. We assme that any node N can choose its power leel, not to exceed some maximm ale p max. Any directed arc (, ) A if and only if p p max. We consider a sorce-initiated mlticast in wireless ad-hoc networks. Any node is permitted to initiate mlticast sessions. Mlticast reqests and session drations are generated randomly at the network nodes. The set of mlticast grop members M ( M = m) and other relay nodes that spport a mlticast session are referred to as a mlticast tree. We assme that no power expenditre is inoled in signal reception and processing actiities. Ths the total power is expended completely on transmission at each node in the tree. Obiosly, leaf nodes do not contribte to this qantity becase they do not relay traffic to any other nodes. Hence, we ealate performance in terms of total RF power from all transmitting nodes reqired to maintain the tree. Any mlticast tree is a rooted tree. We define a rooted tree as a directed acyclic graph with a sorce node s called root with no incoming arcs, and all its other nodes with exactly one incoming arc. A property of rooted tree is that for any node in the tree, there exists a single directed path from s to in the tree. A node with no ot-going arcs is called a leaf node, and all other nodes are internal nodes, or relay nodes. Note that the relay nodes may be mlticast members or may not, and the antenna beam of a relay node shold coer all its children. Therefore, the minimmenergy mlticast problem is to find a mlticast tree with the minimm power consmption. Doing so inoles the choice of transmission power leel, relay nodes, and antenna orientation. Formally, a mlticast tree is modeled by a node-weighted tree T s (N', A', q) rooted at a sorce node s, with a mlticast node set N' N, an arc set A' A, and a node weight fnction defined as q: N' R {}. That is, for each node in N', q is the transmission power of the node reqired by the mlticast tree T s. We define T s (N', A', q) to be a mlticast tree of G(N, A) rooted at s if and only if the following properties are satisfied. ) RTP (Rooted Tree Property): this property reqires T s can span all the mlticast members from node s (M N'); ) WMAP (Wireless Mlticast Adantage Property): this property reqires the node weight fnction to satisfy:, is leaf node q = Max p, is internal node (, ) A' () 3) AOP (Antenna Orientation Property): this property reqires node mst locate within the antenna beam of node, for any (, ) A'. 4. FORMULATION MODEL This is the first time that an accrate definition of mlticast tree is gien in the context of directional antenna applications, pon which Problem MEM shall be formlated as a mixed integer linear programming (MILP) model. The main idea is to extract a sb-graph T s from the original graph G, sch that T s is a mlticast tree with minimm energy consmption. In order to formlate the problem, we define the following ariables: (i) Z is a binary decision ariable which is eqal to one if the arc (, ) is in the sb-graph T s of G, and zero otherwise; (ii) P is a nonnegatie continos ariable which represents the transmission power of the node reqired by the mlticast tree T s ; (iii) F is a nonnegatie continos ariable, which represents the amont of flow going throgh arc (, ); (i) c is a binary ariable which is eqal to one if node is coered by the antenna beam of node, and zero otherwise. We shall proe that if (x) is the optimal soltion of ariable x obtained from this MILP model, then the graph T s (N', A', q) is the optimal tree associated with this soltion. In this graph, N' = { (, ) A' or (, ) A'} is its arc set, A' = {(, ) Z = } is its arc set, and q: A' R {} is a nonnegatie weight fnction defined as q = P. In other words, T s (N', A', q) is a mlticast tree of G with minimm energy consmption. Finally, the Problem MEM can be formlated to minimize total power from all transmitting nodes sbject to the constraints that T s satisfies RTP, WMAP, and AOP. We then hae the following formlation for the Problem MEM as shown below. Inpts: G(N, A, p), M, and s Objectie fnction: Σ N P Constraints: RTP, WMAP, and AOP Otpt: T s (N', A', q) Example Figre. Example 4-node network G4: mlticast grop is {,, 3} and node is the sorce A generic example of a 4-node network G 4 that we consider is shown in Figre. The weight for each arc represents the power reqired transmitting packets on it. G 4 is an asymmetric directed graph. For example, the doble arrow arc (, ) indicates that 36

