On optimizing ow SNR wireess networks using network coding Mohit Thakur Institute for communications engineering, Technische Universität München, 80290, München, Germany. Emai: mohit.thakur@tum.de Murie Médard Research Laboratory for Eectronics, Deparent of Eectrica Engineering & Computer Science, MIT, Cambridge, MA, USA. Emai: medard@mit.edu Abstract The rate optimization for wireess networks with ow SNR is investigated. Whie the capacity in the imit of disappearing SNR per degree of freedom is known to be inear in SNR for fading and non-fading channes, we study the probem of operating in ow SNR wireess network with given node ocations that use network coding over fows. The mode we deveop for ow SNR physicay degraded broadcast channe and mutipe access channe respectivey operates in a non-trivia feasibe rate region. We show that the probem reduces to the optimization of tota network power which can be casted as standard inear muti-commodity min-cost fow program with no inherent combinatoriay difficut structure when network coding is used with non integer constraints (which is a reasonabe assumption. This is essentiay due to the inearity of the capacity with respect to vanishing SNR which heps avoid the effect of interference in a ow SNR physicay degraded broadcast channe and mutipe access environment respectivey. We propose a fuy decentraized Prima-Dua Subgradient Agorithm for achieving optima rates on each subgraph (i.e. hyperarcs of the network to support the set of traffic demands (muticast/unicast connections. Index Terms - Low SNR degraded broadcast channe, network coding, rate optimization, Prima-Dua Subgradient Method. I. INTRODUCTION Wideband fading channes have been studied since the eary 1960 s. Kennedy showed that for the Rayeigh fading channe at the infinite bandwidth imit, the capacity is simiar to the capacity of the infinite bandwidth AWGN channe with the same average received power [1, 2]. The robustness of this resut in the case of with or without channe state information heps us to mode generay the ow SNR wideband wireess networks. Using it as our underying information-theoretic mode to approximate the capacity over a ink, we mode the genera traffic (network demands for this network and show that the inearity of vanishing SNR in SNR per degree of freedom makes for the fundamenta reason for simpicity in our mode. Hence, we caim, it is possibe to do networking over such mode with simpistic and essentiay inear approach. In the context of wideband mutipath reay channe it is shown in [3] that the min-cut coud be achieved using a noncoherent peaky frequency binning scheme. In our mode the rate tupes beong to a non-trivia feasibe region which is made of the convex hu of a tupes, i.e. capacity achieving in the imit of vanishing SNR, for subsequenty defined ow SNR physicay degraded broadcast and mutipe access channe. The traffic mode we use is quite genera. It is divided into two casses: unicast and muticast (broadcast is considered as a specia case of muticast. Where each pair of source and receiver group in the network form a session for a particuar cass of traffic. But the probem of successfuy estabishing muticast connections in wireine or wireess networks has been ong thought to be NP-Compete using arbitrary directed and undirected network modes. With the advent of network coding (ref., [4], [5], [6], the breaking of the fuid mode for data networks i.e. by performing coding over incoming packets, has been abe to intrinsicay circumvent the combinatoria hardness of the muticast fow probem. Later, it was shown that minimum cost setting up of muticast connections bois down to optimizing subgraph over coded packet networks [7]. In our probem, since we consider a ow SNR wireess network with physicay degraded Gaussian broadcast channe (where the number of hyperarcs is equa to n for n receiver nodes, instead of 2 n hyperarcs to optimize the rates over each hyperarc (subgraph to meet the network traffic demands (which we ater show can be casted as a minimum cost muticommodity fow probem for optimizing power over each hyperarc. Aso, we consider intra-session network coding for estabishing traffic demand sessions. This paper is organized as foows. Section II is composed of genera probem formuation, where we define and deveop our underying information theoretic set-up. Section III consists of a proposa with decentraized soution. We present our resuts in section IV that support our theory. Finay, we mention concuding remarks in section IV. II. SET-UP AND PROBLEM FORMULATION In this section we introduce a genera ow SNR channe mode (Gaussian and extend it to physicay degraded Gaussian broadcast channe, then addressing the interference issue in mutipe access. This approach becames our underying mode and we further deveop it to a simpe networking mode. A. Low SNR physicay degraded Gaussian broadcast channe. Consider a genera wideband fading channe where the input waveform is x and the output waveform is y, the fading coefficient matrix is given by h and n is the additive white noise. The channe is given by: y = SNR hx + n. (1
