An Iterative Framework for Optimizing Multicast Throughput in Wireless Networks

Transcription

1 An Iterative Framework for Optimizing Muticast Throughput in Wireess Networks Lihua Wan and Jie Luo Eectrica & Computer Engineering Department Coorado State University Fort Coins, CO Emai: {carawan, Anthony Ephremides Eectrica & Computer Engineering Department University of Maryand Coege Park, MD Emai: Abstract This paper shows that, under certain conditions, a wireess network can be modeed by a directed configuration graph with possibe hyperarc inks if the transmission schedue is given. Assume singe muticast session. The maximum achievabe muticast throughput equas the max-fow min-cut bound of the configuration graph. An optimization framework is proposed to maximize the muticast throughput via iterative updates of the transmission schedue. It is demonstrated that the optima muticast throughput can be obtained without exporing either a arge number of hyperarc inks or a arge number of cuts, athough efficient suboptima agorithm is needed to avoid searching ink combinations and to reduce the compexity further to poynomia in the number of nodes. It is aso shown that, when the configuration graph has hyperarc inks, the minimum cut can no onger be obtained using the we-known fow augmenting path agorithm. An aternative agorithm is proposed. 1 I. Introduction The topoogy of a wireine network can be modeed by a graph, in which, each vertex represents a network node and each edge represents a point-to-point (cabe) ink between two nodes. Extending this graphic mode to a wireess network faces two key chaenges. First, since signa transmitted over the wireess medium can often reach more than one receivers, it is possibe that a wireess node can communicate common information to mutipe receivers simutaneousy [1] using the same power and bandwidth of a point-to-point transmission. To mode such (direct) muticast transmission, it is necessary that a graph representation shoud contain point-to-mutipoint edges (termed hyperarc inks [2]). Meanwhie, a wireess ink can support a positive information rate so ong as the channe gain is not stricty zero. If a inks with nonzero channe gains must be incuded in a graph representation, the tota number of inks is exponentia in the number of nodes 2. Consequenty, optimizing a ink 1 This work was supported by Nationa Science Foundation grants CCF and CCF Forexampe,consideranetworkwith V nodes ocated in an open area. Since each node can transmit common information to any subset of nodes, the tota number of inks (incuding hyperarc inks) equas V 2 V 1. rates simutaneousy can be overy compex even for a moderate-sized network. Second, communication over a wireess ink can be interfered by signas transmitted from the neighboring nodes. Communication rates of the inks are therefore couped. If we ist the rates of a inks as a vector of dimension E, then the cosure of a achievabe rate vectors is characterized by a capacity region in the E dimensiona space. Unfortunatey, obtaining the capacity region, or testifying whether a given rate vector is in the capacity region, can be extremey difficut even for a sma network with three or four nodes [3]. Advances of network coding showed that, if a source node transmits common information to mutipe destinations in a wireine network, the maximum achievabe muticast throughput equas the maximum fow of the minimum cut that separates the source from at east one destination in the topoogy graph [4]. This resut has stimuated a series of consequentia researches on optimizing muticast throughput in wireess and mesh networks [2][5][6][7]. In this paper, we propose an iterative framework to maximize the throughput of a singe muticast session in a wireess network. We show that, given the transmission schedue (defined in Section III) and under certain conditions, a wireess network can be represented by a configuration graph with possibe hyperarc inks. Given the configuration graph, with the hep of network coding, the muticast throughput equas the maximum fow of the minimum cut that separates the source node from at east one destination node. The key idea of the framework is therefore to iterativey update the transmission schedue to improve the max-fow min-cut vaue of the corresponding configuration graph. We show that the proposed framework can address both key chaenges mentioned at the beginning of this section. Efficient network configurations can aso be obtained with a compexity poynomia in the number of nodes. II. Probem Formuations Let V be the node set of a wireess network. We consider a singe muticast session where the source node s V deivers common information reiaby (in information

