Delay Minimization for Relay-Based Cooperative Data Exchange With Network Coding

Transcription

1 1890 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 23, NO. 6, DECEMBER 2015 Delay Minimization for Relay-Based Cooperative Data Exchange With Network Coding Zheng Dong, Son Hoang Dau, Chau Yuen, Senior Member, IEEE, YuGu, Member, IEEE, ACM, and Xiumin Wang Abstract We study the Relay-based Cooperative Data Exchange (RCDE) problem, where initially each client has access to asubsetofasetof original packets, referred to as their side information, and wants to retrieve all other original packets via cooperation. Unlike traditional Cooperative Data Exchange (CDE), in our proposed relay-based model, clients can only cooperate via a relay. The data exchange is completed over two phases, namely Uploading Phase and Downloading Phase. In the Uploading Phase, the clients will encode the original packets and transmit the coded packets to the relay. In the Downloading Phase, the relay will reencode the received packets and multicast the reencoded packets, each to a subgroup of clients. The coded packets in the two phases are carefully selected so that each client can retrieve all original packets with minimum total transmission delay, based on its initial side information and on the coded packets it receives from the relay. In addition, we assume that the bandwidths between the relay and different clients are different, and that the upload/download bandwidths between the relay and the same client are also different. We establish a coding scheme that has the minimum total delay and show that it can be found in polynomial time, for sufficiently large underlying fields. We also design a heuristic algorithm that has a low complexity with binary field size. Our simulations show that the performance of the binary solution is very close to that of the optimal solution. All coding schemes considered in this work are scalar. Index Terms Cooperative data exchange, multicast, network coding, transmission delay. I. INTRODUCTION T HE EVER-GROWING demand for storing and exchanging of large amount of data via handheld wireless devices has put an increasing pressure on the wireless communication infrastructure, especially on cellular and satellite networks. Inspired by the success of the P2P content delivery systems, cooperation among wireless users to exchange data has been proposed in the literature as a promising solution. The Cooperative Data Exchange (CDE) problem, recently introduced by El Rouayheb et al. [1], has attracted considerable attention from the research community over the past few Manuscript received October 11, 2013; revised May 23, 2014; accepted July 17, 2014; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor A. Markopoulou. Date of publication August 08, 2014; date of current version December 15, This work was sponsored by the Singapore University of Technology and Design, Singapore, and in part by the Natural Science Foundation of China under Grant No Z. Dong, S. H. Dau, C. Yuen, and Y. Gu are with the Singapore University of Technology and Design, Singapore , Singapore ( dong_zheng@sutd.edu.sg; sonhoang_dau@sutd.edu.sg; yuenchau@sutd. edu.sg; jasongu@sutd.edu.sg). X. Wang is with Hefei University of Technology, Hefei , China ( wxiumin@hfut.edu.cn). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TNET years [2] [12]. In a CDE instance, a set of clients cooperate via a common broadcast channel. Each client initially possesses a subset of a set of original packets, referred to as its side information. Each client broadcasts some linear combinations of the packets it has to all other clients. It is required that at the end of the communication session, each client is able to retrieve all original packets based on its side information and on the broadcast packets it receives from its peers. As a scheme with fewer transmissions saves more resources (energy, memory, time, etc.), the goal is to keep the total number of broadcast packets at minimum. The scenario described in the CDE problem is also observed in opportunistic wireless networks [13], [14], where wireless devices often opportunistically overhear packets that are not designated to them. These overheard packets then become the side information for the devices. We investigate in this work the so-called Relay-based Cooperative Data Exchange (RCDE) problem, where clients exchange their data indirectly via a relay. The use of a relay is particularly beneficial in practical scenarios where the long distances between the clients do not guarantee reliable connections. Note that the relay does not need to know anything about the original packets. It is simply an intermediate wireless node that receives the information sent from the clients and transmits the processed information back to the clients. Such a relay plays a similar role to an access point in a WiFi network, a base station in a cellular network, or a satellite in a satellite network. The deployment of a relay for data exchange can also be helpful for networks of wireless devices with limited resources, as explained below. In a typical CDE scheme studied by the aforementioned works, either one of the clients must act as a host and perform the following tasks: gathering the information on what other clients have; running a CDE algorithm [3], [8] to find an optimal broadcast scheme; informing all other clients on what to broadcast and on the broadcast scheduling; or all of the clients have to perform the first two tasks and follow the same broadcast scheduling. For certain networks where wireless devices have limited energy, memory, and computational power, the second task can be heavy, especially when some cost criterion is considered, i.e., in the setting of Cooperative Data Exchange with Cost Criterion (CDECC) problem [5], [8] where the searching algorithm has considerably higher complexity [8]. For such networks, a relay can be deployed to alleviate the burden on the wireless devices. The relay can take over all of the three aforementioned tasks, thus allowing the devices to save energy, memory, and other resources IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 DONG et al.: DELAY MINIMIZATION FOR RELAY-BASED COOPERATIVE DATA EXCHANGE WITH NETWORK CODING 1891 TABLE I INPUT PARAMETERS FOR THE RCDE INSTANCE IN FIG. 1.THE HAS COLUMN SPECIFIES THE SIDE INFORMATION OF EACH CLIENT. THE WANT COLUMN SPECIFIES THE MISSING ORIGINAL PACKETS THAT EACH CLIENT WANTS TO OBTAIN. THE COLUMNS / SPECIFY THE DELAYS FOR EACH CLIENT TO UPLOAD/DOWNLOAD A CODED PACKET TO/FROM THE RELAY, RESPECTIVELY Fig. 1. Example for relay-based cooperative data exchange. When designing a