Optimal Multicast in Dense Multi-Channel Multi-Radio Wireless Networks

Transcription

1 Optimal Multicast in Dense Multi-Channel Multi-Radio Wireless Networks Rahul Urgaonkar IBM TJ Watson Research Center Yorktown Heights, NY Prithwish Basu and Saikat Guha Raytheon BBN Technologies Cambridge, MA {pbasu, Ananthram Swami US Army Research Laboratory Adelphi, MD Abstract We study the problem of maximizing the multicast throughput in a dense multi-channel multi-radio (MC-MR) wireless network with multiple multicast sessions. Specifically, we consider a fully connected network topology where all nodes are within transmission range of each other. In spite of its simplicity, this topology is practically important since it is encountered in several real-world settings. Further, a solution to this network can serve as a building block for more general scenarios that are otherwise intractable. For this network, we show that the problem of maximizing the uniform multicast throughput across multiple sessions is NP-hard. However, its special structure allows us to derive useful upper bounds on the achievable uniform multicast throughput. We show that an intuitive class of algorithms that maximally exploit the wireless broadcast feature can result in very poor worst case performance. Using a novel group splitting idea, we then design two polynomial time approximation algorithms that are guaranteed to achieve a constant factor of the throughput bound under arbitrary multicast group memberships. These algorithms are simple to implement and provide interesting tradeoffs between the achievable throughput and the total number of transmissions used. I. INTRODUCTION In dense wireless networks, interference from concurrent transmissions becomes a significant impediment towards achieving acceptable throughput. As shown by the celebrated work in [1], it is necessary to maximize the number of concurrent transmissions in order to achieve optimal capacity scaling with network size. The strategy outlined in [1] relies on transmit power control to reduce interference and maximize spatial reuse. Subsequent work in [2] among others proposes highly sophisticated physical layer techniques based on distributed MIMO that increase capacity. Given the vast literature in this area, we refer to the surveys [3], [4] for an overview of this body of work. However, we note that for many scenarios of practical interest, such fine-grained power control or sophisticated physical layer techniques may not be feasible. A practical way to improve capacity in such settings is to deploy multi channel multi radio (MC-MR) wireless networks. The presence of additional orthogonal channels allows a higher This research was sponsored in part by the U.S. Army Research Laboratory and the U.K. Ministry of Defence and was accomplished under Agreement Number W911NF The views and conclusions contained in this document are those of the author(s) and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Army Research Laboratory, the U.S. Government, the U.K. Ministry of Defence or the U.K. Government. The U.S. and U.K. Governments are authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon. number of concurrent transmissions as compared to the single channel model of [1] and results in higher throughput. Several works have studied the problem of channel assignment, routing, and scheduling in MC-MR wireless networks under different constraints [5] [8]. However, all of these focus on unicast traffic models. We note that the multicast traffic model is prevalent in many settings (especially in military networks) and it introduces new design trade-offs that are quite different from those under the unicast traffic models, particularly in dense MC-MR networks. In this work, we characterize the multicast capacity of a dense MC-MR network that must support traffic from multiple multicast sessions with potentially arbitrary group memberships. We further allow dynamic channel assignments over time. This results in a highly challenging combinatorial problem that requires a joint design of channel assignment, scheduling, as well as routing strategies and becomes intractable for general networks. Therefore, to gain insight, we study a simpler network topology where all nodes are within the transmission range of each other. In spite of its simplicity, this topology is practically important since it is encountered in several real-world settings. Examples include (1) a collection of indoor wireless devices in a smart home, (2) a set of wireless routers that form an intra data center network