Wireless Network Coding and Node-Based Scheduling

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Wireless Network Coding and Node-Based Scheduling

M. Bama, Andrew Thangaraj, Srikrishna Bhashyam Department of Electrical Engineering,

Indian Institute of Technology, Madras, Chennai, India 600036 bama,andrew,[email protected]

Abstract— This paper addresses the extension of network coding to wireless networks in conjunction with medium access control based on conflict-free scheduling. In wireless networks, the maximum possible multicast throughput with network coding has been recently formulated as a linear programming optimization model that incorporates constraints due to interference between nodes. The model uses conflict graphs involving all possible hyperarcs in a graph representing the wireless network to determine an interference-free transmission schedule. Since all hyperarcs are involved, this model is not readily usable for large networks. In this work, we propose a Node-Based Scheduling (NBS) algorithm to select a small set of non-interfering hyperarcs based onreceiver selectivity. Our NBS based linear programming model achieves the same multicast throughput as that of the original model with a significant reduction in the number of variables and the number of constraints. To illustrate our results, we compare NBS - based network coding with optimal scheduled network-coding and routing over a random network and a rectangular grid network. In both cases, we observe that (1) NBS-based network coding achieves the same throughput as optimal network coding, (2) network coding provides about a factor of 2 improvement in throughput over routing, and (3) the complexity of the NBS-based network coding model is about an order of magnitude lesser than that of the optimal model. This suggests that suitably designed sub-optimal schedulers will suffice for realizing the gains of network coding in large regular and random wireless networks.

I. INTRODUCTION

Data communications between nodes of a network is a problem central to all modern communication systems. The flow of information between such networks of nodes has undergone a paradigm change with the recent advent of network coding as introduced in [1].

The multicast capacity of network coding for random networks has been studied in [2] for random wireline networks and random wireless networks. However, [2] does not account for interference between wireless links. Sagduyu et al in [3]

proposed a link scheduling based MAC with network coding, which improves the performance of lossless wireless networks in terms of throughput and energy efficiency over routing algorithms. It has been shown that the throughput optimization and network code design are separable. In [3], new wireless network codes have also been designed. Network codes for lossy networks is considered in [4], which shows that random network codes introduced by Ho et al in [5] work well for lossy networks also.

In [6], Parket alhave formulated an optimization model to

estimate wireless multicast throughput for an adhoc network.

The model considers losses in the network, interference- aware scheduling and broadcast nature of wireless nodes. Our approach follows the model in [6]. The novelty in our work is in the determination of the scheduling of transmissions for minimizing interference and contention. Our approach for the determination of non-interfering subgraphs is node-based, as opposed to the hyperarc-based approach in [6]. We find the sets of active hyperarcs, based on receiver selection ability, i.e., the ability of a Nodeito determine a subset of receivers from among its neighbors for each transmission. This results in a significant reduction in the complexity of optimization enabling determination of maximum multicast throughput in larger networks. Using the proposed Node Based Scheduling algorithm, we achieve the same multicast throughput performance as the optimization model in [6].

For a n node random network with a transmission range, r_n =

qβlogn

nπ , β ∈ Z⁺, the complexity of the optimal model is O(n^β+1) whereas the complexity of our model is O(n^β). While for a 10×3 rectangular grid network, our approach generates only 74 hyperarcs out of a total of 336 hyperarcs but achieves the maximum multicast throughput. In both type of networks, network coding throughput is compared with the multicast routing throughput proposed in [7] for a single multicast session. We observe that the network coding schemes have about a constant factor of 2 gain over the routing schemes. Hence, we remark that the efficient cross layer design of an interference-avoiding scheduler will be crucial in determining and realizing the multicast throughput gains of network coding in large random and regular wireless networks.

The pseudo-random NBS algorithm proposed in this work results in a tractable model for a random wireless network with a moderate number of nodes. Future work promises to extend the results to larger networks and provide asymptotic capacity gains of wireless network coding.

The rest of this paper is organized as follows. Section II describes the model of the wireless network. Proposed NBS algorithm along with the optimization model for network coding is explained in Section III. Complexity analysis for random networks is done in Section IV. The performance of the NBS algorithm on random networks and nr×3 grid networks is detailed in Section V. Section VI concludes the paper.

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II. MODEL OF WIRELESS NETWORK

Consider a wireless packet network consisting of nnodes.