4 node and node can reach each other, while the nidirectional arc (, 4) indicates that only node can reach node 4 since node 4 may not hae enogh power to reach node. We assme the channel loss exponent =, then we can obtain 3 = 3π/4. We assme each node has a fixed antenna beamwidth π. The objectie fnction of G 4 is therefore: minimize: P P 5. LINEAR CONSTRAINTS FOR RTP We want to proide a set of constraints that wold garantee that T s (N', A', q) obtained from the formlation satisfies the rooted tree property. It can be characterized that T s (N', A', q) is a rooted tree spanning all the mlticast members, i.e., M N', by the following properties: RTP (a): Eery node, N' \{s}, has exactly one incoming arc, and node s has no incoming arcs; RTP (b): T s (N', A', q) does not contain cycles. The constrction and interpretation of the linear constraints for these two properties are elaborated in the following lemmas. n k (a) (b) (c) Figre 3. Illstration of constraints (a) any non-mlticast member in V mst hae exactly one incoming arc, (b) a connected component of V may be a simple cycle, (c) a cycle with sb tree leaing ot of it. (Solid nodes indicate mlticast members, and hollow nodes indicate non-mlticast members.) Lemma. T s (N', A', q) is a directed graph in which node s has no incoming arcs, and each other node has exactly one incoming arc, proided Problem MEM satisfies the following constraints: N N N s n Z = ; (3) Z = ; M \{s} (4) Z ; N \ M (5) Z (n ) Z ; N \M (6) N N Proof: Note that Σ N Z and Σ N Z are the in-degree and otdegree of node in T s respectiely. Therefore, the root node s and the other mlticast members satisfy this statement directly from the constraints (3) and (4) respectiely. It remains to proe that any non-mlticast member in T s spporting the mlticast commnications mst hae exactly one incoming arc. Assme N' is a non-mlticast member in T s, indicated by a hollow node in Fig. 3, its incoming degree mst be or from constraints (5). If Σ N Z =, from constraints (6), it follows that Σ N Z =. That means mst be an isolated node as n 3 n shown in Fig. 3a, ths N'. This contradicts the original assmption. Therefore node has exactly one incoming arc. Note that if Σ N Z = for any non-mlticast member in T s, constraints (6) become redndant since the ot-degree of node is at most n. From constraints (3), (4), and (5), we obtain the following conclsion: N Z {,}, N (7) Example Referring to G 4 in Figre, we can now list the first set of constraints corresponding to (3) to (6) for RTP (a) as follows: Z = Z Z 3 = Z 3 Z 3 = Z 4 3Z 4 We shall see in the Lemma that the introdction of ariable F is to help to preent loops in T s, and this ariable only represents fictitios flow prodced by the mlticast initiator s going throgh arc (, ). Lemma. V (N', A', q) does not contain cycles, if Problem MEM satisfies the constraints (3) (5) and the following constraints: F N F N = Z ; N \{s} (8) N Z F (n ) Z ; N \{s}, N (9) Proof: From the constraints in (3), (4) and (5), it follows that the only connected components in T s that might contain cycles cold be composed of either a simple cycle shown in Fig. 3b, or a simple cycle with sb tree leaing ot of it as shown in Fig. 3c. We will show in the following that sch topologies are not feasible for Problem MEM. Assme that the nodes (n l, n,, n k, n kl = n l ), k >, form a simple cycle in T s. Then from constraint (3), node s will neer be inclded in sch a cycle. Constraint in (9) implies that F cold be positie if and only if (, ) A'. Letting F = f, then from the constraints in (8) it follows that r Z i= n n i i n n n r n r F = n n F for r =,, k. Each node n r (r =,, k) is in A' as stated in the assmption, i.e., Z n r n =. Therefore F r n r n = r n n r F Z i = n n = f (r ) for r =,, k. After sbstitting i i F n k n N = f (k ) into constraint (8), for = n l, we obtain F n hand, N N F n F n N = f (k ) f = k <. On the other F n = N Z n from constraints (7). Ths the constraints in (8) are iolated, and therefore simple cycles are not possible in T s. Similar reasoning shows that the topology in Fig. 3c also iolates the constraints in (8), and therefore T s cannot contain cycles. Example 3 37