among multiple servers, and (3) a platoon of soldiers with handhelds in a battlefield. Further, a solution for this network can serve as a building block for more general scenarios that are otherwise hard to analyze. For this network, we show that the problem of maximizing the uniform multicast throughput across multiple sessions is still NP-hard. However, its special structure allows us to derive useful upper bounds on its multicast capacity. Maximizing the multicast throughput for this network involves the following design challenge. When all nodes are within each other s transmission range, multicast transmissions can maximally use the wireless broadcast advantage because a single transmission by a source node reaches all of its group s receivers. Thus, it is tempting to use such single-hop broadcast to schedule all transmissions. However, when nodes belong to multiple multicast groups, we show that the class of algorithms that try to maximize the use of broadcast advantage can surprisingly result in very poor performance in the worst case. To overcome this limitation, in this paper we introduce the novel idea of group splitting where the multicast groups are fragmented into smaller subsets that are then scheduled in an appropriate manner. We design two polynomial time approximation

2 algorithms based on this idea and show that they can achieve performance within a constant factor of the upper bound under arbitrary multicast group memberships. These algorithms are simple to implement and provide interesting trade-offs between the achievable throughput and total number of transmissions used. An interesting feature of these algorithms is that their performance guarantees also hold in the online setting where group memberships are revealed one by one. We highlight the main differences between our approach and prior work on multicast in MC-MR wireless networks. The work in [9] considers the problem of optimal multicast in MC- MR networks assuming a single multicast session and static channel assignment. It uses an Integer Linear Programming (ILP) formulation and proposes heuristics based on LP relaxation. Similar ILP formulations are considered for multicast routing in MC-MR mesh networks in [10] [12]. Our approach is based on a performance ratio analysis and provides rigorous approximation bounds. Refs. [13], [14] study scaling laws for multicast in MC-MR networks using the framework originally proposed in [1]. In contrast, our model is applicable to any finite size network. Further, we note that none of these works considers multiple multicast sessions with arbitrary group memberships. The work in [15] is closest to ours in that it also considers the problem of maximizing multicast throughput across multiple sessions in a single-hop network. However, it only focuses on the single-hop broadcast strategy that can result in very poor performance in the worst case. II. NETWORK MODEL In this section, we discuss the basic network model and assumptions. We consider an MC-MR wireless network of N nodes, indexed 1, 2,..., N, where all nodes are within transmission range of each other. Each node in the network has T 1 identical transceivers. In addition, the network has K different multicast groups, denoted by G 1, G 2,..., G K, where each group is formed by a subset of the N nodes. The network also has a total of C orthogonal frequency channels available, each of bandwidth B Hz. We make the following assumptions in this model. Transmission Model: We assume that the transceivers operate in half-duplex mode where they can either transmit or receive (but not both) on a given channel at any time. A transceiver can be assigned to transmit or receive on at most one channel at any time. Further, we assume a collision model for interference so that at most one transceiver can transmit successfully on a channel at any time. A successful transmission on a channel is received error-free by all the other nodes that have at least one of their transceivers receiving on that channel during the transmission. The link level transmission rate per transceiver (channel) is assumed to be the same for all transceivers (channels) and is given by R bps. Channel Assignment: We consider dynamic channel assignment schemes where a node s transceivers may be assigned different channels over time. This is different from the more commonly studied static channel assignment schemes where a particular assignment, once chosen, is used for all time. It can be viewed as a special case of dynamic channel assignment. Note that in our model if two nodes