communication model, the total delay of the model is an important factor to consider, given the fact that different communication links have different bandwidth, especially in time-sensitive applications. With that in mind, we focus on studying the delay minimization issue for the RCDE problem. All coding schemes considered in our work are scalar. Example 1: As a motivational example, we consider a scenario where three clients,,and cooperate via a relay to obtain all five original packets ( ). The bandwidths of the channels between the clients and the relay are given in Fig. 1. To be general, we assume that the channels are asymmetric, i.e., downloading and uploading bandwidths can be different. We first clarify two categories of packets that we use in this work to avoid any possible confusion (see also [15]). Original/coded/broadcast/multicast packets ( -packets for short): These are symbols over some finite field. When we study the theoretical models of the problems of interest and the coding schemes, we only consider -packets. Channel packets: These are data packets that are sent through the actual physical communication channels. Typically, each channel packet contains a header and a body. The body contains data, which consist of a number of -packets in the first category. For example, if and the channel packet size, excluding the header, is kb bits, then each channel packet consists of -packets. If, in the theoretical model, the first client transmits to the relay, then in reality, the first client transmits the channel packet which is of size 0.5 kb excluding the header, to the relay, where, ( ) are all independent symbols. In fact, the same coding scheme is repeatedly applied for all pairs, to create the corresponding channel packet. The number of channel packets that need to be transmitted is, therefore, equal to the number of -packets that need to be sent according to the coding scheme. For simplicity, suppose that each channel packet is of size 1 kb. Then, the uploading delay (in seconds) for uploading one channel packet from to the relay is the inverse of the uploading bandwidth of the channel between and the relay. The same calculation holds for the downloading delay for to download one channel packet from the relay. Table I summarizes the main input parameters for the scenario in this example. Because of the one-to-one correspondence between the transmitted channel packets and the transmitted -packets in the theoretical coding scheme, we can abuse the language slightly and talk about the delay of an -packet instead. We do so to simplify the presentation, as from now on, we only consider -packets. Note also that it is the number of -packets sent from each client or from the relay that determines the total transmission delay, not the field size. In other words, the total delay depends on the number of packets transmitted and on the identities of the clients who send or receive the packets, not on the particular value of. Therefore, we have the freedom to consider reasonably large as well as very small such as.regarding the computational complexity of the model, however, smaller is often preferred. We present in Fig. 2 two different schemes for the RCDE instance described above. Each scheme consists of two phases: Uploading Phase and Downloading Phase. In the Uploading Phase, the clients upload linear combinations of the original packets they have to the relay. In the Downloading Phase, the relay multicasts linear combinations of what it receives in the Uploading Phase, each to a subgroup of clients. The total delay is calculated by taking the sum of the delays of all coded packets transmitted during the communication session. We should emphasize again that the relay multicasts each packet only to a subgroup of clients that need the packet. Doing so instead of broadcasting the packets to all clients reduces the total delay of the scheme while still guaranteeing the recovery of data for all clients. Muticasting different packets to different groups of clients can be achieved by performing adaptive modulation to vary the transmission rate at the relay (see [16] and [17]). To guarantee that all clients in this group are able to decode the packet, the relay must transmit the packet at the rate equal to the minimum bandwidth between the relay and clients in the group. Therefore, the (downloading) delay of this packet is equal to the maximum delay of a client in that group. Scheme A: No coding is allowed. All five original packets have to be uploaded to the relay and then multicast to clients that need them. A naive greedy strategy for the Uploading Phase simply lets the client with the smallest upload delay send each packet,.inthe Downloading Phase, each original packet is multicast to

3 1892 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 23, NO. 6, DECEMBER 2015 Fig. 2. Two different coding schemes. (a) Scheme A: no coding. (b) Scheme B: with optimal coding, minimum total delay. the subgroup of clients that initially do not have that packet. For example, is multicast to and. As discussed above, the delay of must be s. The total delay of this scheme is 4.25 s. Scheme B: With optimal coding. Clients are allowed to transmit linear combinations of original packets to the relay. For instance, transmits. During the Downloading Phase, the relay may multicast linear combinations of the coded packets it receives in the Uploading Phase to subgroups of clients. For example, it multicasts to,,and with the delay of s. The decoding is as follows. Client obtains, i.e., by subtracting what it has, namely,from what it receives, namely.client obtains,and directly from the channel. Client obtains, and, directly from the channel. The delays during both phases are improved compared to Scheme A. In fact, Scheme B obtains the minimum total delay among all possible coding schemes, as shown later in Section IV-B. Hence, it is desirable to design the optimal coding scheme that leads to the minimum total delay. The main contribution of this paper is to establish an RCDE scheme with minimum total delay, via the use of an optimal solution from the corresponding Cooperative Data Exchange with Cost Criterion problem and an optimal packet assignment. Here, a packet assignment is a table that specifies the group of clients to which each particular packet from the relay is multicast. We show that such an optimal RCDE scheme can be found in polynomial time in (number of original packets) and (number of clients). We also develop an efficient heuristic algorithm to find an RCDE scheme with small total delay over the binary field. The rest of the paper is organized as follows. In Section II, we define our