5 Still referring to G 4 in Figre, we can list the next set of constraints corresponding to (8) and (9) for RTP (b), which is expressed as follows: F F 3 F F 3 = Z Z 3 F 3 F 3 F 3 = Z 3 Z 3 F 4 = Z 4 Z F 3Z Z 3 F 3 3Z 3 Z 3 F 3 3Z 3 Z 3 F 3 3Z 3 Z 4 F 4 3Z 4 6. LINEAR CONSTRAINTS FOR WMAP The constraints for the wireless mlticast adantage property (WMAP) reflect the condition that the power reqired at node is the maximm of the indiidal transmission power to each neighbor from. The following lemma explains how the WMAP, i.e., Eqation (), can be achieed. Lemma 3. T s (N', A', q) satisfies WMAP, if the formlation of Problem MEM incldes the constraint (). P p Z ;, N () Proof: For any node in T s, if is a leaf node, i.e., Z = for all N', then P p Z = ; if is an internal node, then P p Z for all N', i.e., P Max (, ) A p. The eqalities are achieed in the ineqations aboe when the smmation of the ariables P is minimized. Ths Eqation () mst be held by T s. Example 4 The set of constraints for the example G 4 shown in Figre corresponding to the constraints () for WMAP can be expressed as follows: P Z P 5Z 3 P 4.9Z 4 P Z P Z 3 Z 3 7. LINEAR CONSTRAINTS FOR AOP commnication link (, ) exists if and only if c =. Before discssing the constrction of linear constraints for AOP, we first inestigate in more detail the relationship between the antenna orientation and the existence of wireless commnication links c. Let ( < π) be the angle measred conter-clockwise from the horizontal axis to the ector as shown in Fig. 4. Then the angle (, N) can be obtained once their positions are gien. Assme the beam width ( < π) of each node is fixed, and each antenna beam can be pointed in any desired direction by adjsting its orientation continosly to proide connectiity to a sbset of the nodes that are within its commnication range. In Fig. 4, the lighter shaded area is the space coered by the antenna beam of node when it is abot entering the position of node (i.e., for making contact with ), and the darker shaded area is the space jst before the beam is leaing the position of node (i.e., for losing contact with ). Ths it is clear that the wireless link (, ) exists if and only if the antenna orientation is bonded by the two pointing directions = / and = /, indicated by the dotted lines as shown in Fig. 4. c (a) / c π π π (b) / π - / c π π Figre 4. Antenna beam coerage range As defined in Section, the antenna orientation { : N} are continos ariables < π. Let {c : (, ) A} be binary ariables sch that c = if node is coered by the antenna beam of node, and otherwise. In other words, the wireless (c) π - / < π Figre 5. Three possible ale ranges of angle for the constraints between and c 38

6 Assme < /. Then according to the definition of c, the range [ /, /] of shown in the left part of Fig 5a mst be mapped into [, /] [π /, π), since we restrict ariable between and π. Therefore, c = if and only if [, /] [π /, π), which is expressed in Constraint () and depicted in the right part of Fig 5a. Similarly, we hae Constraints () and (3) for and c corresponding to within different ranges as shown in Figre 5b and Figre 5c respectiely. (i) < /:, / or π / < π c = (), otherwise (ii) / < π /:, / / c = (), otherwise c / P c P π π (a) linear constraint between and c (iii) π / < π :, / < π or /π c = (3), otherwise The aboe Constraints () (3) are obiosly nonlinear. In the following three cases, we shall show that these constraints can be linearized. Case : < The right part of Fig. 5a shows the Constraint () in a c plane. We obsere that c can be decomposed into a smmation of two new binary ariables c and c, which are defined in Eqations (4) and (5). c =, /, / < < π c =,, (4) π / < π (5) < π / Eqations (4) and (5) are depicted by thick lines in c plane and c plane as shown in Fig 6a and Fig 6b respectiely. In Fig. 6a, the points (, c ) that satisfy the Eqation (4) mst be within the shaded area between line P P and line, where P = (, /), P = ( /, ), = (π, ), and = ( /, ). It can be clearly obsered that Constraint (7) later coers the shaded area aboe line P P, and Constraint (8) coers the shaded area below line. Since < π and c {, }, the