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Abstract—A joint routing and random network coding ap- proach is introduced for the multi-session networks. With the proposed approach, part of the ...
Joint Routing and Random Network Coding for Multi-session Networks Xiaoli Xu

Yong Liang Guan

School of Electrical and Electronic Engineering Nanyang Technological University Singapore 639801 Email: [email protected]

School of Electrical and Electronic Engineering Nanyang Technological University Singapore 639801 Email: [email protected]

Abstract—A joint routing and random network coding approach is introduced for the multi-session networks. With the proposed approach, part of the information is transmitted using routing, to reduce the inter-session interference, and the rest of the information is randomly combined at the intermediate nodes with proper precoding at the source nodes. The path used for routing and the corresponding routing rate can be determined using simple Fourier-Motzkin eliminations. It can be proved that proposed scheme achieves higher rate region compared with either pure routing or random network coding.

I. I NTRODUCTION Traditionally, network messages are viewed as physical commodities and are routed throughout the network without alteration. However, the emerging field of network coding views the messages as information which can be coded at the intermediate nodes. With network coding, the admissible rate region of the network may be enlarged. For single-session networks, it is well known that the capacity region is given by the celebrated max-flow min-cut theorem [1], and it can be achieved by simple random linear network coding [2]. However, for general multi-session network problems, characterizing the network coding capacity region remains open and it is not yet known how to construct optimal network codes. Some suboptimal but practical inter-session network codes has been introduced in the literature, such as the poison-antidote approach [3] and the pair-wise network coding approach [4]. Among all the suboptimal communication schemes for multi-session networks, routing and random network coding can be viewed as two extremes. For routing, the paths from the source to the corresponding receivers are determined first and then the information is forwarded along the selected paths. Therefore, there is no interference across sessions. Routing is known to be sufficient to achieve the capacity of the singleunicast networks and also proven to be optimal for those networks containing single source and multiple receivers with disjoint demands [5]. On the other hand, random network coding allows each node to randomly mix the incoming data and the information for different sessions are mixed together to the largest extend. By carefully precoding at the source, part of the inter-session interference can be removed. However, the rest of the interference must be decoded at the receivers. Besides the single-session networks, random network coding with proper precoding is also optimal for the networks with

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single source and two receivers [6] [7] and the 3-user degraded 2-message multicast over the combination networks [8]. In this paper, we introduce a Joint Routing and Random Network Coding (J-R-RNC) approach for the multi-session networks which achieves higher rate region compared with either pure routing or random network coding. With proposed JR-RNC, part of the information is routed to the corresponding receivers to reduce the inter-session interference and the rest is transmitted over the network with random linear network coding. The routing paths and the corresponding routing rates are optimized using Fourier-Motzkin elimination to maximize the overall achievable region. The main idea is illustrated with two special classes of the mutli-session networks: the degraded 2-message multicast networks and the 2-unicast networks, and it can be simply extended to any multi-session networks where an achievable region with random network coding can be obtained. The rest of this paper is organized as follows. Section II introduces the network model and the associated notations. In Section III, the idea of J-R-RNC is introduced with the degraded 2-message multicast networks and the 2-unicast networks. The conclusions are given in Section IV. II. N ETWORK M ODEL A noise-free network is represented by a directed acyclic graph G = (V, E), , where V is the set of nodes and E is the set of edges. Let S ⊂ V denote the set of source nodes that generate independent messages and T ⊂ V denote the set of receiver nodes where the messages are desired. For an edge e = (u, v) ∈ E, define tail(e) = u, head(e) = v and the capacity of edge e ∈ E is denoted by Ce . A path in a network from the node u to node v is represented a set of connected edges that start from node u and end at node v. For notational simplicity, the subset of the source nodes {si , i ∈ I}, where I ⊆ {1, 2, ..., |S|}, is denoted as sI . Similarly, tJ  {tj , j ∈ J}, where J ⊆ {1, 2, ..., |T |}. Moreover, denote the cuts between sI and tJ by CI−J and thus the min-cut value is defined as kI−J = minCI−J |CI−J |. If there is only a single source in the network, the index for the source can be omitted and the min-cut value from the source to receiver tJ is denoted by kJ .