2 theoretic sense), possiby through muti-hop paths, to a nodes in a destination set T V. The information rate of such transmission is termed the muticast throughput, and is denoted by R st. A communication ink, e, is defined as the association of one tai (transmitter) node i and a set of head (receiver) nodes J. If we can ist a nodes in J, for exampe J = {a, b}, we aso denote the ink e as e iab. We say e achieves an information rate of r if i communicates common information reiaby and directy to a nodes in J at rate r. We assume the peak transmission power of node i must be kept beow P i. Let P be a V -dimensiona coumn vector whose eements are the peak power bounds of the nodes. Let r be a coumn vector whose eements are the information rates of a feasibe communication inks. Assume channe gains between network nodes are time-invariant. The union of a achievabe rate vectors r, denoted by C r (P ), is defined as the ink capacity region of the wireess network, which is a function of P. We assume the muticast throughput is a function of the ink rate vector r, and formuate the muticast throughput maximization probem as maximize R st (r), s.t. r C r (P ). (1) Note that by formuating the optimization probem (1), we have made two key assumptions. First, writing R st (r) as a function of r assumes information shoud be transmitted reiaby over each ink. This assumption excudes the possibe operation of ampify-and-forward at the reay nodes. Let us consider the four-node network iustrated in Figure 1a. We assume the connections between s and a, b Fig. 1a. a, b, t are connected via wireine inks. Joint decoding at t achieves a higher throughput than decoding information independenty at a and b. Fig. 1b. s, a, b are connected via wireine inks. Jointy encoding information at a, b achieves a higher throughput than encoding information independenty. are wireess, whie a, b and t are connected via noiseess wireine inks. Assume s, a, b have singe antenna each. The channe gains between s, a and s, b are denoted by h sa and h sb, respectivey. Assume h sb >h sa. Let the ambient noise be white Gaussian with zero mean and variance.ifwe regard a and b as two receiving antennas of t and jointy decode the message at t, the achievabe information rate from s to t is given by R st = 1 ( 2 og 1+(h 2 sa + h 2 sb) P ) s. (2) If we assume information must be reiaby decoded at a and b independenty, and then be forwarded to t, the maximumachievaberatefroms to t is given by R st = 1 ) (1+h 2 og 2 P s sb, (3) which is ess than the rate of (2). Second, by presenting each communication ink with a singe tai node, we aso excude possibe joint encoding of common information at different nodes. Let us consider the four-node network iustrated in Figure 1b. We assume s, a and b are connected via noiseess wireine inks, whie the connections between a, t and b, t are wireess ones. Let the channe gains between a, t and b, t be h at and h bt, respectivey. Assume the ambient noise is white Gaussian with zero mean and variance. If we regard a and b as two transmitting antennas of s, and jointy encode and transmit the message at these two nodes, then the achievabe information rate from s to t is given by R st = 1 2 og ( 1+ ( h at P a + h bt P b ) 2 ). (4) If we assume the information must be encoded independenty at a and b, the achievabe information rate becomes R st = 1 ( 2 og 1+ (h2 atp a + h 2 bt P ) b), (5) which is ess than the rate of (4). In addition to the above two assumptions, in this paper, we aso assume reiabe communication over a ink is achieved without channe feedback expoitation. Athough making these assumptions may cause throughput oss, it enabes us to represent a wireess network using a configuration graph given the transmission schedue. This consequenty eads to an efficient cross-ayer optimization framework, as expained in the next two sections. III. Probem Decomposition {( A communication reaization, C k = e,r (k) )}, is defined as a set of ink and rate pairs, where r (k) is the information rate over ink e.wesaye C k if r (k) > 0, and et r (k) =0ife C k. Define the corresponding ink rate vector as r (k). We say the communication reaization C k is feasibe if r (k) can be achieved without invoking the time sharing operation. A transmission schedue, S = {(C k,p k )}, is defined as a set of feasibe communication reaization and time proportion pairs, where p k = 1, and 0 p k 1is the time proportion when communication reaization C k is active. We say C k S if p k > 0. Given transmission schedue S, we can construct a directed configuration graph G(S) = {V,E}, where V and E are the node set and the edge set, respectivey. e E if we can find a C k S such that e C k. We associate with each ink in the configuration graph a configuration rate g (S) = C k S r(k) p k.notethat, whenever we tak about a configuration graph, we aways assume the transmission schedue is specified.