problem rigorously. In Section III, we present necessary and sufficient conditions for the feasibility of an RCDE scheme. The construction of an optimal RCDE scheme is established in Section IV. We propose an efficient heuristic algorithm to find a feasible RCDE scheme with small total delay over the binary field in Section V. Simulation results are discussed in Section VI. The paper is concluded in Section VII. II. PROBLEM DEFINITION Before formulating our main problem of interest, namely the RCDE problem, we first describe the CDECC problem [5], [8], which plays an important role in our study of RCDE. A. Cooperative Data Exchange With Cost Criterion The CDE problem setup includes a set of clients that need to obtain a set of original packets. Each original packet is an element of,a finite field of elements. Each client has access to a subset,where, and requests for the missing original packets in,where. We refer to as the side information, or the Has set, and to as the Want set of. Without loss of generality, suppose that no original packet is known (or not known) by every clients. In a CDE scheme, each client broadcasts some linear combinations of the original packets that it has to other clients via a common broadcast channel. These linear combinations are referred to as coded/broadcast packets. A CDE scheme is called feasible if at the end of the broadcast session, each client is able to retrieve all missing original packets, based on its side information and the received broadcast packets. The goal of the CDE problem [1] [3] is to obtain a feasible CDE scheme with minimum number of broadcast packets. Now suppose that each client is associated with a transmission cost (such as delay, energy, etc.). The goal of the CDECC problem [5], [8] is to obtain a feasible CDE scheme that minimizes the total transmission cost (referred to as an optimal CDE scheme). Such a CDECC instance is abbreviated by.when,thecdecc problem reduces to the CDE problem. A CDE instance is abbreviated by. Polynomial-time randomized and deterministic algorithms to find the minimum number of broadcast packets required for a CDE instance were proposed by Sprintson et al. [2], [3], respectively. The CDECC problem, a generalization to the CDE, wasalsosettledbyozgulandsprintson[5]andbycourtade and Wesel [8]. The generalization of the CDE problem to multihop networks was studied by Courtade and Wesel [6] and by

4 DONG et al.: DELAY MINIMIZATION FOR RELAY-BASED COOPERATIVE DATA EXCHANGE WITH NETWORK CODING 1893 Gonen and Langberg [11]. In contrast to the original one-hop CDE, for the general multihop networks, no polynomial-time algorithms that minimize the number of required transmissions seem to exist. B. Relay-Based Cooperative Data Exchange With Minimum Delay We now discuss the problem of our main interest. The setup of the RCDE problem is very similar to that of the CDE problem, except that the clients now communicate via a relay. More specifically, an RCDE scheme consists of the following two phases. 1) Uploading Phase: The clients upload (transmit) original or coded packets, i.e., linear combinations of original packets, to the relay. 2) Downloading Phase: The relay multicasts coded packets, i.e., linear combinations of the coded packets it received from the Uploading Phase, to the clients. We refer to these packets as multicast packets. An RCDE scheme is called feasible if upon the completion of the two phases, each client is able to retrieve all original packets. We assume that the delay for the transmission of a packet between a client and the relay varies from client to client. For the same client, the delay for uploading a packet to the relay might also be different from the delay for downloading (receiving) a packet from the relay. We use and to denote the delay for uploading and for downloading one packet between client and the relay. This consideration of different delays stems from the fact that the bandwidths of the communication channels (with the relay) of different clients can be different, and that for the same client, the upload and download bandwidths can be different as well. Such an RCDE instance is abbreviated by. We suppose that each client knows andtherelayknows,,and for all beforehand. In practice, this can be achieved via a preliminary communication session, where relevant channel side information is retrieved by all participants and the information about 's is sent to the relay from the clients. Our objective in this work is to find a feasible RCDE scheme that minimizes the total transmission delay. Such an RCDE scheme is called optimal. The total transmission delay is the sum of the total transmission delays of the two phases. Let be the number of coded packets uploaded from, respectively. Then, the total transmission delay of the Uploading Phase is The computation of the total transmission delay of the Downloading Phase is, however, slightly more subtle. Let be the set of multicast packets. We should stress that each packet is only assigned (multicast) to a subgroup of clients. Different RCDE schemes may have different strategies to assign packets to clients, which may result in different total transmission delays. The delay of the packet is defined to be (1) is assigned to (2) Note that the relay must transmit at the minimum rate among all assigned clients so that the client with the smallest bandwidth can manage to decode the packet. Therefore, the largest delay among the designated clients is the bottleneck and dominates the delay for that multicast packet transmission. Hence, given by (2) is indeed the amount of time required for the multicast packet to be successfully received by all assigned clients. Note that after multicasting a packet, the relay can start sending another packet only when all clients that is assigned to already receive successfully. Therefore, the total transmission delay of the Downloading Phase is computed as Thus, the total transmission delay is is assigned to (3) is assigned to (4) For example, let us consider Scheme B in Fig. 2(b). The total transmission delay of the Uploading Phase is For the Downloading Phase Hence, the total transmission delay for this RCDE scheme is In a general transmission scheme, uploading and downloading can be done in an alternating manner, i.e., the following transmission pattern can be repeated again and again until each client can decode all packets: Some clients upload some packets to the relay, and the relay multicasts some packets to some groups of clients. However, we can always separate the transmissions into two separate phases, namely uploading and downloading, without loss of generality. This is because there is no need for the clients to use previously received (multicast) packets from the relay to form the packets they plan to upload to the relay, as the relay already has all of those (multicast) packets. For example, if a client has and receives from the relay previously, then instead of sending to the relay, it is equally good to just send to the relay, which already has in its memory (or can create out of what it has). Therefore, we can always suppose, without loss of generality, that each client only uses its own original side information to create the uploading packets, and hence can group all uploading transmissions into one phase, namely Uploading Phase, and group all downloading transmissions into another phase, namely Downloading Phase, which happens only after the completion of the Uploading Phase. Note that an instance of the RCDE problem cannot be transformed into an instance of the CDE problem with helpers [4]. Indeed, in the setting of [4], a critical assumption is that when