share at least one channel at any time (i.e., the set of channels that their transceivers are TABLE I. A POSSIBLE TDMA SCHEDULE FOR THE EXAMPLE NETWORK. FOR EACH NODE, ENTRIES IN A CELL REPRESENT THE MULTICAST GROUP WHOSE TRANSMISSION THIS NODE PARTICIPATES IN DURING THAT SLOT AS WELL AS THE CHANNEL USED. Node ID slot 1 slot 2 slot 3 slot 4 1 G 1, f 1 G 3, f 1 2 G 1, f 1 G 2, f 2 3 G 1, f 1 G 4, f 1 4 G 2, f 2 G 5, f 2 5 G 1, f 1 G 3, f 1 6 G 2, f 2 7 G 3, f 1 G 4, f 1 8 G 4, f 1 G 5, f 2 tuned to have non-zero overlap), then these nodes can reach each other in one-hop. However, when two nodes do not have any common channel, they will not be directly connected (even though they are in the transmission range of each other). Multicast Groups: We do not make any assumptions on the distribution of the nodes across the multicast groups G 1, G 2,..., G K except that each group has at least 2 nodes. Thus, the group memberships in our model can be arbitrary. For simplicity, we assume that the group memberships are fixed for the duration of interest. If group memberships change over time, the transmission schedule derived from our algorithms can be updated accordingly. We also assume that the group memberships known to us. However, as we will show later, the algorithms that we present in this paper work equally well in the online setting where group memberships are revealed one by one. Multicast Traffic: Each multicast group consists of a source node that generates packets of size D bits at a rate given by λ packets/sec. Each packet must be delivered (possibly over time) to all the other members of that group in order to satisfy the multicast requirement. We say that a group achieves a multicast throughput of µ packets/sec if its source node is able to deliver packets at the time-average rate of µ to all the other nodes in that group. Performance Objective: Given this network model, our objective is to design an algorithm that maximizes the uniform multicast throughput per group. That is, we want to maximize µ such that every group G 1, G 2,..., G K is able to achieve a multicast throughput of µ packets/sec. We call this the uniform multicast throughput maximization problem. As an illustration of the basic model and objective, consider an example network with 8 nodes and 5 multicast groups with memberships given by G 1 = {1, 2, 3, 5}, G 2 = {2, 4, 6}, G 3 = {1, 5, 7}, G 4 = {3, 7, 8} and G 5 = {4, 8}. Suppose T = 1 so that each node has only 1 transceiver and there are 3 available channels f 1, f 2, f 3. For simplicity, assume unit packet size and transmission rates. Also assume that the lowest id node in each group is the source node for that group. In Table I, we show one possible TDMA schedule for this network that achieves a uniform multicast throughput of 1/4 packets/sec by repeating this schedule over time. While our performance objective is seemingly straightforward, the underlying problem is actually quite challenging. This is because in order to maximize the multicast throughput across all groups, we need to optimize along several dimensions. First, given the MC-MR network, our algorithm must optimize for channel assignments. This is further complicated

3 by the fact that we allow the channel assignments to be dynamic over time. Second, the interference and collision constraints necessitate the design of scheduling policies to share the channels between different users. Finally, while all nodes are within each other s transmission range, the channel assignment may effectively result in a multi-hop topology. In such cases, multi-hop multicast routing will be required to ensure that all group members receive the packets generated by the source. Therefore, in general the overall problem consists of channel assignment, transmission scheduling, as well as routing decisions and is combinatorial in nature. A. NP-hardness Here, we show that the problem of maximizing the uniform multicast throughput is NP-hard. We assume that T = 1 and show that any general instance of the bin packing problem can be reduced to an instance of the uniform multicast throughput maximization problem. Theorem 1: The uniform multicast throughput maximization problem is NP-hard. Proof: See Appendix. The reduction in the proof above considers the more restricted class of single-hop broadcast strategies. In general, more sophisticated strategies involving group splitting and routing can be used to achieve higher throughput as compared to single-hop broadcast. This makes the overall problem highly challenging. In order to approach this problem