ICON 2013

III. J OINT ROUTING AND R ANDOM N ETWORK C ODING In this section, we propose a joint routing and random network coding (J-R-RNC) scheme for the multi-session networks. The main idea is illustrated using two classes of networks, the 3-user degraded 2-message multicast networks and the 2-unicast networks. It can be shown that the achievable region with the proposed scheme is strictly larger than that achieved by pure routing or random network coding. A. J-R-RNC for the degraded 2-message multicast networks In this section, we consider the 3-user degraded 2-message multicast problem, as shown in Fig. 1, where a common source sends two messages X1 and X2 to three receivers, denoted as t1 , t2 and t3 , at rate R1 and R2 , respectively. The first two receivers, known as “public receivers”, want to recover message X1 , while the third one would like to recover both messages. This model is motivated by the application of multicasting real-time video to two sets of receivers of different level of priority. This problem has been investigated in [9] where an outer bound of the capacity region has been presented. They propose a coding scheme which is claimed to achieved this outer bound. However, the network that will be presented in Example 1 is the counter-example which shows that their outer bound is actually not tight in general and thus may not be achievable.

If the field size used for the random network = coding is large enough, rank(Hj )  we have   T T T H H kj , j = 1, 2, 3, rank = k12 , and 1 2    T T T T = k123 , where k12 is the rank H1 H2 H3 min-cut values between the source s and receivers {t1 , t2 } and k123 is the min-cut value between the source and all the receivers. Therefore, with random network coding, we can obtain an achievable region for this network in terms of its min-cut values, which is given by R1 ≤ min {k1 , k2 } R1 + R 2 ≤ k 3

(1)

2R1 + R2 ≤ k1 + k2 + k123 − k12 . However, the achievable region given in (1) is not optimal in general. This is mainly because, with random network coding, the inter-session interference is maximized, which is undesired. Note that since X2 is demanded only by the third receiver, it is viewed as interference at the other two receivers. Therefore, the achievable region may be enlarged if we can reduce the inter-session interference caused by X2 to t1 and t2 , which can be achieved by sending part of X2 to t3 by routing. Assume that there are totally q paths from s to t3 , denoted as P1 , ..., Pq , and the routing rate assigned for path Pi is denoted by αi , for i = 1, ..., q. Note that the routing rate assignment is feasible if and only if it satisfies αi ≥ 0, i = 1, ..., q  αi ≤ Ce , ∀e ∈ E,

(2) (3)

{i:e∈Pi }

where the second constraint given in (3) follows from the capacity of edge e. After subtracting the amount of resource used for routing, we may re-label the capacity of each edge in G by  Ce = Ce − αi , ∀e ∈ E. (4) {i:e∈Pi }

Fig. 1.

The three-user degraded two-message multicast network.

If all the intermediate nodes in the network performing random linear network coding, the problem reduces to degraded broadcast networks over linear deterministic channel. Denote the transfer matrix from source s to receiver ti by Hi , for i = 1, 2, 3. It has be shown in [10] that the capacity of this linear deterministic channel is given by R1 ≤ min {rank(H1 ), rank(H2 )} R1 + R2 ≤rank(H3 )

⎤⎞ H1 2R1 + R2 ≤rank(H1 ) + rank(H2 ) + rank ⎝⎣H2 ⎦⎠ H3

H1 . − rank H2 ⎛⎡

Denote the resulted network by G and the min-cut from the source s to receiver tJ in G by kJ . The rate pair (R1 , R2 ) is achievable in G by applying random network coding with proper precoding if it satisfies R1 ≤ min {k1 , k2 } R1 + R2 ≤ k3

(5)

  2R1 + R2 ≤ k1 + k2 + k123 − k12 .

By including the message sent by routing, the overall achievable region is thus given by R1 ≤ min {k1 , k2 } q   R1 + R 2 ≤ k 3 + αi i=1   2R1 + R2 ≤ k1 + k2 + k123 − k12 +

(6) q  i=1

αi .

With Fourier-Motzkin elimination, the achievable region given in (6) can be maximized subject to the constraints given in (2) and (3). Example 1: Consider the 3-user degraded 2-message multicast network shown in Fig. 2, where each edge in the network has unit capacity. Since k1 = 1, k2 = 2 and k3 = k12 = k123 = 3 for this network, the achievable region by random network coding is R1 ≤ 1 2R1 + R2 ≤ 3.

(7)

Fig. 2. (a)A 3-user degraded 2-message multicast network example.(b)The achievable region by random network coding and J-R-RNC.

To obtain an achievable region by J-R-RNC, we first list all paths from s to t3 , which are P1 = {e1 , e10 } P2 = {e1 , e4 , e11 , e13 }

R1 ≤ 1 3R1 + 2R2 ≤ 6.