point set defined by Eqation (4) and the point set defined by the constraints (7) and (8) are the same in a c plane as shown in Fig. 6a. Since P may be any point between (, ) and (, ), we may choose their middle point withot loss of generality. Similarly, the points (, c ) defined by the Eqation (5) can be rewritten in constraints (9) and () with the help of Fig. 6b. In smmary, the nonlinear Constraint () is linearized sing Constraints (6) to (). π P P π (b) linear constraint between and c Figre 6. Illstration of constraint linearization ( < /) c = c c <, (6) c > (7) c (8) c > (9) c () Case : < π In this case, c can be decomposed into another linear combination of binary ariables c and c, gien in Eqation () and (). Following a similar step as aboe and the help of Fig. 7, we linearize Constraint () into Constraints (3) to (7). c =, / < π (), < / c =, /, / < < π c = c c, < π () (3) 39

7 c c P P / P P π π π c (a) linear constraint between and c c (a) linear constraint between and c P / P P P π π π c c c (b) linear constraint between and c Figre 7. Illstration of constraint linearization ( / < π /) > > c Case 3: π < π (4) (5) (6) (7) Very similar to Case, linear constraints (3) to (34) illstrated in Fig. 8 can be obtained for Eqation (3). c =, / - π, / - π < < π c =,, / < π < / (8) (9) c = c c, π - < π (3) c > (3) c (b) linear constraint between and c Figre 8. Illstration of constraint linearization (π / < π) 8π 8π (3) c > (33) c (34) So far, the linear constraints hae been completely constrcted to characterize the relation between antenna orientation and the existence of wireless commnication links c, pon which the Antenna Orientation Property (AOP) can be easily achieed by the following lemma. Lemma 4. T s (N', A', q) satisfies AOP, if the formlation of Problem MEM incldes the following constraints: Z c ;, N (35) Proof: Arc (, ) exists in T s, i.e. Z =, only if node locates within the antenna beam of node, i.e. c =. This is eqialent to Constraint (35). Example 5 Referring the example of the 4-node network shown in Fig., we consider the additional constraints imposed on ariables Z once taking into accont AOP in a context of directional antenna applications. As we assmed before, each node has a fixed 4

8 beamwidth π. We only discss the linear constraint formlation abot ariables Z and Z as examples. For constraints of Z, since = 5π/4, = π, and / < π /, this belongs to Case. For constraints of Z, since = π/4, = π, and < /, it belongs to Case. We list the constraints for AOP corresponding to Z and Z expressed as follows: Z c c Z c c c > / 5π 3/ 5 c > / 3π / 4 c / 3π c / 5π 8 / c > / 7π / c > / π c / π 8 c / 7π 8. PROBLEM MEM FORMULATION Or preios deriation on the linear constraints can now help s to rewrite the problem formlation in Section 4 as an MILP model. This is shown in Fig. 9, in which the coefficients A i, B i, C i, D i, (i =, ), and E are gien in Table. A < A B B C C D Table. Vales of coefficients 4 π <π - π - <π 8π 8π 4 π D E In this formlation, Z, c, and c are integer ariables; P, F, and are continos ariables. Note that c helps to simplify the form of constraints, bt they are not independent ariables. The nmber of ariables in the formlation is approximately 4n n, and the nmber of constraints is of the order of O(n ). Recall that in Lemmas to 4, we proed that any soltion, which satisfies the constraints in (37) to (5), shold be a mlticast tree. In order to proe that the formlation of Problem MEM soles the minimm-energy mlticast roting problem, it remains to show that eery mlticast tree can be expressed by the ariables P, Z, F, c, c and in Problem MEM. This is achieed by the following theorem. Theorem. The formlation of Problem MEM soles the minimm-energy mlticast problem. Proof: Assme that T s = (N', A', q) is an arbitrary mlticast tree of G. The soltion corresponding to T s is obtained by setting ariables in the following order: ) If (, ) A' then we set Z = ; ) We set all other Z = otherwise; 3) If is an internal node of T s, we set P = Max p ; : (, ) A' 4) We set all other P = otherwise; 5) If Z =, we set F = ; 6) If Z is not eqal to zero, the corresponding F are assigned the ale of the cardinality of T, i.e., F = T, where is the set of nodes that belong to the sb-tree that