in a tractable manner, we first develop a simple upper bound on the maximum uniform multicast throughput. This bound will be useful in characterizing the performance of our algorithms in the rest of the paper. We will also show some special cases where this bound is tight. For simplicity, in the rest of the paper, we assume normalized and idealized transmission rates so that one packet can be transmitted per channel per slot. III. AN UPPER BOUND Let d n represent the number of multicast groups that node n belongs to. We call d n the participation degree of node n. Define d max to be the maximum participation degree across all nodes, i.e., d max= max d n (1) n {1,2,...,N} As an example, d max = 2 in the example network in Table I. Then, we have the following. Lemma 1: The uniform multicast throughput µ achievable under any algorithm satisfies [ T µ min, C ]. (2) d max K Proof: We first show that µ T/d max. Consider any node n with participation degree d n. This node must send and/or receive packets from at least d n different flows using its T transceivers. Since each transceiver has a unit capacity and it operates in half-duplex mode, the maximum traffic rate per flow cannot exceed T/d n. Applying this bound over all nodes, we have that µ T/d max. In order to show that µ C/K, Fig. 1. The conflict graph for the example network in Table I. note that the overall network must support K different flows using at most C unit capacity channels. Thus, the maximum traffic rate per flow cannot exceed C/K. The overall bound in (2) follows by taking the minimum of these two bounds. The two bounds in (2) can be interpreted as two different operating regimes of the network: transceiver-limited and channel-limited. The transceiver-limited case corresponds to when there are sufficient number of channels but not enough transceivers while the other one corresponds to the complementary case. In the rest of the paper, we will focus on the transceiver-limited case so that the effective upper bound is µ T/d max. We will assume that C T K/d max throughout. The channel-limited case is interesting in its own right but beyond the scope of this paper. Before proceeding, we define the conflict graph representation of our model for the case T = 1 which will be useful in subsequent sections. Definition 1: Given multicast groups G 1, G 2,..., G K, the conflict graph K is defined as a graph containing K vertices such that each vertex k K corresponds to multicast group G k and there is an edge connecting any two vertices j and k if multicast groups G j and G k have at least one common member, i.e., G j G k. As an example, the conflict graph for the network in Table I is shown in Fig. 1. We call this the conflict graph because the groups corresponding to any two adjacent vertices cannot be scheduled simultaneously. A. Performance Ratio In general, the bound µ T/d max is not necessarily tight. However, it is quite useful in characterizing the performance of the algorithms in the subsequent sections. Note that this bound depends only on T (which is a constant) and d max (which is a function of the group memberships). It does not directly depend on parameters such as the number of nodes N and the number of multicast groups K. Note also that the bound is identical for two potentially very different multicast group membership distributions as well as network sizes as long as d max is the same. We will characterize the performance of our algorithms in terms of a performance ratio which is defined as follows. Fix a value of T and d max. Definition 2: For any algorithm A and an instance X of our model with maximum participation degree d(x), let µ A (X) denote the achievable uniform multicast throughput. Then the performance ratio Φ A (T, d max ) of the algorithm A

4 Fig. 2. Example of a conflict graph that is an even cycle. Even and odd numbered groups can transmit in consecutive slots. is defined as Φ A (T, d max ) = T/d max arg max X:d(X)=d max µ A (X) For a given T and d max, Φ A (T, d max ) represents the worst possible ratio between the upper bound T/d max and the achievable uniform multicast throughput under algorithm A over all possible network instances that have maximum participation degree of d max. By definition, we have that Φ A (T, d max ) 1 for all A, T, d max. Also, a performance ratio of Φ A (T, d max ) means that the algorithm A can always achieve a fraction 1/Φ A (T, d max ) of the upper bound T/d max for any network instance with T transceivers per node and maximum participation degree of d max. Since the optimal uniform multicast throughput is upper bounded by T/d max, it follows that algorithm A can always achieve a fraction 1/Φ A (T, d max ) of the optimal capacity. Note that for a given T and d max, there may not exist any algorithm that has Φ A (T, d max ) = 1 because the upper bound T/d max may not be tight. B. Special Cases Here, we provide some special cases where the upper bound µ = T/d max is achievable. 