Delay Minimization for Relay-based Cooperative Data Exchange with Network Coding

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1 Delay Minimization for Relay-based Cooperative Data Exchange with Network Coding Zheng Dong,XiuminWang,S.H.Dau, Chau Yuen Singapore University of Technology and Design, Singapore Hefei University of Technology, China {dong zheng, sonhoang dau, Abstract In this paper, we consider the problem of minimizing the delay for data exchange among a group of wireless clients, where each client initially holds a subset of the packets and needs to get all the packets held by other clients. It is assumed that clients cannot communicate with each another directly, they can only exchange packets through a wireless relay. To minimize the total transmission delay during data exchange process, we need to determine at every client, which packets to be uploaded and how to encode the packets. It is also important for the relay node to decide how to encode multiple packets from different clients and select the transmission rate in the downloading process, such that every client can decode all required packets in shortest delay. We first formulate theoretically the above problem of minimizing the total transmission delay as an integer programming, and show that its complexity is NP hard. We then propose an efficient heuristic algorithm, which consists of two processes: uploading process, i.e., how to select and encode the packets from the clients to the relay, and downloading process, i.e., how the relay encode packets and select transmission rate for broadcast to all clients. For each process, theoretical formulation has been derived to minimize their transmission delay, and efficient algorithms are proposed separately. Finally, simulation results demonstrate the effectiveness of the proposed algorithm in reducing the total data exchange delay. I. INTRODUCTION In resent years, the rapid growth in the popularity of mobile devices, such as smart phones and tablets, are challenging our abilities to deliver data. Especially, the last mile wireless link between the base station and the mobile devices is the major bottleneck in scaling the throughput with the increasing number of devices. One solution to address these issues is to allow wireless clients to cooperatively exchange data among themselves, named cooperative data exchange [1]. In a cooperative setting, every client initially holds a subset of packets, and needs the packets held by other clients. Specifically, [2]- [6] study how to utilize the network coding to minimize the number of transmissions [2], the transmission delay [3] or to improve the security [4] during the data exchange process. Most literatures on coded cooperative exchange assume that the clients can communicate with each other via a wireless broadcast channel. However, in a practical system, it is possible that the clients cannot communicate with each other directly, especially for a distance communication. In this paper, we consider a more practical scenario, where there is a wireless relay node to help the clients exchange the packets. The main differences of our cooperative data exchange with a relay node and traditional network model are that: a) unlike the index coding problem: the relay node in our problem does not need to know and does not have all the packets; b) unlike the traditional cooperative data exchange problem: the clients cannot communicate with each another directly. For this relay-based cooperative data exchange problem, we need to determine how many packet to be sent from every client to the relay node and how to encode the packets at the client, such that the relay node has enough information to broadcast to the clients with the minimum transmission delay. In addition, most literatures assume that the clients always transmit packets at the same transmission rate. However, wireless systems are capable of performing adaptive modulation to vary the link transmission rate in response to the signal to interference plus noise ratio at the receivers [8]- [1]. Transmission rate diversity exhibits a rate-range tradeoff: the higher the transmission rate, the shorter the transmission range for a given transmission power. Thus, considering the heterogeneous transmission rate (i.e., bandwidth) on wireless links, how to select the transmission rate is also an important issue to minimize the transmission delay [7]. 1 P1 P2 1K/s 2K/s P2.5K/s P3 1K/s 2.25K/s.5K/s Fig. 1. A simple example of the relay based cooperative data exchange. Each node communicates with the relay in different transmition rate. The relay can encode the received packets and broadcast them to minimize the system delay. Consider a simple example in Fig.1, there are one relay and three clients. Initially, each client has a subset of packets, e.g. T 1 has p 1,p 2, and needs p 3 ; T 2 has p 2,p 3, and needs p 1 ; T 3 has p 1, and needs p 2,p 3. Suppose that the size of each packet is B = 2kB, and the maximum transmission rates from the relay to T 1, T 2, T 3 are 2kB/s, 1kB/s and.5kb/s, P /13/$ IEEE