5 1894 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 23, NO. 6, DECEMBER 2015 one client broadcasts, all other clients can receive the packet. In our setting, only communication between clients and relay is possible. Hence, our setting is more restrictive in this sense. On top of that, in our setting, we account for the cost (delay) for each client to transmit or receive a packet (to or from the relay), while in the setting of [4], only the cost of transmitting a packet is taken into account. Hence, our setting is more general in this sense. These two differences clearly distinguish the two problems. Remark 2: We later argue that the CDECC and the Uploading Phase of the RCDE are in fact equivalent. Since we are able to prove in the following sections that the two phases of the RCDE can be optimized separately, an optimal solution from the CDECC can be used as an optimal solution for the Uploading Phase of the RCDE. III. FEASIBILITY OF AN RCDE SCHEME VIA NETWORK CODING In this section, we establish a necessary and sufficient condition for the feasibility of an RCDE scheme via network coding. Let be an instance of the RCDE problem as described in Section II-B. Consider an RCDE scheme where we have the following. is the counting vector of the Uploading Phase: transmits packets to. is the packet assignment matrix for the Downloading Phase, which is an binary matrix, where: is the total number of coded packets multicast from the relay to the clients in the Downloading Phase; if and only if the th multicast packet is assigned to. For example, for the RCDE instance given in Fig. 1,,,and,,.Let us consider the RCDE scheme described in Fig. 2(b) (Scheme B). The counting vector is since the three clients upload one, two, and one coded packets to the relay, respectively. The assignment matrix for this scheme is As the first multicast packet is transmitted to all clients, the first row of is an all-one vector. Since the second multicast packet is transmitted to and,wehave and. As the last multicast packet is transmitted to only, we have and. Note that in the above notation of an RCDE scheme, the coded packets transmitted during the two phases are not included. The reason is that when the field size is sufficiently large, the feasibility of an RCDE scheme depends only on the counting vector and the assignment matrix, as proved later in Lemma 3. As long as and satisfy certain properties, then the coded packets during the two phases can be determined by a network flow algorithm (e.g., [18]) in polynomial time. (5) (6) Fig. 3. Network when,,and are given by Table I, (5), and (6), respectively. In the following, we first associate each RCDE instance and RCDE scheme to a network. We then prove necessary and sufficient conditions for the feasibility of the RCDE scheme by examining relevant cuts in the corresponding network. For an RCDE instance and an RCDE scheme,we define a network as follows. The set of nodes of consists of the following: one source node, which possesses all original packets ; original packet nodes, each corresponds to an original packet; client nodes, each corresponds to a client; a relay node that corresponds to the relay; multicast packet nodes, each corresponds to a multicast packet in the Downloading Phase; sinks, each corresponds to a client and demands all original packets. The set of (directed) edges of following: with capacity one for all ; with capacity infinity if and only if ; with capacity for all ; with capacity infinity for all ; with capacity one for every ; with capacity one if and only if. consists of the For example, for the RCDE instance giveninfig.1andthe RCDE scheme given in (5) and (6), the network is depicted in Fig. 3. The network is called solvable if the source is able to multicast original packets to all sinks simultaneously by using a linear coding scheme (see [19]). Lemma3: AnRCDEscheme is feasible if and only if the network is solvable. Proof: The proof can be found in the Appendix. We present in the following lemma necessary and sufficient conditions for the feasibility of an RCDE scheme.the first condition is a necessary and sufficient condition for the counting vector to be a feasible solution for the corresponding CDE problem. This condition for the CDE problem was established by Courtade and Wesel [20]. The fact that this condition for CDE also holds for RCDE stems from a simple observation that a coding scheme for the Uploading Phase is also a CDE scheme, and vice versa. It is not clear, however, that the Uploading Phase and the Downloading Phase can be treated separately regarding the feasibility. This

6 DONG et al.: DELAY MINIMIZATION FOR RELAY-BASED COOPERATIVE DATA EXCHANGE WITH NETWORK CODING 1895 lemma provides an affirmative answer, i.e., the two phases can be indeed treated separately. Lemma4: AnRCDEscheme is feasible if and only if it satisfies the following conditions. (C1) (CDE condition) TABLE II INPUT PARAMETERS FOR THE RCDE INSTANCE for all (7) In the setting of CDE, this condition requires that for every group of clients, the number of packets they receive from the clients outside the group is at least as many as the total number of original packets that they collectively miss. (C2) (Packet assignment condition) for every where: is the number of 1-entries in Column of the matrix, denotes the cardinality of the set. Note that according to the RCDE scheme, receives precisely multicast packets from the relay. Therefore, this condition requires that each client receives the same number of multicast packets as the number of original packets that it lacks. Proof: The proof can be found in the Appendix. According to Lemma 4, we achieve an optimal solution