1. When T = d max, µ = T/d max = 1 and can always be achieved. To see this, consider the following channel assignment algorithm. First, assign a unique channel f k to each multicast group G k, i.e., all members of that group are assigned this channel. Then for each node n, the total number of channels it gets assigned to is equal to its participation degree d n. Since d n d max = T, each node has a sufficient number of transceivers to assign to its channels subject to the constraints of the model. Note that under this algorithm, the channel assignment is fixed for all time and each multicast transmission happens as a single broadcast. 2. When T = 1 and the conflict graph K is an even cycle, then d max = 2. This is because a node can belong to at most two groups. Otherwise, there would be a clique of size 3 or more in K. Thus µ = 1/2. Next, note that when K is an even cycle, it can always be partitioned into two equal sized subsets of nodes that do not conflict with each other. Fig. 2 illustrates an example. By scheduling these two subsets in consecutive slots, we can achieve µ = 1/2. 3. When K = d max and T = 1, the bound µ = 1/d max can be achieved simply by scheduling one multicast group at a time. In the rest of the paper we will assume T d max since the case T = d max is already covered as a special case. (3) IV. SINGLE-HOP BROADCAST In this section, we study the performance of a simpler class of algorithms for the uniform multicast throughput maximization problem where each multicast transmission is restricted to a single-hop broadcast. Intuitively, this strategy is appealing because (1) it fully exploits the broadcast advantage, and (2) uses the smallest number of transmissions to deliver any multicast packet. Further, it also models the behavior of the multicast algorithms proposed in prior works [9] [12], [15] since the basic multicast structures of Steiner tree or connected dominating set reduce to a single node in our network model. Note that the single-hop broadcast constraint subsumes any need for routing, making the design space considerably smaller. However, as we will show, even with this simplification the problem remains NP-hard. In order to analyze the performance of single-hop broadcast algorithms, we first define an independent set over the multicast groups G 1, G 2,..., G K. Definition 3: An independent set I over the multicast groups G 1, G 2,..., G K is a set of groups such that no node is a member of more than T of these groups. Note that any independent set is a feasible transmission schedule under the single-hop broadcast algorithms. This is because all multicast groups in an independent set can be scheduled to transmit concurrently on orthogonal channels. Similarly, any feasible transmission schedule under the singlehop broadcast policies can be mapped to an independent set. Note that when T = 1, the independent sets defined above correspond to the independent sets of the conflict graph K defined in Sec. III. Let I denote the set of all independent sets over the multicast groups G 1, G 2,..., G K. Also, let I(G k ) denote the set of all independent sets that include the multicast group G k. The achievable uniform multicast throughput under the singlehop broadcast algorithms can be then characterized as follows (proof omitted for brevity). Theorem 2: A uniform multicast throughput of µ is achievable under the single-hop broadcast algorithms iff there exist non-negative real numbers x I [0, 1] for each I I such that x I k {1, 2,..., K} (4) I I(G k ) x I µ I I Thus, under the single-hop broadcast constraint, the uniform multicast throughput maximization problem becomes equivalent to finding the maximum value of µ for which (4) is satisfied. The optimal values of x I can be interpreted as the fraction of time independent set I is activated. Thus the overall problem is equivalent to finding the optimal time-sharing between all independent sets of the multicast groups and is an NP-hard problem. One may consider designing approximation algorithms for this problem. However, we next show that the performance ratio of this class of algorithms is unbounded for all T and d max. Theorem 3: The performance ratio of the class of singlehop broadcast algorithms is unbounded for all T and d max.