2 respectively; the maximum transmission rates from T 1, T 2, T 3 to the relay are 1kB/s,.5kB/s and.25kb/s, respectively. Considering a simple solution that simply select the node with largest transmission rate to transmit the packet: the relay gets p 1,p 2,p 3 from T 1,T 2 respectively (according to the upload bandwidth), and sends p 1 to T 2, sends p 2 to T 3, sends p 3 to T 1,T 3. The whole delay in this scheme is 18s. Alternatively, the relay can send an encoded packet that will maximize the number of clients which can decode, thus the relay should get p 1 p2 from T 1, p 3 from T 2 and send p 1 p2 to {T 3,T 2 }, p 3 to {T 1,T 2 }. T 2 will decode p 1 through p 2 (p1 p2 ) ;T 3 will decode p 2 through p 1 (p1 p2 ). In this network coding based scheme, the whole delay is 14s, which is 4s less than previous solution. In addition, from this example, it is clear that the relay need not know all the packets p 1, p 2, and p 3, and yet manage to assist all the clients to obtain every packets. In this paper, by considering the heterogeneous transmission rates, we deal with the relay based cooperative data exchange problem in wireless networks using network coding, such that every client can eventually get all the packets with the minimum delay. Our contributions can be summarized as: We theoretically formulate the problem of minimizing the total transmission delay for the cooperative data exchange with a relay node, as an integer programming. We simplify the proposed problem by separating it into two processes: uploading and downloading processes. We propose a graph model, and model the downloading process with minimum transmission delay as a clique partition with minimum weigh problem. Based on the graph model, we propose two efficient encoding algorithms to solve the uploading and downloading processes respectively. We compare the performance of the proposed coding approach with other schemes in the literature. The simulation results show that the proposed transmission strategy can reduce the overall data exchange delay. The remainder of this paper is organized as follows. In Section II, we will give the problem statement and formulate the problem in a general case. The graph model is given in Section III, and based on the graph model, the general formulation is divided into two processes. Simulation results are shown in Section IV. Finally, we conclude the paper in Section V. II. PROBLEM DESCRIPTION AND FORMULATION In this section, we first describe our coding based cooperative data exchange problem. Then, we formulate the defined problem as an integer programming. A. Problem Description Consider a set of m packets in P = {p 1,p 2,,p m } to be delivered to n clients in X = {x 1,x 2,,x n }. Suppose that initially, client x i holds a subset of packets in H(x i ) P, and the clients collectively hold all the packets, i.e., x H(x i X i)=p. The objective of the data exchange is to let every client get all the packets in P eventually. Different from the existing works, we assume that the clients cannot communicate with each other directly. This assumption is possible especially for a practical wireless network where the clients may be long distance away from each other. Without loss of generality, we assume that there is a wireless relay node R to help the clients exchange the packets, i.e., the clients first upload the packets to the relay node R, which then broadcasts the packets to all other clients. Note that, it is possible that R may generate new encoded packets for broadcasting based on the packets uploaded by the clients. Suppose that u i and d i are the maximum rates for the transmissions from client x i to relay node R, and from R to client x i respectively. With the help of the relay node R, our objective is to find a network coded data exchange scheme that satisfies the following conditions: each client x i X can finally receive/decode all the packets in P from its received packets. The total transmission delay is minimum among all the transmission schemes that satisfy the first condition. To realize the above objective, we need to determine how many packets should be sent by the clients and the relay node R, which transmission rate should be used at the sender, and how to encode the packets before sending. To simplify the presentation, we use n i to denote the number of packets in H(x i ), i.e., n i = H(x i ), and H(x i ) = P \H(x i ) denotes the packets required by client x i. B. Problem Formulation In this section, we formulate the problem defined above. The whole cooperative data exchange consists of two processes: uploading process and downloading process. In the uploading process, the clients send packets (where the packets can be original data packet or network coding encoded packets) to the relay node R, while in the downloading process, the relay node R broadcasts the packets received from the uploading process, and may re-encode multiple received packets with network coding before broadcast to the clients in the downloading process. For the simplicity of presentation, we say that the packets are sent round by round and in each round, only one packet is transmitted. Before formulating the problem, we first define the following variants, which is set to if the condition is not met. x i,h =1if client x i sends a packet in the h-th round of uploading process. x i,h =1if x i receives and decodes a new native packet from the h -th round of downloading process. y j,h =1if packet p j is encoded in the packet sent by the h-th round of uploading process. y j,h =1if packet p j is encoded in the packet sent by the h -th round of downloading process. z h,h =1if the packet sent in the h-round of uploading process is encoded in the h -th round of downloading process.