Fig. 1. (a: Two receiver physicay degraded Gaussian broadcast channe with Z 1 N(0, N 1 and Z 2 N(0, N 2 N 1. (b: Rate region for the channe in (a, dotted ine denotes the fatness of the rate region curve in the imit of vanishing SNR with C 1 and C 2 as max rates for each receiver respectivey. (c: Decomposition into hyperarcs {(s, (d 1, (s, (d 1, d 2 } with their common rates for the case in (a with receivers d 1 and d 2 (corresponding to better and worse respectivey. The capacity of the channe, for both Gaussian channes and fading channes increase subineary with the increase in signa to noise ratio (SNR but in the ow SNR regime the capacity in the imit is inear in SNR for fading and non-fading channes: C(SNR = SNR + o(snr(nats/s/hz. (2 Ceary at ow SNR imit, the signa-to-noise ratio per degree of freedom (SNR approaches unity in the imit [2, 8, 9]. We consider ow SNR physicay degraded Gaussian broadcast channe, et s ook at the standard mode of a singe sender and 2 receivers with noise variances N 1 and N 2 respectivey (ref. Fig 1(a. The capacity region is given by: r 1 < C( λ 1P (λ 2 P, r 2 < C(. (3 N 1 λ 1 P + N 2 where C(x = W(n(1 + x, λ 1 + λ 2 = 1 and s the tota power (ref. Fig. 1(b. As the channe in consideration is a degraded broadcast channe, the high-resoution receiver (in this case r 1 aways get enough information to decode for the second receiver and then cances it out from the received signa to decode information for itsef. The rate region defined in (3, when ooked under the ow SNR ens comes across as a rather simper picture. For the power imited ow SNR regime, the effect of the power aocated for the better receiver, as the contribution to the tota noise experienced by the worse receiver is negigibe (ref. Fig. 1(b, for the rate region for ow SNR in the imit. So, for the ow SNR physicay degraded Gaussian broadcast channe, the rate for the worst receiver can be approximated as r 2 C( λ 2P N 2. (4 Generaizing the same idea for the case of a given source i with power and n receiver nodes, where the receiver set J = (1,..., n can be broken into n subsets as J k = (1, 2,..., k for k [1, n]. The rate region defined for each hyperarc (i, J k in the ow SNR imit is given as r ij k C( (λ k, k [1, n] N 2 (5 (λ k, k [1, n]. L i L k α/2 2 N 2 (6 where, n λ k 1, which when combined appropriatey gives k=1 the rate region of the set J k. The equation (6 comes from the fact that the SNR is inear in the imit of disappearing SNR per degree of freedom, where L i for a i [1, n] is the ocation of the node and α is the path oss exponent. We formaize the above mentioned concepts and motivate our next definition. Let λ k = J k, k. Definition. 1: For a given sender i with tota power and a receiver set J = [1, K] in ow SNR physicay degraded Gaussian broadcast channe, the set J can be decomposed into K hyperarcs where each hyperarc is defined as the connection from the sender i to the receiver set J k = [1, k], where k [1, K], with individua receiver rates (r ij k, J k equa to an associated common rate (r ij k of this hyperarc. The rate for each receiver n the hyperarc is defined as r ij k = r ij k = J k L i L α/2 2 N 2, where, k J k, k J and the set J k ranges from best to worst receiver (ref. Fig 1 (c. B. Interference issues in mutipe access at ow SNR. Now, et s consider the case of mutipe access where more than one node tries to access the channe at the given instance. Let there be U nodes in the system at an instance, and u U of them are trying to access the channe at this instance, if node i u intends to communicate with node j U among others in u, the signa to interference and noise ratio (SINR, denoted as µ ij experienced at node j is given by: µ ij = 2N 0 + v u,v i L i L j α/2. (7 P v L v L j α/2 2N 0 Note that, since every node in u is interested ony in a common receiver, we aocate the whoe power of the node over this singe hyperarc, so k = 1 and J 1 = for every transmitter. But as we are operating in the ow SNR regime, the intuition suggests that the effect of the interference shoud be negigibe. We straightforwardy incude it in our assumption, thus we define the rate (denoted with R experienced at the receiver j as: ( L R ij = Wn 1 + i L j α/2 2N 0 + (8 P v L v L j α/2 2N 0 v u,v i Wn ( 1 + L i L j α/2 2N 0 (9 W (. L i L j α/2 2N 0 (10 The approximation (9 comes from the fact that the contribution of other signas being transmitted from other sources in the system with ow SNR channe to the interference is negigibe and the approximation (10 comes from the inearity of SNR in the imit of disappearing SNR per degree of freedom (ref. Fig 2(a and 2(c. In Fig. 2(b, we can see that the SNR curve approaches the capacity curve in the imit, corroborating