5 TABLE II. EXAMPLE GROUP MEMBERSHIP WHEN T = 1 THAT RESULTS IN A CONFLICT GRAPH THAT IS A CLIQUE AS SHOWN IN FIG. 3. Multicast Group Group Membership G 1 {1, 2, 3, 4, 5} G 2 {1, 6, 7, 8, 9} G 3 {2, 6, 10, 11, 12} G 4 {3, 7, 10, 13, 14} G 5 {4, 8, 11, 13, 15} G 6 {5, 9, 12, 14, 15} TABLE III. EXAMPLE GROUP MEMBERSHIP WHEN T = 2 SUCH THAT AT MOST 2 GROUPS CAN BE SCHEDULED SIMULTANEOUSLY. THE MAXIMUM POSSIBLE THROUGHPUT UNDER SINGLE-HOP BROADCAST FOR THIS NETWORK IS 2 7. Multicast Group Group Membership G 1 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} G 2 {1, 2, 3, 4, 5, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} G 3 {1, 6, 7, 8, 9, 16, 17, 18, 19, 26, 27, 28, 29, 30, 31} G 4 {2, 6, 10, 11, 12, 16, 20, 21, 22, 26, 27, 28, 32, 33, 34} G 5 {3, 7, 10, 13, 14, 17, 20, 23, 24, 26, 29, 30, 32, 33, 35} G 6 {4, 8, 11, 13, 15, 18, 21, 23, 25, 27, 29, 31, 32, 34, 35} G 7 {5, 9, 12, 14, 15, 19, 22, 24, 25, 28, 30, 31, 33, 34, 35} Fig. 3. The conflict graph corresponding to the multicast groups in Table II. The maximum possible throughput under single-hop broadcast is 1 6. Proof: Given a T and d max, in order to show that performance ratio is unbounded, we will construct a network instance with K multicast groups such that the maximum possible uniform multicast throughput under the single-hop broadcast constraint is T K. We will show that this holds for all T values of K. Thus, by letting K go to infinity, K approaches 0 and we have that the performance ratio is unbounded. For simplicity, we first consider the case T = 1 and d max = 2. Fix K > 1. We form K multicast groups using N = K(K 1) 2 nodes as follows. First, group G 1 is formed by assigning to it nodes 1, 2,..., K 1. Then, each of these nodes is assigned to be a member of one group in G 2,..., G K in ascending order. For example, node 1 is in G 2, node 2 is in G 3, and so on. This ensures that G 1 conflicts with each one of G 2,..., G K in the conflict graph K. Further, nodes 1,..., K 1 each have degree 2 and they will not be assigned to any other group. Next, this process is repeated by adding new nodes K, K + 1,..., 2K 3 to G 2 and then assigning them to be a member of one group in G 3,..., G K in ascending order. This ensures that G 2 conflicts with each one of G 3,..., G K in the conflict graph K. By repeating this procedure, we can construct an instance where d max = 2 but all groups conflict with each other. This means that the resulting conflict graph K is a clique of size K. Under the single-hop broadcast algorithms, at most one group can be activated at any time. Thus, the maximum possible uniform multicast throughput for this network is 1 K. An illustration of this construction is given in Table II for K = 6 and the resulting conflict graph is shown in Fig. 3. For the case when T = 1 and d max > 2, we can simply make node 1 a member of some additional groups such that its participation degree becomes equal to d max. The resulting conflict graph will continue to be a clique of size K. The general case T > 1 can be treated similarly and we omit the details for brevity. An illustration for T = 2 and K = 7 is given in Table III. Theorem 3 implies that any attempt to develop approximation algorithms for the class of single-hop broadcast algorithms may not be very useful since the performance ratio will remain unbounded. In the worst case, even the optimal performance of the single-hop broadcast algorithms scales as 1/K. Note also that the 1/K scaling can always be achieved by scheduling only one group at a time. In this sense, single-hop broadcast can result in the worst possible scaling. Intuitively, the construction in the proof of Theorem 3 suggests that this is due to the high node degrees in the resulting conflict graph that prohibits concurrent scheduling of multiple groups. One possible way to overcome this is to use group splitting, i.e., scheduling subsets of multicast groups instead of scheduling them in their entirety. This can potentially reduce the degree of the resulting conflict graph between these subsets. The approximation algorithms we present in the next section are based on this insight. V. APPROXIMATION ALGORITHMS In this section, we present two approximation algorithms that use group splitting to overcome the limitation of singlehop broadcast algorithms. These algorithms construct a TDMA frame and schedule transmissions such that each multicast group can deliver one packet per frame. Both of these algorithms use a common group splitting