by optimizing each phase separately. More specifically, provided that the counting vector of the Uploading Phase satisfies the CDE condition and the assignment matrix satisfies the packet assignment condition, the whole scheme is feasible. This separation between the two phases allows us to optimize the total delay of an RCDE scheme by optimizing the total delay in each phase separately, as discussed in Section IV. IV. OPTIMAL SOLUTION FOR THE RELAY-BASED COOPERATIVE DATA EXCHANGE WITH MINIMUM DELAY In this section, we show that an optimal RCDE scheme, which has the minimum total transmission delay among all RCDE schemes, can be found in polynomial time. Our main idea is to employ an optimal counting vector for the corresponding CDECC problem in the Uploading Phase and an optimal packet assignment scheme in the Downloading Phase. This idea works because we already show in Section III that the two phases can be treated separately regarding the feasibility. The actual coded packets received and multicast by the relay are then decided by a network information flow algorithm (e.g., [18]). A. Our Proposed Solution We present our proposed solution for the RCDE problem. Let be an instance of the RCDE problem as described in Section II-B. Without loss of generality, we henceforth assume that for the Uploading Phase is equal to the sum of the delays of the uploading packets. For CDECC, the total transmission cost of a coding scheme is also equal to the sum of the costs of packets broadcast from clients. Therefore, the delay of the uploading packets in the Uploading Phase of the RCDE problem can be considered as a type of cost in the corresponding CDECC problem, and the total delay for the Uploading Phase of the RCDE problem can be treated as the total transmission cost in the CDECC problem. Hence, an optimal coding scheme with minimum total transmission cost for the corresponding CDECC problem provides an optimal coding scheme with minimum total delay for the Uploading Phase in the RCDE problem, and vice versa. We present our construction of an optimal RCDE scheme. Construction 1: An optimal RCDE scheme can be constructed as follows. is the counting vector in an optimal CDE scheme for the CDECC instance, i.e., incurs a minimum total delay (cost) among all feasible counting vectors for.,where and if if. According to this scheme, each client receives exactly the first packets among all multicast packets. The total number of multicast packets in the Downloading Phase,, is as small as possible. The RCDE scheme described above is feasible. Indeed, we have the following. As is a counting vector in an optimal CDE scheme for the corresponding CDECC problem, by [8, Theorem 1], satisfies the CDE condition (C1) stated in Lemma 4. satisfies the Packet assignment condition (C2), according to its construction. Therefore, according to Lemma 4, is a feasible RCDE scheme. By Lemma 3, a network information flow algorithm (e.g., [18]) can be employed to produce relevant coded packets for the two phases. An example of such an RCDE scheme is given in Fig. 1. We present another example. Example 5: The input parameters for the RCDE instance are given in Table II. According to our proposed RCDE scheme in Construction 1, we can obtain the optimal solution, where and are given by (9) We also consider the corresponding CDECC instance. In an RCDE scheme, the total delay (8) (10)

7 1896 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 23, NO. 6, DECEMBER 2015 Note that can be produced by an algorithm for CDECC, for instance, the one in [8]. To compare, we also consider another scheme where and are given by (11) Fig. 4. is turned into in three steps. (a) after Step 1. (b) after Step 2. (c) after Step 3. The total delay of the Uploading Phases in both schemes is that is the counting vector of a CDE scheme for the corresponding CDECC instance.aswealready choose to be the counting vector for with minimum cost, it holds that The total delay of the Downloading Phase for is In the remainder of this section, we prove that (12) The total delay of the Downloading Phase for (6.5 s) is smaller than (6.83 s). In fact, as we always has min- The total delay for the scheme the total delay for the scheme prove in Section IV-B, the scheme imum total delay. is B. Optimality Now we prove the optimality of from Construction 1 in terms of the total delay. Let be an arbitrary feasible RCDE scheme for the RCDE instance Our goal is to show that. For a counting vector, we denote the total delay of the Uploading Phase where is employed by For an assignment matrix delay of the Downloading Phase where,wedenotethetotal is employed by We also refer to as the total delay of. According to the discussion in Section III, the feasibility of implies which, together with the preceding inequality, establishes the optimality of. By Lemma 4, the feasibility of implies that for every. Since flipping a 1-entry into a 0-entry does not increase, we may assume that for every. In Lemma 6, we show that is not smaller than.first,weillustratetheidea of Lemma 6 via an example. Consider the RCDE instance discussed in Example 5. Let and be the assignment matrices given in (10) and (11). We now show that using an algorithmic approach. We modify through several steps so that finally, is turned into. Moreover, in every step, is never increased. Step 1: We permute the second and the third row of.obviously, remains unchanged. The matrix now is given in Fig. 4(a). We can see that the first columns of and are now the same. Step 2: We shift the only 1-entry in the second column of all the way up to the first row, by swapping and. The matrix now is given in Fig. 4(b). As, the multicast packet, which corresponds to the first row of, still remains to be after the aforementioned swap. As is now zero, the delay of the third packet is decreased to.thesearethe only changes in the total delay of after this step. Therefore, is not increased [in fact, it is decreased by (s)]. Now the first two columns of and are the same. Step 3: We first swap and. The delay of the second multicast packet is still after the swap. The delay of the forth multicast packet, from, is now decreased to.next, we swap the third and the fifth row of. The total delay of is unchanged. The matrix now is given in Fig. 4(c). The first three columns of and are the same. Their fourth columns are also identical. Lemma 6: Let be an binary matrix where for every.then. Proof: The proof can be found in the Appendix. Theorem 7: The RCDE scheme obtains the minimum total delay among all feasible schemes for the RCDE instance. Its total delay is the sum of the total delay of the two phases, where we have the following.