3 In the configuration graph, a cut γ m is defined as a partition that divides the node set V into two disjoint subsets V (m) and V r (m).wesayγ m is an s t k cut if s V (m) and t k V r (m).wesayγ m is an s T cut if it is an s t k cut for at east one t k T. A ink e crosses cut γ m if the tai node i satisfies i V (m), and at east one head node j J satisfies j V r (m).thecut vaue of γ m, aso denoted by γ m, is the sum configuration rates of a inks crossing γ m.letγ(s) be a coumn vector whose eements are the vaues of a the s T cuts. Denote the m th eement of γ(s) by[γ(s)] m. As shown in [4], given S, via discarding and network coding information, the maximum achievabe muticast throughput equas R st =min [γ(s)] m. (6) m Exampe 1: Consider a three-node wireess network where s wants to deiver common information to two destination nodes t 1 and t 2. Assume we have three feasibe communication reaizations: C 1 = {(e st1, 4)}, C 2 = {(e st2, 4)}, C 3 = {(e st1t 2, 3)} 3. Given transmission schedue S = {( ) ( ) ( )} C1, 1 3, C2, 1 3, C3, 1 3, we can form a configuration graph iustrated in Figure 2, in which the configuration rates are g st1 = 4 3, g st 2 = 4 3, g st 1t 2 =1.LetT = {t 1,t 2 }, the three s T cuts are iustrated by the dashed ines in Figure 2. The cut vaues are γ 1 = 7 3, γ 2 = 7 3, γ 3 = 11 3, respectivey. Given S, the maximum achievabe muticast throughput equas R st1t 2 =min(γ 1,γ 2,γ 3 )= 7 3. Fig t 1 e st1 e st 1 t 2 est A three-node network with three s T cuts. Exampe 2: In this exampe, the network has four nodes. The source node s wants to transmit information to the destination node t. Assume we have the foowing three feasibe communication reaizations: C 1 = {(e sab, 3)}, C 2 = {(e sb, 3), (e at, 3)}, C 3 = {(e bt, 3)}. Giventhetransmission schedue S = {( C 1, 1 3), ( C2, 1 3), ( C3, 1 3)},wecan form a configuration graph iustrated in Figure 3, in which the configuration rates are g sab =1,g sb =1,g at =1, g bt = 1. It is easy to verify that the maximum achievabe throughput from s to t is R st =2. Let the union of γ(s) (taken over a S) be defined as the s T cut capacity region C γ (P ). The optimization probem (1) can be rewritten as max min[γ(s)] m, s.t. γ C γ (P ). (7) S m 3 These information rates can appear in a practica system if s is equipped with mutipe antennas. t 2 Fig A four-node network with ink capacities. IV. The Cross-ayer Optimization Framework Let λ be a coumn vector of the same dimension of γ with non-negative rea-vaued eements. Optimization probem (7) can be equivaenty written as max γ C γ 1 min λ T γ. (8) λ,λ 0, m [λ]m =1 Since (7) is a convex optimization probem, the throughput achieved by equiibriums of (8) must be unique. This optima throughput can be obtained numericay via iterativey carrying out the foowing two steps. The Basic Iterative Framework Step 1: Update λ by λ = λ δ 1 γ,whereδ 1 > 0isthe step size. Project λ to the constraint set to satisfy λ 0, m [λ] m =1. Step 2: Update γ by γ = γ + δ 2 λ,whereδ 2 > 0isthe step size. Project γ to the constraint set to satisfy γ C γ. Unfortunatey, one can rarey impement this basic agorithm in a practica system due to the two key chaenges mentioned in Section I. Particuary, the cosed-form expression of C γ is often not avaiabe; optimizing a s T cuts together can aso be overy compex since the number of s T cuts is exponentia in the number of nodes. In the foowing two sections, we show these difficuties can be addressed by revising the two steps correspondingy. A. Revision on Step 1 To avoid optimizing a s T cuts together, it is necessary to upper bound the number of nonzeros eements in λ. In this section, we show this can be done without sacrificing the optimaity of the soution. Let I min (γ) denote the number of cuts that achieve the minimum vaue in γ. LetΓ be the set of s T cut vectors that achieve the optima throughput of (7). Define κ as the minimum number of minimum cuts among a the optima s T cuts, i.e., κ =min I min(γ). (9) γ Γ Athough we have κ = 1 for most of the wireine networks, the foowing proposition shows that κ > 1 often hods for muti-hop wireess networks. Proposition 1: Let κ be defined by (9). If κ =1, then R st is maximized by a transmission schedue S that contains ony a singe communication reaization C (S = {(C, 1)}); the communication reaization C ony contains a singe ink e st, i.e., the source node s directy muticast common information to a destination nodes in T.