phase where an attempt is made to greedily schedule an entire multicast group in the smallest available slot in the TDMA frame. However, if this is not feasible, the group is fragmented into smaller subsets and these are assigned different slots. Note that the resulting fragmentation requires relaying between these fragments to complete the multicast packet delivery and these two algorithms differ in how this is done. We first describe the common group splitting phase. We assume for now that T = 1 and design a basic TDMA schedule. This same schedule can be used for the case T > 1 as shown later. A. Group Splitting Phase In the group splitting phase, we construct a basic TDMA frame as follows. Starting with the lowest id multicast group, greedily assign each member node of that multicast group to the lowest available time slot for that node in the TDMA frame. Note that all nodes of the first multicast group get assigned to slot 1. However, starting with the second group, it may happen that not all nodes of a multicast group can get the same slot number assigned to them for that group. This naturally leads to a group splitting because our scheduling strategy will schedule all nodes that are assigned a particular slot to transmit/receive on their respective multicast group. To illustrate the basic TDMA frame construction as well as the subsequent relaying mechanism of our approximation

6 TABLE IV. TDMA FRAME FOR THE EXAMPLE NETWORK (5) UNDER ALGORITHM 1. TABLE V. TDMA FRAME FOR THE EXAMPLE NETWORK (5) UNDER ALGORITHM 2. Node ID slot 1 slot 2 slot 3 slot 4 slot 5 slot 6 1 G 2 G 4 G 5 G 2 G 4 G 5 2 G 4 G 6 G 6 G 4 3 G 1 G 2 G 3 G 2 G 3 4 G 1 G 6 5 G 1 G 3 G 4 G 3 6 G 5 G 5 7 G 3 G 3 8 G 2 G 4 G 6 G 6 Node ID slot 1 slot 2 slot 3 slot 4 slot 5 slot 6 slot 7 1 G 2 G 4 G 5 G 2 G 4 G 4 G 5 2 G 4 G 6 G 6 G 4 3 G 1 G 2 G 3 G 2 G 3 4 G 1 G 6 5 G 1 G 3 G 4 G 3 G 3 G 4 6 G 5 G 5 7 G 3 G 3 8 G 2 G 4 G 6 G 6 algorithms, we will use an example network with parameters N = 8, K = 6 and d max = 3. The groups are given by G 1 = {3, 4, 5}, G 2 = {1, 3, 8}, G 3 = {3, 5, 7} G 4 = {1, 2, 5, 8}, G 5 = {1, 6}, G 6 = {2, 4, 8}. (5) Assume the lowest id node in each group is the source node for that multicast session. The first 3 slots of the TDMA frame in Table IV form the basic TDMA frame resulting from the procedure described above for this example. For a node n, the entry in any slot t represents the id of the multicast group that this node would be transmitting to/recceiving from in slot t. Notice that all nodes of G 1 are assigned slot 1. However, for G 2, nodes 1 and 8 are assigned slot 1 while node 3 gets slot 2. This means that node 3 would not be able to receive the multicast transmission of G 2 from source node 1 in slot 1. We say that G 2 has become fragmented and additional slots are needed to allow relaying of packets between these fragments. The two algorithms we next present schedule additional relay transmissions are that required after the basic TDMA frame has been constructed. Before analyzing their performance, we note an important property of the basic TDMA frame which follows from the greedy construction where, for each node, its lowest available slot is assigned first. Property 1: The length of the basic TDMA frame is d max. In order to calculate the achievable multicast throughput under the approximation algorithms, we will calculate bounds on the number of additional relay slots that are needed to complete all multicast transmissions. The achievable multicast throughput is then lower bounded as follows. Lemma 2: If the additional number of slots needed to schedule all relay transmissions to complete multicast delivery is, then a uniform multicast throughput of 1/(d max + ) is achievable. Proof: Note that each group transmits one packet in every TDMA frame. By periodically repeating this frame, we can achieve a uniform multicast throughput that is equal to the inverse of the total frame length. The bound follows by noticing that the total frame length is upper bounded by (d max + ). B. Algorithm 1 In Algorithm 1, we start with the basic TDMA frame. Then for each multicast group that was split during the basic TDMA frame construction, we select the lowest id node from each fragment as the relay node for that fragment. All the chosen relay nodes for this group are then assigned the lowest available common slot to complete the multicast transmission. Table IV shows the resulting TDMA frame for the example given in (5). Note that G 1 does not require any relay slots since it is not split. However, G 2 is split into