(10)

Specifically, the rate pair (R1 , R2 ) = (1, 23 ) is achieved with (α1 , α2 , α3 , α4 , α5 ) = (0, 0, 0, 12 , 0), i.e., we first route X2 from s to t3 via P4 at the rate of 12 and then applying random network coding with proper precoding on the rest of the network to achieve (R1 , R2 ) = (1, 1). Note that the achievable region by proposed J-R-RNC given in (10) actually matches the capacity region of this network and is thus optimal. Lemma 1: The new achievable region specified by (2),(3) and (6) is no smaller than that achieved by random network coding in (1) and that achieved by pure routing. Proof: Note that random network coding is a special setting of the proposed J-R-RNC by choosing αi = 0, i = 1, ..., q. Since the proposed scheme optimize the achievable region over all feasible αi , it is guaranteed to include all the achievable rate pairs by random network coding. Moreover, if a rate pair (R1 , R2 ) is achieved by pure routing, there exists paths from s to t3 of capacity at least R2 . After subtracting those resources used for routing X2 , the mincut from s to ti for i = 1, 2, 3 should be no smaller than R1 since the routing capacity from s to t123 is upper bounded by min{k1 , k2 , k3 }. Therefore, choosing the same routing paths for X2 as that in this routing solution and performing random network coding over the rest of graph, it is guaranteed to achieve rate pair (R1 , R2 ). It is observed that the number of variables involved in formulating the new achievable region equals to the number of possible paths from s to t3 , which may quite large. Without loss of optimality, we can simplify the formulation by only considering the paths satisfying the following condition: ∃0 < α ≤ min Ce such that

P3 = {e2 , e5 , e11 , e13 } P4 = {e2 , e6 , e14 }

e∈P

α + min{|C1 | − α|C1 ∩ P |} + min{|C2 | − α|C2 ∩ P |} C1

P5 = {e3 , e8 , e14 }

C2

− min{|C12 | − α|C12 ∩ P |} + min{|C123 | − α|C123 ∩ P |}

Denote the routing rate along path Pi by αi for i = 1, .., 5. The nontrivial constraints on αi as given in (2) and (3) are αi ≥ 0, i = 1, ..., 5 α1 + α2 ≤ 1 α3 + α4 ≤ 1

By performing Fourier-Motzkin elimination over (8) and (9), we can obtain the achievable region on the rate pair (R1 , R2 ), which is given by

(8)

α2 + α3 ≤ 1 α4 + α5 ≤ 1 The new achievable region, as that given in (6), is R1 ≤ min{1 − α1 − α2 , 2 − α2 − α3 − α5 } R1 + R2 ≤ min{3 − α2 , 3 − α4 } (9) 2R1 + R2 ≤ min{6 − α2 + α4 , 7 − 2α2 + α4 − α5 }− min{3, 3 − α2 + α4 }.

C12

> k1 + k2 + k123 − k12 .

C123

(11) Such a choice follows the intuition that the first two inequalities given in (1) matches the outer bound and thus cannot be improved. Therefore, the achievable region can be improved only if the third inequality in (1) can be relaxed. Example 2: Consider the network shown in Fig 2. Although there are 5 possible paths from s to t3 , the only path that satisfies the condition given in (11) is P4 . Therefore, the previous procedures in obtaining the new achievable region can be significantly simplified by only considering routing along the path P4 . Unless the achievable region by random network coding in (1) is optimal, the new achievable region by proposed J-RRNC in (6) will strictly improve over it if there exists at least

one path from s to t3 in the network that satisfies the condition given in (11). B. J-R-RNC for 2-unicast networks In this subsection, we consider the 2-unicast networks as shown in Fig. 3. There are two sources and two receivers in the network where each source wants to communicate with its corresponding receiver while causing interference to the other pair. The capacity region for general double-unicast networks is still unknown.

obtained, which is given by R1 R2 R1 + R2 R1 + R2

≤k1−1 ≤k2−2 ≤k12−1 + k2−12 − k2−1 ≤k12−2 + k1−12 − k1−2

R1 + R2 ≤ (k12−2 + k2−12 − k2−2 − k1−2 − k2−1 )+ + (k12−1 + k1−12 − k1−1 − k1−2 − k2−1 )+ + k1−2 + k2−1 2R1 + R2 ≤k12−1 + k1−12

(13)

+ (k12−2 + k2−12 − k2−2 − k1−2 − k2−1 )+ R1 + 2R2 ≤k12−2 + k2−12 + (k12−1 + k1−12 − k1−1 − k1−2 − k2−1 )+ ,

Fig. 3.

where (x)+  max{x, 0}. The achievable region given in (13) is larger than that derived in [13]. The details for deriving (13) is omitted here for brevity. Similarly as for the 2-message degraded multicast networks, we need to choose the proper routing paths such that the overall achievable region can be enlarged. Assume that there are m 1 paths from s1 to t1 , denoted as P11 , ..., Pm and n paths from 2 2 s2 to t2 , denoted as P1 , ..., Pn . Let the routing rate assigned to path Pi1 and Pj2 be αi1 and αj2 , respectively, we have the following constraints:

The double-unicast network.