We assume that all the nodes are identical i.e. they have uniform transmission power, transmission range, interference range and omnidirectional antennas. Such a network is modeled as a graph, G= (V, E), whereV ={1,2, . . . , n}is the set of vertices andE⊆ {(i, j) : i, j∈V}is the set of edges.

Each wireless node has inherent wireless multicast advan- tage i.e. each node has the capability for omnidirectional transmission, and can reach more than one receiver in its transmission range in a single hop. This is modeled as a hyperarc as in [6]. Notation (i, Ji) indicates that Node i reaches receivers in Ji (Ji⊂V) in a single hop. We consider a single multicast session with a source node, s ∈ V and multiple sinks{r1, r2. . . rm} ∈V.

III. NODE-BASED SCHEDULING ALGORITHM AND NETWORK CODING

Scheduling is an interference avoidance scheme which defines a maximal independent set (or a non-interfering subgraph) of non-interfering hyperarcs that can be active simul- taneously. We follow the protocol model of [8] to define non-interfering links. Slotted transmission is assumed so that one of the non-interfering subgraphs is activated at every slot boundary to transmit packets.

A. Selection of non-interfering subgraphs

The node-based scheduling (NBS) algorithm determines non-interfering subgraphs by selecting random nodes and performing receiver selection at each node. For a Node i, receiver selection is basically discarding of interference-prone receivers and retaining receivers that do not interfere with the previously selected nodes. The output of the NBS algorithm is a hyperarc set A and M non-interfering subgraphs A_k (1≤k≤M). The subgraphsA_kcan be used for simultaneous transmission in a network represented by a directed graph G = (V, E). We assume there are no incoming links to the source node,s∈V. LetJ_i={j∈V : (i, j)∈E}denote the neighbors of NodeiinG. Initialize the firstnnon-interfering subgraphs as follows:

A_k ={(k, J_k) :k∈V}, 1≤k≤n.

Consider each hyperarc of the source node {(s, K) :K⊆ J_s, K 6= ∅} as a virtual node in a new directed graph G⁰ = (V⁰, E⁰). Denoting the virtual node corresponding to a hyperarc(s, K)asv_K, the nodes and edges ofG⁰ are defined as

V⁰ ={V \s}[ ([

K

vK), E⁰= (E\ {(s, j)∈E})∪E₂⁰, where

E₂⁰ = [

K⊆Js

K6=∅

{(vK, k) :k∈K}.

In other words, the graph G⁰ is constructed by replacingsin G with the virtual nodes representing the hyperarcs from s

s

j1 j2 j1 j2

G⁰ G

v{j₁} v{j₂} v{j₁,j₂}

Fig. 1. Construction ofG⁰

as depicted in Fig. 1. We let d_v_i_j (inG⁰) =d_sj (inG) and d_v_i_v_j = 0.

We now present an algorithm that provides one pseudo- random non-interfering subgraph of the network. Let J_i⁰ = {j ∈V⁰ : (i, j)∈E⁰} denote the neighbors of Nodei inG⁰. LetRi={j ∈V⁰:dij ≤ transmission range}denote the set of nodes present in the transmission range of Node i. Note that Ji⊂Ri in a general graph G⁰.

1) Arrange the nodes in V⁰ in a random order. Initialize Ak ={(v, J_v⁰)}, where v ∈V⁰ is the first node in the random arrangement. SetT =v,N=J_v⁰, andi= 2.

The setT denotes the set of transmitting nodes selected inAk. The setN denotes the set of active neighbors of T inAk.

2) Select the i-th node m in the random arrangement for possible inclusion inAk. The setAkis altered under the following conditions:

a) If m∈N, then the selected node is not included inAk. Go to Step 3.

If m /∈N, proceed to (b) to determine a suitable non-interfering hyperarc emanating from m for possible inclusion inA_k.

b) Determine the set of neighbors ofmthat are within the transmission range of nodes inT i.e. compute D1={j ∈J_m⁰ :dtj ≤, trans. range for somet∈T}.

c) Determine the set of nodes inNthat are within the transmission range of Nodem asD2=Rm∩N d) If|D1| ≤1 and|D2| ≤1, then

• Let t ∈ T be the transmitter connected to the single node inD2.