3 Suppose that H h (x i ) is the set of the packets owned by client x i before the h -th round of downloading process. Thus, initially, H 1 (x i )=H(x i ). We then formulate the problem of minimizing the total transmission delay as follows: m n min x i,h 1 m + max u i 1 i n (x i,h 1 ) (1) d i h=1 i=1 Subject to : h =1 n x i,h 1, 1 i n (2) i=1 m 1, if ( (z h,h y j,h ))%2 = 1 y j,h = (3) h=1 { 1, if pj H c h h (x i) j,i = (4) T h = {p j y j,h =1, 1 j m} (5) T h = {p j y j,h =1, 1 j m} (6) { 1,Th H(x i) x i,h (7) m x i,h = 1, if c h j,i y j,h = H h (xi) 1 j=1 (8) H h (x i )=H h 1(x i ) T h 1, if x i,h =1 (9) m x i,h = m H(x i) i {1,,n} (1) h =1 In the above formulation, the objective is to minimize the delay for cooperative data exchange process, where the first term of the objective denotes the transmission delay for the uploading process and the second term means the transmission delay for the downloading process. The constraint in Eq.2 denotes that in each round of uploading process, at most one client can send a packet (where the packets can be original data packet or network coding encoded packets) to the relay node. The constraint in Eq.3 shows the condition that packet p j can be encoded (under XOR operation) in the packet sent by the h -th round of downloading process. c h j,i in Eq.4 denotes whether packet p j is available after the h -th round of downloading, T h in Eq.5 represents the set of the native packets combined in the h-th round of uploading process, and T h in Eq.6 indicates the set of the native packets encoded in the h -th round of downloading process. The constraint in Eq.7 shows that client x i can send the packet pj T h p j only if all the native packets in T h are available at x i. The constraint in Eq.8 denotes that client x i can decode a new native packet only if all the other native packets encoded in the transmitted packet are available at x i. The constraint in Eq.9 updates the packets in the buffer of client x i. Finally, Eq.1 denotes that after downloading process, all clients should have all packets. We then discuss how to get the sender and the encoded packet for each transmission with the above integer programming. For the h-th round of uploading process, x i will be selected as the sender if x i,h =1, and the packet sent by x i is pj T h p j.fortheh -th round of decoding process, the relay node R sends the packet pj T h p j, and the transmission rate used is d i, where d i =argmax di {x i,h 1 d i }. III. GRAPH MODEL AND PROBLEM ANALYSIS Although the integer programming in Eq.1 can get the optimal solution with minimum transmission delay, it is infeasible to be solved in polynomial time. To reduce the complexity, instead of considering the uploading and downloading processes simultaneously, we would like to minimize the transmission delay for these two processes separately. A. Graph Model Before considering the transmission delay for uploading and downloading processes, we first introduce a graph model G(V,E), which is helpful for the following algorithm design. For each packet p j H(x i ), we add a corresponding vertex v i,j V (G), i.e., V (G) = {v i,j p j H(x i ), x i X}. For any two vertices v i,j,v i,j V (G), there is an edge (v i,j,v i,j ) E(G) if and only if any of the following two conditions is satisfied. 1) j = j, which means both client x i and x i requires the same packet. 2) j = j, p j H(x i ) and p j H(x i ), which means packet p j required by x i is available at client x i, while p j required by x i is available at client x i. In other words, E(G) = {(v i,j,v i,j ) j = j or j = j,p j H(x i ) and p j H(x i )}. Fig. 2 shows the corresponding graph model of the example in Fig. 1. v 13 v 21 v 32 v 33 Fig. 2. The graph model corresponding to the example in Fig 1. According to the work in [11], we know that every clique in graph G(V,E) indicates a feasible encoded packet. Suppose that C = {v i1j 1,v i2j 2,...,v ik j k } is a clique in G(V,E). Let P C = {p j v i,j C} and R C = {x i v i,j C}. According to [11], we know that Lemma 1. After receiving the encoded packet pj P C p j,the client x i R C can decode the native packet p j,for v i,j C. Thus, each clique partition of the graph represents one of the feasible transmissions that can satisfy the requirements of all the clients. B. Formulation for Downloading Process Before discussing the uploading delay, we need to know which packets are required at the relay node such that it has enough packets to broadcast. Thus, we would like to first introduce the downloading process. After deriving the required transmitted packets for downloading phase, we then know the packets exactly required by relay node R.