8 DONG et al.: DELAY MINIMIZATION FOR RELAY-BASED COOPERATIVE DATA EXCHANGE WITH NETWORK CODING ) The total delay of the Uploading Phase is the minimum cost of an optimal scheme for the corresponding CDECC instance, which can be found in polynomial time [8]. 2) The total delay of the Downloading Phase is calculated as follows (13) whereweset. Proof: The optimality of is proved by Lemma 6 and the preceding discussion. The assertion on the total delay of the Uploading Phase follows from (12) and Lemma 6. To prove that (13) holds, we show that there are multicast packets that have delay. Obviously, the first multicast packets have delay, due to (2) and (8). By the definition of, for each, there are precisely multicast packets that are assigned to but to none of the clients with. Due to (2) and (8), these are the only multicast packets that have delay. Hence, (13) follows. One important idea behind the packet assignment is that a coded packet sent from the relay need not be received by all clients but a subgroup of clients; provided that each client receives as many coded packets as its missing original packets, then the scheme can be made feasible via network coding. Given that the feasibility is satisfied, the total delay can then be minimized without much difficulty. This idea was used, as far as we know, first by Lun et al. [21]. In their work, the objective is to find a minimum-cost subgraph of a network that allows given multicast connections to be established. However, they studied the nonintegral setting where each packet is split into an infinite number of subpackets, while we are interested in the integral setting with indivisible packets. From a practical point of view, solutions to the integral setting are often preferred due to their simplicity in implementation, lower complexity in computation, and smaller buffer required at clients. In general, the integral setting might be harder to tackle than the nonintegral setting. However, in our case, because of the special objective function (the total delay), the optimal solution for the integral setting can be found in polynomial time. V. HEURISTIC SOLUTION FOR THE BINARY ALPHABET In previous sections, we provide an optimal RCDE scheme with the assumption that field size is as large as the number of clients. As computation over large field sizes can be rather expensive, an efficient scheme over smaller fields might be preferred in the scenarios where the wireless devices have very limited power, memory, or computational capability. In this section, we provide a heuristic solution for an RCDE scheme over. An additional requirement is that no buffering is allowed at the clients. In other words, every time a client received a coded packet from the relay, it must be able to extract an original packet from the coded packet, using its side information. This Fig. 5. Graph where isgivenbytablei. type of instantaneously decodable network coding scheme has been investigated in several works in literature (for instance, see Traskov et al. [22] and Sorour and Valaee [23] [25]). In this section, we adopt a graph-based solution by modeling the problem with a graph. We first determine which coded packets the relay should multicast in the Downloading Phase, using a clique-based scheme. Each multicast packet can be instantaneously decoded by relevant clients to retrieve one of their missing original packets. We then assign uploading tasks to suitable clients in a greedy manner in order to supply the required multicast packets for the Dowloading phase. A. Downloading Phase We first associate an RCDE instance with a graph the vertex set is where For any two vertices and of,thereisan(undirected) edge if any of the following two conditions is satisfied. 1), that is, both clients and want the same packet. 2),,and, that is, the packet required by is available at client and the packet required by is available at client. For example, when is given by Table I, the corresponding graph is given in Fig. 5. Suppose that is a clique (a complete subgraph) in. Let for some and for some. According to [26], after receiving the coded packet, the client can decode the original packet that it does not have, for all. Thus, each clique partition of the graph represents a coding scheme that satisfies the demands from all clients. To guarantee that all clients in can receive the coded packet, the relay must transmit the packet at the rate according to the minimum bandwidth between the relay and the clients in the group. Therefore, the delay of this coded packet is. For each clique in, we can assign as the weight of. A clique partition with the lowest total weight corresponds to a coding scheme that has the minimum total transmission delay for the Downloading Phase. However, finding such an optimal clique partition is hard because this problem reduces to the clique cover problem when for all, which is a well-known NP-hard problem [27]. We therefore develop a heuristic algorithm to obtain a suboptimal coding scheme for the Downloading Phase.

9 1898 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 23, NO. 6, DECEMBER 2015 Algorithm 1 Input:, ; 1: ; 2: while do 3: ; 4: Find a clique in ; 5: Compute for ; 6: ; 7: Remove from 8: end while Output: The set of multicast packets the set of cliques ; and The algorithm, namely Algorithm 1, finds a clique partition in and produces the corresponding coded packets that the relay multicasts in the Downloading Phase. In Step 3, a clique of is found by a simple greedy algorithm. This algorithm starts with an empty set of vertices and keeps adding a vertex of at each step; the vertex to be added must satisfy the following two requirements. 1) Adding to still produces a clique of. 2) is the minimum among all vertices that satisfy the first requirement. If there are more than one vertex satisfying the above two conditions, then a random selection is made. If this scenario happens, the output of Algorithm 1 is not unique. The output of this algorithm is a set of multicast packets that the relay transmits in the Downloading Phase. For example, if is given in Fig. 5 and are given by Table I, then Algorithm 1 may produce the following set of coded packets: The correponding cliques are,,and. The corresponding delays are,,, respectively. Algorithm 1 obviously runs in polynomial time. B. Uploading Phase Given that the set of multicast packets in the Downloading Phase is known, we need to decide which coded packets are transmitted from which clients so as to supply to the relay. The goal is to keep the total transmission delay in the Uploading Phase low. Ideally, for each packet, we would want to determine a set of clients and corresponding coded packets with minimum delay that collectively generate. However, this is a hard problem, as even when, it reduces to the set cover problem, another well-known NP-hard problem [27]. We propose a greedy algorithm to find a suboptimal solution for this problem in Algorithm 2. The main steps in this algorithm are explained as follows. The for-loop simply examines each clique one by one, and the inner while-loop finds a reasonable way for clients to supply the corresponding multicast packet to the relay. Note that it is not always the best to choose a client with shortest uploading delay since a client with longer uploading Algorithm 2 Input:, ; 1: for do 2: Compute for some for ; 3: while do 4: ; 5: for do 6: ; 7: ; 8: if then 9: ; 10: ; 11: end if 12: end if 13: ; 14: transmits to the relay; 15: Delete from ; 16: end while 17: end for delay may encode and upload multiple packets in one transmission, which can well lead to shorter overall delay. Therefore, for each client, we judge its relevance to via the quantity. Here, measures how many original packets involved in the linear combination that forms that has. A client that shares more common original packets with and has shorter uploading delay will have higher relevance to.at every iteration of the while-loop, we choose the client with the highest relevance, that is, the client with the largest,to upload a coded packet to the relay. This packet is the XOR-sum of the original packets in. When the while-loop terminates, is the XOR-sum of the coded packets generated within this loop. VI. SIMULATIONS In this section, we set up a simulation environment and conduct several experiments to demonstrate the effectiveness of our coding schemes in a network where every node has a different transmission delay in receiving/sending a packet from/to the relay. The sets are randomly generated. The uploading delays ( ) are randomly selected from the interval, and the downloading delays ( )are also randomly selected from the interval. The three coding schemes are the following. Trivial scheme with no coding (TSNC): TSNC does not consider any encoding of the packets. In the Uploading Phase, the clients with the minimum uploading delays are chosen to send each original packet to the relay. In the Downloading Phase, each original packet is multicast to a subgroup of clients that initially do not have that packet. An example of such a scheme is Scheme A in Fig. 2. The optimal RCDE scheme (OPTC) established in Section IV. The binary RCDE scheme (BINC) obtained by the heuristic algorithm established in Section V.