• UpdateAk andN as, A_k = A_k\(t, J_t), A_k = A_k[

{(t, J_t\D₂),(m, J_m⁰ \D₁)}, N = {N\D2} ∪ {J_m⁰ \D1}.

In determining a suitable non-interfering hyperarc above, only a subset of nodes are selected as receivers for each newly added node. Nodes inD1 are not selected as receivers for node m, and nodes in D2 are not selected in N. For a Node m, ifJ_m⁰ =Rm, thenD1=D2

e) UpdateT =T∪ {m}.

3) If i <|V⁰|, incrementiby 1 and go to Step 2.

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The above algorithm is repeated several times to obtain distinct non-interfering subgraphs A₁,A₂,· · ·,A_M that are maximal in the sense that A_i *A_j. All distinct hyperarcs present in SM

k=1Ak are collected to form the hyperarc setA. The steps 2(b),(c),(d) select nodes that do not interfere significantly with nodes previously chosen. We know that the benefits of network coding can be improved by considering the broadcast nature of wireless nodes. We desire to make use of it and to keep the complexity of the optimization model low. So, we limit the size of D₁ andD₂ to 1. The inclusion of all hyperarcs from the source as virtual nodes results in a good chance for the selection of all hyperarcs from the source in the non-interfering subgraphs produced by the above algorithm. We have observed that the virtual nodes play a crucial role in obtaining maximum throughput.

B. Linear Programming Model

We now follow the linear programming formulation pre- sented in [6] for determining the maximum multicast throughput from a source to multiple sinks. As before, we assume that the wireless network is represented as a directed graph G = (V, E). We construct non-interfering subgraphs Ak

(1 ≤ k ≤ M) with hyperarcs from the set A as discussed in Section III-A. An important difference in our optimization when compared to [6] is that we consider only hyperarcs generated in the NBS algorithm.

Let z_iJ be the average broadcast rate at which packets are injected into the hyperarc (i, J)∈A. Let L packets per unit time be the link capacity. The unreliability of wireless links results in a packet loss in the link with probability q. If a packet is injected on to the hyperarc (i, J)at an average rate ziJ, it is shown in [4] that the packets reach a subsetK⊆Jat an average rateziJ Kgiven byziJ K =ziJ(1−q)^|K|q^(|J|−|K|), assuming losses in each link of a hyperarc are independent.

As in [6], we assume that the subgraph A_k is used in a particular slot of a slotted transmission schedule. Let λk be the fraction of time for which the subgraph Ak is active.

We denote the average multicast throughput as f, the set of destinations as T (T ⊂ V), and the information flow rate in link (i, j) of hyperarc (i, J) towards sink t ∈ T as x^(t)_{iJ j}. We maximize f over a rate vector z ∈ [0, L]^|A| with linear constraints. The optimization problem, which is the same as that of the model stated in Section II of [6], is given below

maximize f subject to X

k

λ_kc_k(i, J)−z_iJ ≥0, ∀(i, J)∈A.

X

k

λk≤1.

X

{L⊆J:L∩K6=∅}

ziJ L−X

j∈K

x^(t)_{iJ j} ≥0,

∀(i, J)∈A, K⊆J, t∈T.

X

{J:(i,J)∈A}

X

j∈J

x^(t)_{iJ j}− X

{j:(j,I)∈A,i∈I}

x^(t)_jIi=







f ifi=s

−f ifi=t 0 otherwise

,

∀i∈V, t∈T.

ziJ ≥0, x^(t)_{iJ j} ≥0, λk ≥0;

It is shown in [6] and [4] that a feasible solution of the above optimization problem always results in a valid network code. Once a feasible rate vectorz^∗ is determined, a network code can be designed, theoretically, to achieve the maximum multicast throughputf.