4 As described in Section III-A, a clique partition in the graph represents a feasible transmission solution that can satisfy all the requirements of the clients. For Lemma 1, to make sure all the clients in R C can receive the transmitted packet pj P C p j, the minimum transmission rate that can be used by the relay node R should be d i = min{d i v i,j C}. Thus, the problem of minimizing the total transmission delay during downloading process becomes to find a clique partition C = {C 1,C 2, } such that C k C max{ 1 d i v i,j C k } is minimum among all possible clique partitions. We refer the above delay minimizing problem as Minimum Delay Clique Partition () problem. In computational complexity theory [12], finding a minimum clique cover is a graph-theoretical NP-complete problem. Hence, the is NP hard. Let a k i,j be 1 if vertex v i,j V (G) belongs to the k-th clique in the found clique partition C, otherwise, let it be. We also let b i,k be 1 if for j, there is at least one vertex v i,j in clique C k. Then, we can formulate the above problem as follows. min C Subject to : C k C C k C max{ b i,k d i } (11) a k i,j =1, v i,j V (12) a k i,j + a k i,j 1, (v ij,v i,j ) E, k (13) a k i,j {, 1},v i,j V (14) { m 1, if b i,k = j=1 ak i,j 1 (15) In the above formulation, the objective is to minimize the transmission delay for downloading process. The constraint in Eq.12 denotes that each vertex can only belong to one clique. Eq.13 means that if there is no edge between vertices v i,j and v i,j, these two vertices cannot belong to the same clique. After getting the clique partition with the above formulation, we can derive the encoded packets that the relay node should send and the transmission rate it should use. For example, if the clique C k = {v i1,j 1,v i2,j 2, } is selected in the clique partition, the encoded packet pj P k p j will be sent by R with transmission rate max vi,j C k {d i v i,j C k }. In Algorithm.1, it finds a clique partition in G and produces the corresponding coded packets that the relay multicasts in the Downloading Phase. C. Formulation for Uploading Process In order to optimize the downlink delay, we assume all the clients finish the uplink process, and the relay has enough buffer to store all these uploaded packets, and to perform necessary encoding before the downloading process. After knowing the packets that is required at the relay (as discussed in previous Section III-B), we then consider to minimize the transmission delay for uploading process. Ideally, for each packet p i P, we would want to determine a set of clients and corresponding coded packets with minimum delay that collectively generate p i. However, this is a hard problem, as even when u 1 = u 2 = = u k,it reduces to the set cover problem, another well-known NP-hard 1.max 2.for (v ij G) do 3. if (d i >max) then 4. max = d i 5. k i 6. t j 7.end for 8.U {v kt } 9.while(N(v kt ) \ U ) do 1. C d for(v i j N(v kt) \ U) if ({v i j } U can construct a clique) then 13. C d C d {v i j } 14. end if 15. end for 16. J {max(d i ),v i j C d} 17. Randomly select v i j J 18. U U {v i j } 19. k i 2. t j 21.end while Fig. 3. Algorithm 1: The Downloading Algorithm problem. Algorithm is a greedy algorithm to find a suboptimal solution for this problem. The main steps in this algorithm are explained as follows. 1.max 2.m 3.Compute P C for C; 4.while(P C φ) 5. P i = H(x i) P C ; 6. S i = T i u i ; 7. for i=1 to n do 8. if (S i >max) then 9. max = S i 1. m = i 11. p code p i1 p i2... p ik where p it P i, 1 t k; 12. X m upload p code 13. Delete P i from P C; Fig. 4. Algorithm 3: The Uploading Algorithm IV. SIMULATION In this section, we set up a simulation environment and conduct several experiments to demonstrate the effectiveness of our algorithm. The simulation environment which is shown in the square box in Fig.1 consists of a relay and m clients. We randomly generate a set of available packets in H(x i ) and the wanted packets in R(x i ) at client x i X, where H(x i ) R(x i )=P. The downlink transmission rate from