contains node and is formed by remoing arc (, ) from the tree T s, and it is clear that T < n; s 7) We set to be any direction sch that the antenna beam of node can coer all the nodes { Z = }; 8) We set c by Eqations (4), (), or (8) corresponding to different ale of ; 9) We set c by Eqations (5), (), or (9) corresponding to different ale of. It is a straightforward exercise to show that those P, Z, F, c, c and ales aboe satisfy the RTP, WMAP, AOP and Integrality Property constraints in Problem MEM. Or analytical model can be easily applied in some specialized cases. Broadcast can be considered as a special case of mlticast when M = N. Therefore, the constraints (39) and (4) in the formlation (see Fig. 9) disappear, and the constraint (4) can be simplified as constraints (5). s s T s F F = ; N \{s} (5) N N Omni-directional antenna scenarios can also be considered as a special case of directional antenna model when the optimized ariable for each node is set to a constant π. Variables, c, c, and c, and constraints (44) (49) all disappear, since c = is always tre. 9. CONCLUSION In this paper we present a constraint formlation for the joint minimm-energy mlticast and antenna orientation optimization problem in mlti-hop ad hoc wireless networks. Based on the analysis on the properties of minimm energy mlticast tree, the problem can be characterized in a form of mixed integer linear programming problem, and we proceed to proe the correctness of this formlation. To or best knowledge, these are the first work sing integer programming to formlate the problem in a context of directional antenna applications. Many application scenarios can be soled efficiently based on the formlation sing branch-and ct or ctting planes techniqes. The optimal soltions can be sed to assess the performance of heristic algorithms for mobile networks by rnning them at discrete time instances. 4

9 Objectie fnction: minimize P (36) N Sbject to: (I) Rooted Tree Property N N N N Z = ; (37) s Z = ; M \{s} (38) Z ; N \ M (39) Z F N (n ) Z ; N \M (4) F N N = Z ; N \{s} (4) N Z F (n ) Z ; N \{s}, N (4) (II) Wireless Mlticast Adantage Property p P p Z ;, N max (43) (III) Antenna Orientation Property c > A B ;, N (44) c C D ;, N (45) c > A B ;, N (46) c C D ;, N (47) Z c c E ;, N (48) < π; N (49) (IV) Integrality Property Z {, }; c {, }; c {, },, N (5) Figre 9. MILP model for Problem MEM A major challenge, and a topic of contined research, is to extend or analytical model to large-scale networks with thosands of nodes. A near optimal soltion can be fond in a polynomial time sing the Lagrange Relaxation and sb-gradient techniqes [8] based on or formlation. Frthermore, it is important to deelop the distribted algorithms of MEM to cope with the dynamic topologies.. APPENDIX: NOTATION A an arc set corresponding to the nidirectional wireless commnication link; A' the arc set of mlticast tree T s (N', A', q), A' A; F the nonnegatie ariables, which represent the amont of flow prodced by the mlticast initiator going throgh (, ); G a directed graph modeling the wireless ad hoc network with a node set N, an arc set A, and an arc weight fnction p; M a set of mlticast members, M = m and M N'; N a node set in a two-dimensional plane, N = n; N' the node set of mlticast tree T s (N', A', q) inclding all the mlticast nodes M N' N; P the nonnegatie continos ariables that represent the transmission power of the node reqired by the mlticast tree T s ; T s a mlticast tree of G (N, A, p) rooted at a sorce node s; Z the binary decision ariables that are eqal to one if arc (, ) exists in the sb-graph T s of G, and zero otherwise; c the binary ariables sch that c = if node is coered by the antenna beam of node and otherwise; p max the maximm power leel that node can choose; p a weight fnction p: A R that presents the power needed for the link from node to node, which is always greater than zero; q a weight (power) fnction q: N' R {} that can be zero; r the distance between node and node ; (.) an optimized soltion; the propagation loss parameter the angle measred conter-clockwise from the horizontal axis to the ector from node pointing to node ( < π); the beamwidth of node s antenna ( < π); 4

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