IV. COMPLEXITYANALYSIS

In this section, we analyze the complexity reduction ob- tained using the optimization model based on the NBS algorithm for a n node random network. Nodes are placed uniformly at random in an operational area independent of each other. Such a random network is modeled as a random geometric graph, G = (V, E) where V = {1,2, . . . , n} is the set of vertices and E = {(i, j) : dij ≤ rn} is the set of edges, where rn is the transmission range of a node in a n-node random network. For nnodes in a disk of area A, it is shown in [9], that the network is connected as n → ∞, if r_n =

q(logn+γ_n)A

πn , (γ_n → ∞ as n → ∞). We take

Constraint Name Number of Constraints Scheduling Constraints E[Nh(n)] + 1 Capacity Conservation

constraints for a lossless network E[N_h(n)]

Flow Conservation constraints n

Total number of constraints n+ 2E[Nh(n)]

Variable Name Number of Variables

Number ofλi M

Broadcast rate variables E[Nh(n)]

Number of flow variables

(for a hyperarc) logn

(for all sinks) |T|logn

(for all sinks and all hyperarcs) |T|lognE[Nh(n)]

Total number of variables M+E[Nh(n)](1 + |T|logn) TABLE I

AVERAGENUMBER OFCONSTRAINTS ANDVARIABLES

the area A = 1, the transmission range as r_n =

qβlogn πn

and the probability that any two nodes are connected as pn =πr_n² =β^log_nⁿ (ignoring the edge effects). We compare the complexity of the model in [6] and the NBS based model in terms of the number of constraints and variables used. In Table I, we have tabulated the average number of variables and constraints used in the optimization model of [6]. We notice these numbers are proportional to the expected number of hyperarcs for thennode random network. In our optimization model, we use hyperarcs generated in the NBS algorithm.

Hence, the complexity of our model can be found using Table I with the expected number of hyperarcs replaced by the number of hyperarcs generated by the NBS algorithm.

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5 10 15 20 25 30 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Number of nodes, n

Average Throughput

NBS based Opti. Model Original Opti. Model Routing

(a) Multicast throughput ofn-node random networks

5 10 15 20 25 30

10¹ 10² 10³ 10⁴

Number of nodes, n log(No of hyperarcs), log(E[Nh])

E[N_h] in a random n/w E[N_h] in NBS Derived bound n^1.5 curve fit

(b) Average number of hyperarcs inn-node random networks Fig. 2. Performance of NBS forn-node random networks

A. Expected number of hyperarcs of random networks Let d_i be the degree of Node i andN_hi(n) = 2^dⁱ−1 be the number of hyperarcs associated with Node i. It is known that for a random geometric graph, if the nodes are uniformly placed, the node degree distribution is binomial. i.e.

P(d_i=k) = n−1

k

p^k_n(1−p_n)^n−1−k

The expected number of hyperarcs for a n node random network is E[N_h(n)] =Pn

i=1E[N_hi(n)]. Since all the node degree distributions are identical, we see that

E[Nh(n)] = nE[Nh1(n)]

= n

n−1

X

k=0

n−1 k

p^k_n(1−pn)^n−1−k(2^k−1)

= n (1 +pn)ⁿ⁻¹−1

≤ n(exp(pn(n−1))−1) (Using1 +pn ≤e^pⁿ)

≤ O(n^β+1) (Usingpn=βlogn n )

SinceE[N_h(n)]isO(n^β+1), the number of constraints and the number of variables also grow asO(n^β+1)withnin [6].

In the next section, we will estimate E[N_h(n)] for the NBS algorithm.

B. Expected Number of hyperarcs in the NBS algorithm In the NBS algorithm, each subgraph is a set of non- interfering hyperarcs. We bound the expected number of hyperarcs generated in NBS, using the following observations made on the values ofNhi(n)in simulations.

• Observation 1:Number of hyperarcs associated with the source node (Node 1) isN_h1(n) = 2^d¹−1

In the NBS algorithm, we introduce2^d¹−1virtual nodes for each hyperarc of the source. Since the number of nodes inG⁰is|V⁰|=n+2^d¹−1, the expected number of virtual nodes and total nodes inG⁰ aren^βandn+n^β−1 respectively. So, the probability that a subgraph Ak is initialized with a virtual node is very high asnincreases.

• Observation 2: Number of hyperarcs associated with nodes other than source node is Pn

i=2Nhi(n) ≤ c1Pn

i=2di. This was true for c1 = 2 in all our simulations.

Therefore,

N_h(n) ≤ 2^d¹−1 +

n

X

i=2

c₁d_i

E[N_h(n)] ≤ E[2^d¹−1] +

n

X

i=2

c₁E[d_i]

≤ (1 +pn)ⁿ⁻¹+c1(n−1)βlogn

≤ exp(p_n(n−1)) +c₂(n−1) logn

≤ n^β+c2(n−1) logn

≤ O(n^β)

In the above equations, c₁ andc₂ are constants. Therefore, we see that our approach reduces the complexity by a factor of n. Note that the above two observations can be enforced by modifying the NBS algorithm, if desired.