5 The Number of Clients The Number of Clients The Number of Packets The Number of Packets (a) (b) (c) (d) Fig. 5. Experimental Results With different number of packets and clients the relay to x i is randomly selected in [rmin; rmax], and the uplink transmission rate from x i to the relay is randomly selected in [umin; umax]. We consider two baseline algorithms for comparison. One is the traditional scheme without network coding (), and the other is a simulation for the cooperative data exchange (). uploads the wanted packets from the clients which possess the highest upload bandwidth and broadcast them without network coding. is similar with the algorithm mentioned in [1]. In each of the clients uploads the packets or network encoded packets in its possession via the relay node. The relay does not encode the packets which it receives. While in our proposed, both clients and relay can perform network coding on the packets. A. The Impact of the Number of Clients m We investigate the impact of the number of destinations on the performance of whole transmission in Fig.5. In this simulation, we set n = 2, [rmin; rmax] = [1, 5], [umin; umax] = [5, 25] in Fig.5(a) and [rmin; rmax] = [1, 1], [umin; umax] =[5, 5] in Fig.5(b). Fig.5 (a) and (b) shows the coding gain with the increase of m when the number of packets n is fixed at 2. With the increase number of clients, the delay of the whole scheme is increasing. The reason is that there are more packets being required by more clients. We can also see that the system delay is smaller in Fig.5(b) than Fig.5(a), this is because higher transmission rates in Fig.5(b) incur less transmission delay. B. The Impact of the Number of Packets n Finally, we investigate the impact of the total number of packets on the performance of the whole transmission. We set total number of clients m = 1, [rmin; rmax] = [1, 5], [umin; umax] = [5, 25] in Fig.5(c) and [rmin; rmax] = [1, 1], [umin; umax] =[5, 5] in Fig.5(d). Again, in Fig.5, we can see that, with the increase number of packets, the overall delay is increased. The reason is that there are more wanted packets which should be delivered to the clients. And also, comparing the two figures, we can notice that when the transmission rate is higher, the overall delay will be lower. And our proposed always achieve a lower delay than and. V. CONCLUSION In this paper, we consider a relay based cooperative data exchange problem in wireless networks using network coding with the assumption that each clients has different transmission rates. We prove that the problem is NP-hard. In order to provide a practical solution with manageable complexity, we divide the whole transmission scheme into two processes: uploading and downloading, by using the graph model. Simulation results demonstrate the proposed algorithm effectively reduces the delay. In the future work, we consider a model where the data exchange can be achieved with the help of relay and the exchange of packets among the clients directly. ACKNOWLEDGMENT This research is partly supported by the International Design Center (grant no. IDG31112 & IDD11111). It is also supported in part by the Fundamental Research Funds for the Central Universities (No. 212HGBZ64). REFERENCES [1] S. El Rouayheb, A. Sprintson, and P. Sadeghi, On coding for cooperative data exchange, Proc. IEEE ITW, 21. [2] J.S. Park, D. S. Lun, F. Soldo, M. Gerla, and M. Medard, Performance of network coding in ad hoc networks, Proc. IEEE Milcom 6. [3] Nebojsa Milosavljevic, Sameer Pawar, Salim El Rouayheb, Michael Gastpar, Kannan Ramchandran, Data Exchange Problem with Helpers, Proc. IEEE ISIT 212. [4] M. Yan; A. Sprintson. Weakly Secure Network Coding for Wireless Cooperative Data Exchange, Proc. IEEE GLOBECOM 211. [5] D. Lun, N. Ratnakar, R. Koetter, M. Medard, E. Ahmed, and H. Lee. Achieving minimum-cost multicast: a decentralized approach based on network coding, Proc. IEEE INFOCOM, Miami, Florida, Mar. 5. [6] X. Wang, C. Yuen, S. H. Dau, Delay Minimization for Network Coded Cooperative Data Exchange with Rate Adaptation, VTC-Fall 213. [7] X. Wang, C. Yuen, Y. Xu, Joint Rate Selection and Wireless Network Coding for Time Critical Applications, Proc. IEEE WCNC 212. [8] Y. Kim and G. D. Veciana, Is rate adaptation beneficial for inter-session network coding?, IEEE Journal on Selected Areas in Communications, vol. 27, pp , Jun. 9. [9] K. Chi, X. Jiang, and S. Horiguchi, Joint design of network coding and transmission rate selection for multihop wireless networks, IEEE Trans. on Vehicular Technology, vol. 59, pp. 2435/444, 21. [1] T. Kim, S. Vural, I. Broustis, D. Syrivelis, S. Krishnamurthy, and T. La Porta, A framework for joint network coding and transmission rate control in wireless network, Proc. IEEE INFOCOM 21. [11] Z. Dong, C. Zhan, and Y. Xu, Delay aware broadcast scheduling in wireless networks using network coding, in Proc. of the Second Int. Conf. on Network Security, Wirless Comm. and Trusted Computing, 21. [12] Richard M. Karp (1972). Reducibility Among Combinatorial Problems. In R. E. Miller and J. W. Thatcher (editors). Complexity of Computer Computations. New York: Plenum. pp.85/3.