10 DONG et al.: DELAY MINIMIZATION FOR RELAY-BASED COOPERATIVE DATA EXCHANGE WITH NETWORK CODING 1899 total transmission delay can be found in polynomial time. Such a scheme consists of two parts: an optimal solution of the Cooperative Data Exchange with Cost Criterion is used for the Uploading Phase, and an optimal packet assignment is used for the Downloading Phase. We also propose a greedy algorithm to find a feasible RCDE scheme with small total delay when the binary field is used (only simple binary XOR operation is used for encoding). Simulation results show that the performance of the proposed binary solution is close to that of the optimal solution. APPENDIX Fig. 6. Number of clients varies while the number of original packets is kept fixed at 20., (in milliseconds). Fig. 7. Number of original packets varies while the number of clients is kept fixed at 10., (in milliseconds). All three solutions are implemented in MATLAB. For OPTC, we use the algorithm introduced by Courtade and Wesel [8] (based on Submodular Function Optimization) to find the optimal counting vector for the Uploading Phase. We compare these three coding schemes in terms of the total transmission delay, which is the sum of the total transmission delays for the Uploading Phase and for the Downloading Phase. For each setting, we present the average result of 100 samples. In Fig. 6, the number of original packets is kept fixed at 20, while the number of clients varies from 4 to 12. In Fig. 7, the number of clients is kept fixed at 10, while the number of original packets varies from 5 to 45. As expected, the optimal scheme OPTC always results in the lowest total transmission delay, and the trivial scheme with no coding TSNC always requires the highest total transmission delay. This is consistent with our theoretical results established in previous sections of this paper. We also observe that the proposed BINC has only a slightly higher delay compared to OPTC, but requires a lower computational complexity due to simple and fast binary XOR encoding. VII. CONCLUSION We study in this work the delay minimization issue for the Relay-based Cooperative Data Exchange problem. Our main contribution is to prove that minimizing the delay for the Downloading Phase and the Uploading Phase can be done separately. As a result, a feasible RCDE scheme with minimum A. Proof of Lemma 3 Assume that there exists a feasible RCDE scheme where:, ; uploads packets to the relay,, in the uploading phase; the relay transmits the multicast packet ( )to if and only if. Then, can multicast packets to all sinks in simultaneously using the following linear coding scheme. sends to for every. ( )sends to ( ) if there is an edge connecting and. sends ( )to for every,whichis possible as the capacity of the the edge is infinity. sends to for every,whichis possible as the capacity of the edge is. sends to for every. ( ) sends to ( )ifthereisanedge connecting and. Indeed, notice that each sink ( ) acquires exactly the same set of data as the client does with the RCDE scheme, namely As each client can recover all original packets in the RCDE scheme, so can each sink in the network. Assume that the network is solvable. By definition, there is a linear coding scheme that allows to multicast packets to all sinks simultaneously. By applying an invertible linear transformation if necessary, we can suppose that sends to for every. For each,let (some can be zero) be the packets sent through the edge. For each,let be the packet sent through the edge. Then, each is a linear combination of the vectors in the set (14) Note that via the edge with infinite capacity, the sink receives a subspace of the space spanned by.moreover, as edges and all have capacity one, each sink obtains a subspace of the space spanned by the set Therefore, having access to is sufficient to recover all,.

11 1900 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 23, NO. 6, DECEMBER 2015 We now let be the packets that transmits to the relay in the Uploading Phase, and,,be the packets that the relay multicasts in the Downloading Phase. Each is transmitted to ifandonlyif.by(14),the packets that the relay multicasts are indeed linear combinations of the packets it receives in the Uploading Phase. Moreover, each client has access to and hence can recover all original packets. Thus, the scheme is feasible. B. Proof of Lemma 4 According to Lemma 3, it suffices to show that the network is solvable if and only if (C1) and (C2) are satisfied. It is well known from the network coding literature (for instance, see [19]) that is solvable if and only if every cut between the source and a sink has capacity at least. We refer to the latter as the min-cut condition for. Recall that a cut of a network is a partition of the set of nodes into two parts, namely and. We are only interested in cuts that separate the source and some sink, i.e., contains the source and contains some sink. Let denote the capacity of an edge in a network. Then, the capacity of a cut is defined as Since separates and,wehave. Therefore for every. Thus, (C2) holds. If Direction: We now suppose that (C1) and (C2) hold, and we aim to show that any cut that separates the source and a sink must have capacity at least. Let be any cut of,where and for some.if,then trivially. Therefore, we suppose that. As,wehave. We consider the following two cases. Case 1: : Let As,wehave, and hence.for,as if,wehave whenever. Therefore Only If Direction: Suppose that the min-cut condition for holds, we aim to show that (C1) and (C2) are also satisfied. We first prove that (C1) is satisfied. Take an arbitrary subset. Consider the cut where where in the last inequality, (C1) is applied for. Case 2: : Let and contains the remaining vertices of the network. Then Recall that.since and, we have. Moreover, as whenever, we conclude that all such belong to. Therefore Note that since, contains a sink for some. Therefore, by the min-cut condition,. Hence Since we have Thus, (C1) is satisfied. We now prove that (C2) is also satisfied. For each, let us consider the cut where and contains the remaining vertices. Then