4 Let κ>0 be an integer. For a given γ, etimin κ (γ) be the set of indices corresponding to the first κ sma-vaued cuts in γ. Inotherwords,foram / Imin κ (γ) andn Imin κ (γ), we have [γ] m [γ] n. In the case when mutipe cuts achieve the same cut vaue, we assume the choice of Imin κ (γ) is deterministic with respect to γ. The foowing proposition shows that (8) can be written in another equivaent form. Proposition 2: For a κ κ, a equiibriums of the foowing optimization probem achieve the optima throughput of (8), max γ C γ min λ, λ 0, [λ]m =1 m [λ] m =0, m I κ min (γ) λ T γ. (10) Based on Proposition 2, given κ κ,wecanrevise Step 1 of the iterative agorithm as foows. Step 1 (revised): For a m Imin κ (γ), update [λ] m by [λ] m =[λ] m δ 1 [γ] m,whereδ 1 > 0 is the step size. For a m Imin κ (γ), set [λ] m = 0. Project λ to the constraint set to satisfy λ 0, m [λ] m =1. B. Revision on Step 2 Since the cosed-form expression of C γ is often not avaiabe, it is necessary to keep track on the transmission schedue corresponding to the cut vector γ to ensure that γ is in the cut capacity region. Consequenty, the task of Step 2 in the agorithm is to update the transmission schedue S such that λ T γ(s) can be improved. Since such update shoud be incrementa, we can further assume the update is driven by a singe communication reaization in the sense that one shoud either add a new feasibe communication reaization into S, or increase the time proportion of an existing communication reaization in S (and then scae the time proportions of other communication reaizations in S accordingy). Before impementing such revision to Step 2, we have to answer two key questions. First, whether improving λ T γ(s) is aways possibe by updating S with a singe communication reaization. Second, since the number of feasibe inks (incuding hyperarc inks) can be exponentia in the number of nodes, whether it is possibe to expore ony a poynomia number of inks to find the optima communication reaization. Unfortunatey, we can ony get a positive answer to the two questions under the foowing additiona assumption. Assumption 1: For any communication reaization, we assume information transmitted over a ink is decoded without expoiting codebook information of other inks 4. 4 Take a mutiaccess scheme for exampe. Assumption 1 prevents the use of joint mutiuser decoding such as the maximum ikeihood and the decision feedback mutiuser detection agorithms. However, it does not excude interference avoidance methods such as the decorreation detection and the minimum mean square error (MMSE) detection, since these detectors ony expoit the channe gain information of other inks, but not their codebooks. In the foowing proposition, we show that, under Assumption 1 (presented beow), not ony the transmission schedue update is aways possibe, we aso do not need to activate any hyperarc ink with more than κ receivers, where κ is defined in (9). Proposition 3: Let Assumption 1 be enforced. Let κ (and I min (γ) ) be defined by (9) under Assumption 1. For any feasibe communication reaization C, define γ(c) as the s T cut set vector corresponding to (the configuration graph derived from) transmission schedue S = {(C, 1)}. Given λ, and a cut vector γ corresponding to transmission schedue S. Assume λ T γ(s) is stricty ess than the optima throughput of (7). Then we can aways find a communication reaization C, which does not contain any hyperarc ink with more than κ receivers, such that the foowing inequaity is satisfied, λ T γ(c) > λ T γ(s). (11) Based on Proposition 3, given κ κ,wecannowrevise Step 2 of the iterative agorithm as foows. Step 2 (revised): Given λ, γ and its corresponding transmission schedue S. Among a communication reaizations with the number of receivers of each of their inks being no more than κ, find the communication reaization C that maximizes λ T γ(c). If C S, add the communication reaization and time proportion pair (C, δ 2 )intos, whereδ 2 > 0 is the step size. If C S, increase its time proportion by δ 2.Thenscaeathetime proportions of communication reaizations in S so that their sum equas 1. Even though the revised Step 2 ony invoves a poynomia number of inks, since a communication reaization can simutaneousy activate mutipe inks and the number of ink combinations is exponentia in the number of nodes, the compexity of the revised Step 2 is sti exponentia in the number of nodes. A simpe but suboptima approach to avoid such exponentia compexity is to activate inks sequentiay to construct the communication reaization C mentioned in the revised Step 2, as opposed to search it exhaustivey. Unfortunatey, we have to skip further discussions due to page imitations. Note that Assumption 1 does not affect the vaidity of Proposition 2 as ong as κ and I min (γ) are aso derived under Assumption 1. C. Discussions In both two steps of the revised agorithm, vioating κ κ may resut in a suboptima soution. Consider the network given in Exampe 1. If we et κ = 1, it is easy to see the agorithm wi either find γ 1 or γ 2 as the minimum cut. The best communication reaizations that maximize γ 1 and γ 2 are C 1 = {(e st1, 4)} and C 2 = {(e st2, 4)}, respectivey. Consequenty, the iterative agorithm wi converge to transmission schedue S = {( ( C 1, 2) 1, C2, 2)} 1 with the corresponding muticast