If pure routing is admitted, this problem reduces to finding the maximum two-commodity flow in a directed graph, which is as difficult as linear programming [11]. On the other hand, if random network coding is performed at all the intermediate nodes, this problem reduces to the 2-user linear deterministic interference channel. Denote the transfer matrix from si to tj by Hij for i, j ∈ {1, 2}. The capacity of the 2-user linear deterministic interference channel has been independently obtained in [12] and [13], which is given by R1 ≤rank(H11 ) R2 ≤rank(H22 )  R1 + R2 ≤rank H11 

{i:e∈Pi1 }

αi1



+

H12



+ rank



H12 H22

− rank(H12 )

− rank(H12 ) − rank(H21 )

  H11 2R1 + R2 ≤rank H11 H12 + rank H21

H21 H22 + rank − rank(H12 ) − rank(H21 ) 0 H12

  H12 R1 + 2R2 ≤rank H21 H22 + rank H22

H11 H12 − rank(H12 ) − rank(H21 ) + rank H21 0 (12)

By relating the rank of the transfer matrices with the mincut values of the network, an easily computable achievable region in terms of the min-cut values of the network can be

(14)

αj2

≥0, j = 1, . . . , n

(15)

αj2

≤Ce , ∀e ∈ E.

(16)

{j:e∈Pj2 }

After subtracting the resource used for routing, we can obtain a new network G with the capacity for link e reduced to:   αi1 − αj2 , ∀e ∈ E. (17) Ce = Ce − {i:e∈Pi1 }

 H11 H21 H22 + rank − rank(H21 ) H21



H21 H22 H11 H12 + rank R1 + R2 ≤rank H21 0 0 H12 R1 + R2 ≤rank



αi1 ≥0, i = 1, . . . , m

{j:e∈Pj2 }

Denote the min-cut between the set of source sI and the set  . We can obtain an achievable rate of receivers tJ in G by kI−J  region on G by random network coding by substituting the min-cut values into (13), denoted as R (R1 , R2 ). Adding the rates achieved by routing for each session, achievable m the new  n rate region is thus given by R (R1 + i=1 αi1 , R2 + j=1 αj2 ). Follow the similar procedures as that for degraded multicast networks, with FM elimination, the new achievable region can be optimized subject to the constraints given in (14)-(16). Example 3: Consider the double-unicast network shown in Fig. 4, where each edge has unit capacity. With random network coding, the achievable region is given by R1 ≤ 2 R1 + R2 ≤ 3.

(18)

By applying the proposed J-R-RNC, the achievable region can be improved to R1 ≤ 2 R1 ≤ 3 R1 + R2 ≤ 4.

(19)

For instance, the rate pair (R1 , R2 ) = (2, 2) is achieved by routing X1 through the path P11 = {e2 , e12 }, routing X2 through P12 = {e4 , e10 } each at unit rate and then applying random network coding over the rest of the graph.

Fig. 4. (a) An example 2-unicast network. (b) The achievable region with network coding and J-R-RNC.

C. Extension to General Multi-session Networks Proposed J-R-RNC can be applied to any multi-session network where an achievable region with random NC can be obtained. By routing part of those information that are not required by all receivers, the inter-session interference can be reduced and thus the achievable region may be enlarged compared to the one achieved with random network coding. The routing path and the corresponding rate can be determined based on the optimization result using simple Fourier-Motzkin elimination, similar as that introduced in previous two subsections. IV. C ONCLUSION In this paper, we have proposed a new coding scheme for the multi-session networks by integrating the random network coding and traditional routing. It is shown that by routing some information through properly chosen paths, the intersession interference can be reduced and thus we may obtain a larger achievable rate region compared with the one achieved by random network coding. ACKNOWLEDGEMENT The work of Xiaoli Xu and Yong Liang Guan was supported by the Advanced Communications Research Program DSOCL06271, a research grant from the Directorate of Research and Technology (DRTech), Ministry of Defence, Singapore. R EFERENCES [1] R. Ahlswede, N. Cai, S. Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inform. Theory, vol. 46, pp. 1204–1216, Jul. 2000. [2] T. Ho, R. Koetter, M. Mdard, M. Effros, J. Shi, and D. Karger, “A random linear network coding approach to multicast,” IEEE Trans. Inf. Theory, vol. 52, pp. 4413–4430, Oct. 2006.

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