probem. For that, we define another graph G = (N, A, which is simpy the equivaent directed graph of G = (N, A with arcs instead of hyperarcs. This graph can be easiy obtained by decomposing the hypergraph appropriatey. Let s define the term (ref. [7] for detaied notation expanation: x ij = ((i,j A J x. (11 which simpy describes the way to add a the fow entering a node on a incoming hyperarcs, corresponding to the graph Fig. 2. (a: Two sender case for the ow SNR mutipe access channe, where Z N(0, N. (b: Rate region for case in (a, the dotted ine denotes the respective SNR s µ 1 and µ 2 for two senders and the arrow shows that in imit of disappearing SNR, the SNR curve touches the capacity curve. (c: As the effect of interference is negigibe, the case is (a can be approximated as individua hyperarcs. our assumption that the SNR equas capacity in the imit of disappearing SNR per degree of freedom. We woud ike to point out here that in [3], it was shown by the authors that with the hyperarc mode the non-coherent peaky frequency binning scheme is capacity achieving for the mutipath fading reay channe in the imit of arge bandwidth for ow SNR regime, hence achieving the min-cut. C. Low SNR network rate optimization. Let us represent the wireess network as a directed hypergraph G = (N, A, where N is the set of nodes and A is the set of hyperarcs, where each hyperarc emanates from a node and a terminates at a group of nodes, which we aso refer to as the broadcast group of the hyperarc. A hyperarc represents a subgraph which when combined in the appropriate way resuts in the origina hypergraph of the probem. Note that we consider muticast in our muticommodity fow optimization mode (as opposed to ony unicast, thanks to network coding. It s important to note that the common rate associated with each hyperarc r ij k, is the capacity of the hyperarc, beacause this is the rate that can be guarantied to a the receivers for a given transmit power. Aso (as defined in the previous subsection, r ij k, is a nonnegative function of the transmit power J k of the hyperarc (i, J k. Now that we consider a network with more than one sender, update of notations is required. For a sender i N, that is capabe of reaching (1 i,.., K i nodes, where each N \i, the K hyperarcs are denoted by (i, J ki, (1 i,.., K i. Imagine a set of traffic demands where m = 1,..., M sessions need to be estabished, each with t m = 1,..., T m set of receivers, in a given wireess network that experiences ow SNR channe physicay degraded Gaussian broadcast channe and that is represented by the hypergraph G = (N, A. We know from the definition of hyperarc that a singe node can ie on mutipe hyperarcs, therefore, we need a way to carefuy count the incoming information and outgoing information to appy the aw of fow conservation to the hypergraph and finay be abe to cast the probem as a fow optimization G = (N, A. Notice that x ij defined in the previous section, x ij is not the same as r J i can be interpreted as the fow between i and receiver of the hyperarc J ki, and it cannot exceed the common rate (r associated to the hyperarc which is aso the hyperarc capacity, for each. P L i L ki α/2 2N 0 Let, r = = γ. Then, the minimum cost optimization probem for the ow SNR network can be formuated as: minimize (A (i,j A y (m max( t m ij (m, (i, A, m (12 Jki M z = y (m, (i, J ki A (13 z γ ij, (i, Jki A (14 K i =1 i, i N. (15 where is given i, (m F (m, and F (m a J J k bounded poyhedron made of fow conservation constraints: (m (m = s i(m, i (J (i,j A i N, t m, m (m = (J (J,i A (J J ( A J (m, (i, J ki A, t m, m (16 (17 (m 0, (i, J ki A, m, t m [1, T m ]. (18 As opposed to standard muticommodity fow probem in which fows are simpy added over a ink, the constraint (12 in fact catches the essence of network coding by taking ony the maximum among a the fows of a session (note that we ony consider intra-session network coding. Since F (m is the poyhedron formed by the aw of fow conservation, constraint (17 transates the fow conservation aws from the underying directed graph A to the hypergraph A (the wireess network by adding the fows on a hyperarcs between node i and J ki j i.e. fow in (i, J ki j A is the sum of a the fows on the hyperarcs (i, J ki, J ki J ki j.