5 throughput being R st1t 2 = 2. This is suboptima since R st1t 2 = 3 can be achieved by transmission schedue {(C 3, 1)}. The same exampe aso shows that vioating κ κ in Step 2 can ead to a suboptima soution. Since the vaue of κ is unknown before soving the optimization probem, a practica way to avoid requiring κ is to initiaize κ with a sma vaue and then, upon convergence of the agorithm, increase κ to check whether higher throughput can be achieved. V. Finding The Minimum s T Cut In the revised Step 1 of the iterative agorithm, given a transmission schedue with s T cut vector γ, we need to find Imin κ (γ), which is the indices of the first κ sma-vaued s T cuts. This needs to be done without exporing a s T cuts since the number of s T cuts is exponentia in the number of nodes. The core of this probem is to find the minimum s T cut given a configuration graph. Due to the compication brought by hyperarc inks, even if there is ony one destination node t, the minimum s t cut can no onger be obtained using the we-known fow augmenting path agorithm [8]. Consider the network given in Exampe 2 whose configuration graph is iustrated in Figure 3. Consider the s t path consists of edges e sab and e at. Assign the path with a fow of 1. It is easy to see there is no fow augmenting path (see definition in [8]). However, the achieved rate, R st =1 is not maxima since we can simutaneousy assign fow 1 to path (e sab,e bt ) and fow 1 to path (e sa,e at )toachieve R st = 2. This shows that Coroary 5.2 of [8] does not hod for configuration graphs with hyperarc inks. Since given the transmission schedue the ink rates are decouped, the maximum fow can be derived using the agorithm proposed by Lun et a in [2], which aso gives the minimum s T cut as a byproduct. Specificay, given the configuration graph G(S) =(V,E) corresponding to transmission schedue S. Letg be the configuration rate of e E. Lets be the source node, T = {t 1,t 2,...} be the destination node set. We formuate the foowing optimization probem, with j, e E,j J, t k T, and R st being the variabes. maximize R st s.t. g j, e E,t k T j J j jii = σ (t k) i, {J e E} j J {j e ji E,i I} j 0, e E,j J, t k T (12) where R st if i = s σ (t k) i = R st if i = t k T (13) 0 otherwise As shown in [2], (12) is a convex optimization probem, and hence can be soved efficienty with a poynomia compexity in the number of nodes and the number of inks. The optima R st is the maximum achievabe throughput from s to T given the transmission schedue S. To achieve this muticast throughput, the actua fow over ink e equas j J x(t k) j. max tk T Consequenty, we can construct an auxiiary configuration graph G(S) from G(S) by assigning g max tk T j J x(t k) j as the configuration rate to e. A ink is removed if its configuration rate equas zero. Since R st equas the maximum muticast throughput, G(S) mustbe disconnected. Suppose in G(S), the node set V is divided into K 2 disjoint subsets {V 0,V 1,V 2,...,V K 1 } each beongs to one connected subgraph, but there is no ink connecting any of the two subgraphs. Assume V 0 contains the source node s and V k, k =1,...,K 1 each contains at east one destination node. Reca that a cut of the graph is defined as a partition that divides the node set V into two disjoint subsets V, V r. We can form a cut by assigning V k, k =0,...,K 1, to V or V r according to the foowing rues. Minimum Cut Formuation: Assign V 0 to V, i.e., V V 0. Choose an integer 1 m K 1. Assign V m to V r. For a 1 k K 1, k m, assign V k to either V or V r. Proposition 4: Any cut formed by the Minimum Cut Formuation is a minimum s T cut of graph G(S). Due to page imitations, we skip the demonstration that, based on the above minimum s T cut agorithm, we can aso obtain the first κ sma-vaued s T cuts with a poynomia compexity in the number of nodes and the number of inks. References [1] J. Wiesethier, G. Nguyen, and A. Ephremides, Energy-efficient Broadcast and Muticast Trees in Wireess Networks, Mobie Networks and Appications, Vo. 7, pp , [2] D.Lun,N.Ratnakar,M.Medard,R.Koetter,D.Karger,T.Ho, E. Ahmed, and F. Zhao, Minimum-cost Muticast over Coded Packet Networks, IEEE Trans. Inform. Theory, Vo. 52, pp , Jun [3] R. Etkin, D. Tse, and H. Wang, Gaussian Interference Channe Capacity to Within One Bit, submitted to IEEE Transactions on Information Theory. [4] R. Ahswede, N. Cai, S. Li, and R. Yeung, Network Information Fow, IEEE Trans. Inform. Theory, Vo. 46, pp , Ju [5] Y. Wu, M. Chiang, and S. 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