V. RESULTS ANDDISCUSSION

To evaluate the NBS algorithm, we performed simulations over two types of networks (a) random networks withnnodes and (b) rectangular grid networks with a n_r×3 array of nodes. The multicast throughput and complexity of NBS are compared with optimal routing [7] and network coding [6].

A. Random networks

We consider an unit square region wherennodes are placed uniformly at random independent of each other. Each node has a transmission rangernas described in Section IV withβ = 2.

Number of nodes is varied fromn= 6ton= 30in steps of2.

Nodes are labelled from1ton. Node1is taken as the source node. We consider two randomly chosen receivers among the remaining n−1 nodes for multicast. For each n, the NBS algorithm is run 3000 times to find non-interfering subgraphs A_k. The multicast throughput is averaged over 500 different network realizations and over all possible choices of sink for each realization.

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Sink2 Sink3 Sink1

Source

(a) An example - 4 x 3 grid network

3 4 5 6 7 8 9 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Number of rows, n_r

Average Multicast Throughput

NBS based Opti. Model Original Opti. Model Routing

(b) Multicast throughput ofnr×3grid networks

3 4 5 6 7 8 9 10

10¹ 10² 10³

Number of rows, n_r

log (Number of hyperarcs)

NBS based Opti. Model Original Opti. Model

(c) Average number of hyperarcs innr×3grid networks Fig. 3. Performance of NBS fornr×3grid networks

It is compared with the optimization model in [6] and with the multicast routing.Multicast routing throughput is found based on the technique proposed in [7]. From Fig. 2(a), we observe that the throughput performance of our model is almost identical to the optimization model in [6]. We also notice that the network coding throughput is always higher than the routing throughput. Infact, we obtain a gain factor around 2.

From Section IV, we know that the average number of hyperarcs for an-node random network isn³and the average number of hyperarcs generated in the NBS algorithm is bounded byn². In Fig. 2(b), the average number of hyperarcs is plotted against the number of nodes for the two optimization models. The estimated upper bound is also shown in the figure. We notice our algorithm has a significant reduction in complexity. Indeed, the closest curve fit for the average number of hyperarcs generated using NBS is n^1.5. If we include the edge effects in the nodes coverage area, the probability any two randomly chosen nodes are connected,will be,pn≤πr²_n=c3πr²_n,0.25≤c3≤1. Therefore, the average number of hyperarcs of a n node random network is n^c³^β+1 and the average number of hyperarcs generated in the NBS algorithm is bounded byn^c³^β. Edge effects become significant when the number of nodes in a network is small.

B. Rectangular grid networks

As a next step, we describe the multicast throughput results on a n_r×3 rectangular grid network. The j-th node of the i-th row is placed at the coordinates(i, j)for i= 1,2, . . . , n and j = 1,2,3. The nodes are identical with a transmission range, r= 1. The number of rows nris varied from 3 to10.

The center node in the first row is considered as the source node. The three nodes in the nr-th row are taken to be the sinks. For nr= 4, the grid is shown in Fig. 3(a).

In Fig. 3(b), we present results regarding the performance of our proposed node-based interference-free subgraph selection algorithm assuming that the link error probability qequals 0.

The subgraph selection algorithm is run for 500 times. The maximum multicast throughput for pure routing schemes is 1/3 for all values of nr. With network coding, a multicast

throughput of 2/3 can be readily achieved for all values ofnr

by using flows down the first and the third column of the grid.

Fig. 3(c) shows the number of hyperarcs of a grid network used in the two optimization models. From the figure, we observe that there is a significant decrease in the number of variables and the number of constraints used in the optimization process.

VI. CONCLUSION

In this paper, we have considered the problem of reducing the complexity of optimization in determining the maximum multicast throughput in large wireless networks. We proposed a node-based scheduling algorithm which results in a significant reduction in variables and constraints used in the optimization model. We have analyzed the performance of our algorithm for random networks and rectangular grid networks and shown that our simpler algorithm results in almost the same multicast throughput as the optimization model in [6]

with significantly lower complexity.The simplicity of our proposed optimization model promises possible determination and realization of capacity of network coded multicast in large wireless networks.

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