12 DONG et al.: DELAY MINIMIZATION FOR RELAY-BASED COOPERATIVE DATA EXCHANGE WITH NETWORK CODING 1901 Thus, in both cases, Lemma 4. C. ProofofLemma6. We complete the proof of Since for all,wehave The idea is to repeatedly modify the matrix through steps, so that at the final step, is turned into, and after each step, the total delay of A can only decrease or remain the same. As the total delay never goes up during the whole process, we conclude that. Hereafter,wesaythatthetwocolumnvectors and are almost identical if their first coordinates are identical and the last coordinates of are all zeros. Step 1: As, we can permute the rows of (if necessary) so that the first columns of and are almost identical. As permuting rows does not affect the total delay, after Step 1, the total delay of remains the same. Step ( ): Suppose that up to Step,the first columns of and are almost identical. In this step, we modify so that the th columns of and become almost identical. Intuitively, we shift all of the 1-entries in the th column of upward as much as we can and prove that during the process, the total delay of is not increased. Let s.t. In other words, denotes the set of upper rows of,and each of these contains at least a 1-entry that is located within the first columns. Note that consists of the first rows of. In opposite, denotes the set of remaining lower rows of, where all entries in these rows that are located within the first columns are zeros. The following modifications to do not increase and, at the same time, keep the first columns of unchanged. (M1) Modify the entries in the th column that are located within the first rows. As the first columns of and are almost identical, the delays (w.r.t. ) of the first multicast packets are from the set.since for all,any change in the th column within the first rows does not affect the delays of the corresponding packets. (M2) Turn a 1-entry in the th column that is located within the last rows into a 0-entry. The delay of the corresponding multicast packet is changed from to for some.as, the packet delay is not increased. (M3) Permute rows in. It is obvious that permuting rows in does not affect. Moreover, by definition of and, permuting rows within does not affect the first columns of. With (M1) (M3) in mind, we now apply some modifications to. Within the first rows, in the column of,weswap pairs of 0- and 1-entries such that the 0-entries are below all the 1-entries. Due to (M1), remains unchanged. We next consider two cases. (C1) The th column of has no 1-entries in the last rows. Then, we are done for Step since now the th column of is already almost identical to that of. (C2) The th column of has some 1-entries in the last rows. We now examine only the entries in the th column of. a) If there are as many 0-entries in the upper part as 1-entries in the lower part, then we can shift the 1-entries all the way up by applying appropriate entry swaps; doing so makes the th columns of and almost identical. By (M1) and (M2), is not increased. b) If there are fewer 0-entries in the upper part than 1-entries in the lower part, we first shift as many of the 1-entries as we can from to ; then all entries in are one. By (M1) and (M2), is not increased. Finally, we permute rows in so that in the th column of,the 1-entries lie above all the 0-entries. Then, the th columns of and are almost identical. Moreover, by (M3), is unchanged. Step : The previous steps guarantee that is not increased and all columns of are almost identical to that of. Therefore, the last rows of are all-zeros. In this step, we remove the last rows of,toturn into. Certainly, remains unchanged in this step. ACKNOWLEDGMENT The authors thank X. Xu and T. 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Dong, C. Zhan, and Y. Xu, Delay aware broadcast scheduling in wireless networks using network coding, in Proc. 2nd NSWCTC, 2010, vol. 1, pp [27] R. M. Karp, Reducibility among combinatorial problems, Complexity Comput. Comput., vol. 40, no. 4, pp , and combinatorics. Son Hoang Dau received the B.S. degree in applied mathematics and informatics from Vietnam National University, Hanoi, Vietnam, in 2006, and the M.S. and Ph.D. degrees in mathematical sciences from Nanyang Technological University, Singapore, in 2009 and 2012, respectively. He is currently a Postdoctoral Research Fellow with the SUTD-MIT International Design Centre, Singapore University of Technology and Design, Singapore. His research interests include coding theory, network coding, distributed storage systems, Chau Yuen (S'02 M'08 SM'12) received the B.Eng. and Ph.D. degrees in electrical and electronic engineering from Nanyang Technological University, Singapore, in 2000 and 2004, respectively. In 2005, he was a Postdoctoral Fellow with Lucent Technologies Bell Labs, Murray Hill, NJ, USA. In 2008, he was a Visiting Assistant Professor with Hong Kong Polytechnic University, Kowloon, Hong Kong. From 2006 to 2010, he was with the Institute for Infocomm Research, Singapore, as a Senior Research Engineer. Since 2010, he has been an Assistant Professor with the Singapore University of Technology and Design, Singapore. Dr.YuenservesasanAssociateEditorfortheIEEETRANSACTIONS ON VEHICULAR TECHNOLOGY. He received the IEEE Asia Pacific Outstanding Young Researcher Award in Yu (Jason) Gu (S'07 M'10) received the Ph.D. degree in computer science from the University of Minnesota, Twin Cities, Minneapolis, MN, USA, in He is currently an Assistant Professor with the Pillar of Information System Technology and Design, Singapore University of Technology and Design, Singapore. He is the author and coauthor of over 80 papers in premier journals and conferences. His publications have been selected as graduate-level course materials by over 20 universities in the US and other countries. His research includes networked embedded systems, wireless sensor networks, cyber-physical systems wireless networking, real-time and embedded systems, distributed systems, vehicular ad hoc networks, and stream computing systems. Dr. Gu is a member of the Association for Computing Machinery (ACM). Zheng Dong received the B.S. degree in computer science from Wuhan University, Wuhan, China, in 2007, and the M.S. degree in software engineering from the University of Science and Technology of China, Hefei, China, in He is currently a Research Assistant with the SUTD-MIT International Design Centre, Singapore. His research interests include network coding, wireless sensor networks, and real-time and embedded systems. Xiumin Wang received the B.S. degree in computer science from Anhui Normal University, Wuhu, China, in 2006, and the joint Ph.D. degree from the School of Computer Science and Technology, University of Science and Technology of China, Hefei, China, and City University of Hong Kong, Hong Kong, in She did her postdoctoral work with the Singapore University of Technology and Design, Singapore, from 2011 to Currently, she is with the School of Computer and Information, Hefei University of Technology, Hefei, China. Her research interests include wireless networks, routing design, and network coding.