As we can see, the above mentioned probem is a convex optimization probem. The ony noninear constraint is (12, and coud be readiy repaced by the set of inear inequaity constraints y (m ( ij (m, t m [1, T m ]. The modified probem resuts in a standard inear muticommodity fow probem with inear objective and inear constraint set. minimize (i,j A (B y (m ( ij (m, t m, m, (i, J ki A (19 M z = y (m, (i, J ki A (20 z γ ij, (i, Jki A (21 K i =1 i, i N. (22 where (m F (m, and F (m is a bounded poyhedron made of fow conservation constraints. Note that we optimize the power over each hyperarc, to determine the optima rates for each hyperarc that satisfies the network demands, we simpy need to mutipy the optima power with γ. We wi prefer to sove the probem by proposing a decentraized agorithm for generay understood and appreciated reasons. III. DECENTRALIZED ALGORITHM For deveoping a decentraized soution for probem (B we need to understand the structure of the prima probem first and transform it into a separabe form. We know that the objective function is a inear and increasing in its domain and so are the constraints. Taking the Lagrangian dua of the probem (B we get the dua optimization probem as: maximize ( ζ i (C (i,j Aq ij ki + i N where, q = q (λ, ν, µ, ζ, x, y, z, P = min M t (m F (m T m (λ (λ, µ 0 (23 ( + (m(x ij (m y (m+ M ν ( y (m z + µ (z γ ij + ζ i. (24 The dua probem is ceary hyperarc separabe and coud be soved in a decentraized manner. But the dua probem is not differentiabe at a the points in the dua domain, this is due to the fact that there might not be a unique minimizer of q for every dua point as the objective function is a minimum over sum of inear functions for fixed dua variabes. To sove the dua probem (C, we need to sove its subprobem (24. The subprobem (24 (and the dua probem (C coud be soved with a ot of techniques, [10, Chapters 8-10], [11-Chapters 5-6, 12-Chapters 6] using some subgradient based technique but they do not necessariy yied the prima soution (which is of our interest here. There are however, methods for recovering prima soutions from the dua optimizers. We wi take a different technique than the above mentioned approaches but before ets oonto some inter-dependence characteristics of the dua and prima probem structures. Simpy having convex prima probem in hand does not guarantee strong duaity, but with some constraint quaifications we can assert that strong duaity hods or not. One such simpe constraint quaification technique is caed Sater s condition. In our case it can be easiy seen for constraint (12 (or 19 of probem (A (or (B, there exist a vector { (m} for which the inequaity can be strict. Let us represent the set of prima vectors as p = {x,y,z,p} S 1 where S 1 is the feasibe set for the prima probem, and simiary we can do it for the dua probem, d = {λ, ν, µ, ζ} S 2. As we can see that the prima and dua optima are equa (thanks to strong duaity, we can express our probem in the standard sadde point form max min φ(p,d = d S 2 p S 1 min maxφ(p,d, where function φ is the Lagrangian dua p S 1 d S 2 of the probem (B. This impies that for (C, we get the hyperarc separabe sadde-point form maxq = min maxφ(p,d. (25 d S 2 p S 1 d S 2 Now we are in the position where we can sove the probem, separabe in hyperarcs using any sadde-point optimization method for non-smooth functions. For our probem set up, we propose a Prima-Dua Subgradient Agorithm by Nesterov for nonsmooth optimization [ref. 13]. Nesterov s method generates a subgradient scheme inteigenty based on Dua-Averaging method which beats the ower case compexity bound for any back-box subgradient scheme. The agorithm works in both prima and dua spaces, generating a sequence of feasibe points, and utimatey squeezing the duaity gap to zero by finay approaching the optima soution. A positive consequence of the Prima-Dua approach is that at each iteration we get a pair of points (p,d which are prima and dua feasibe, hence, we get the prima feasibe soution with essentiay no extra effort. As opposed to many subgradient type methods where there needs to be a method for prima recovery, speciay for arge and i-posed probems. A. Prima-Dua Subgradient Agorithm. Since the dua function is hyperarc separabe, we can optimize the power over each hyperarc separatey and add each of the optima soutions to construct the optima soution of the dua probem (C, utimatey achieving the prima optima soution for probem (B. The agorithm is as foows: 1 Initiaization: Set s 0 = 0 Q. Choose θ > 0. 2 Iteration (k 0:
Compute g k = φ(p k, d k. Choose σ k > 0 and set s k+1 = s k + g k. Choose θ k+1 θ k Set y k+1 = θ k S k arg max x Q ( k i=0 σ i g(y i, y i y where (g p, g d is the set of prima and dua subgradients and σ k, s k and S k are aggregated sequence of points. IV. SIMULATIONS We now show the resuts of our simuations that support the caims of the agorithm presented. We soved the dua probem in a decentraized way by soving it for every hyperarc separatey and then adding up the respective soutions to construct the dua optima soution of the probem (C, which when optima is the prima optima soution for probem (A in our case. The setup consists of uniformy paced nodes on a chosen area of a a m 2, with given node ocations. We start our simuations with smaer networks of ony 4 nodes on a 10 10 m 2 area with the area size increasing as the number of nodes in the networncrease to keep the node density/area in a controed range. Each node has a singe hyperarc and it can communicate with a the nodes in the network, this is just a simpe generaization of our case where a node can communicate with ony a subset of tota nodes in the network. For each network we randomy choose a set of m muticast sessions and T m set of receivers for each session respectivey with the required rate demand associated with each session that need to be estabished, but making sure the the traffic demands are the respective min-cut for each session to make the probem feasibe. In Figure 3, we compare the optima soution approximations of the Prima-Dua Subgradient Method for probem (C with the standard infeasibe path foowing method for probem (B. It can be seen that the our proposed agorithm gives cose approximations of the prima soution of the probem (B. Note that the path foowing method is directy appied to the prima probem and the Prima-Dua subgradient method is appied to the dua probem, to compute the dua soution of the probem (C, which wi be give us the cose approximation to the prima soution of probem (A. V. CONCLUSION We deveop an efficient optimization mode that provides an achievabe rate region. And we do this by showing that rate optimization for the Low SNR physicay degraded broadcast wireess network can be formuated as a standard inear muticommodity fow probem for optimizing power over each hyperarc using network coding. Our mode is reieved from interference reated issues, this is due to the fact that the capacity of the ow SNR wideband channe is essentiay inear in SNR per degree of freedom for vanishing SNR in the imit, which reieves the system from interference and reated issues. Our mode operates in the non-trivia feasibe rate region that achieves capacity in the imit of disappearing SNR with appropriate encoding scheme. Fig. 3. Y-axis denotes 2 items, optima prima costs computed using the infeasibe path foowing method when appied directy to prima probem (B and the optima cost for the dua (prima optima soution to (B when Prima-Dua Subgradient Agorithm is appied to the dua probem (C. We use a prima-dua agorithm to construct a decentraized soution for soving the probem, which has apparent advantages for recovering the prima soution than standard projected subgradient methods. In the simuation resuts shown, we don t present the gains of routing using network coding over simpe routing. But there is aready a vast iterature estabishing this fact. Finay, we beieve that reaizing ow SNR networks is a worthwhie attempt as the inearity of SNR in the imit 0 provides a fundamenta simpicity for networking to be done. Insights revea interesting and promising work coud be buid up and bended with our simpe mode (e.g. mobiity, reiabiity etc, which remains to be expored in this scenario. REFERENCES [1] R. S. 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