fragments {1, 8} in slot 1 and fragment {3} in slot 2. Algorithm selects node 1 from the first fragment and node 3 from the second fragment and schedules them to perform relay transmission in slot 4. Similarly, nodes 3, 5, 7 get selected as relay nodes for the fragments of G 3 and are assigned slot 5 for relay transmission. Since Algorithm 1 schedules all relay nodes for a group in the same additional slot, it follows that the end-to-end path length from a multicast source node to any member is at most 2 hops. Let 1 denote the number of additional relay slots used by by Algorithm 1 for any network instance. Then, we have the following. Theorem 4: 1 is upper bounded by d max (d max 1). Proof: This bound is obtained by calculating the maximum possible number of fragments that can conflict with any given fragment under the assignment used by Algorithm 1. The full proof is provided in [16]. Theorem 4 together with Lemma 2 implies that Algorithm 1 can always achieve a uniform multicast throughput of 1/d 2 max for any network instance with maximum node participation degree d max. Using the definition (3), we get Φ Algorithm1 (1, d max ) d max. (6) Given d max, the performance ratio of Algorithm 1 is upper bounded by a constant and is independent of K and N. C. Algorithm 2 Algorithm 2 also starts with the basic TDMA frame and the set of relay nodes per group fragment are chosen in the same way as in Algorithm 1. However, the relay assignment is done differently. For each group, we pick relay nodes from two consecutive fragments and assign them a common additional relay slot. Table V shows the resulting TDMA frame for the example given in (5). As before, G 1 does not require any relay slots since it is not split. Since G 2 has only 2 fragments, the relay assignments are identical to Algorithm 1. However, G 3 has 3 fragments. Thus, unlike Algorithm 1 where the corresponding relay nodes 3, 5, 7 get assigned to slot 5, here nodes 5 and 7 are first assigned slot 4 and then nodes 3 and 5 are assigned to slot 5. Since Algorithm 2 schedules only two relay nodes for a group in the same additional slot, the total number of additional slots needed for a group is at most (d max 1) and the end-toend path length from a multicast source node to any member can be at most d max hops. Let 2 denote the number of

7 TABLE VI. SUMMARY OF PERFORMANCE BOUNDS. Perf. Ratio Total Transmissions Max. Path Length Algorithm 1 d max K(d max + 1) 2 hops Algorithm 2 4 K(2d max + 1) d max hops Single-Hop B/cast K 1 hop additional relay slots used by by Algorithm 2 for any network instance. Then, we have the following. Theorem 5: 2 is upper bounded by 3d max. Proof: This bound is also obtained by calculating the maximum possible number of fragments that can conflict with any given fragment under the assignment used by Algorithm 2. The full proof is provided in [16]. Theorem 5 together with Lemma 2 implies that Algorithm 2 can always achieve a uniform multicast throughput of 1/4d max for any network instance with maximum node participation degree d max. Thus, its performance ratio can be bounded as Φ Algorithm2 (1, d max ) 4. (7) Thus, the performance ratio of Algorithm 2 is constant and independent of K, N, or d max. D. Discussion When compared with the single-hop broadcast algorithm, both Algorithm 1 and 2 guarantee finite and bounded performance ratios and achieve a constant fraction of the upper bound on maximum uniform multicast throughput under arbitrary group memberships. Note that the performance ratio bound of Algorithm 1 depends on d max and thus could be poor when this is large (say d max = 10). On the other hand, Algorithm 2 has a constant performance ratio bound of 4. Intuitively, this is because the relay node fragments that Algorithm 2 needs to schedule are of size 2, thereby allowing a more efficient packing in the TDMA frame. As shown earlier, the maximum path lengths resulting from the assignments under these algorithms can be bounded. This can be used to bound the total number of transmissions that happen in a TDMA frame for each algorithm. Table VI summarizes these performance bounds. It can be seen that Algorithm 1 and 2 offer a trade-off between the total number of transmissions used and the performance ratio. Algorithm 2 uses more transmissions in the worst case but yields a superior performance ratio bound. It should also be noted that the construction of the schedule under both algorithms makes use of the membership information of only one group at a time. This means that the same algorithms can be used in the online setting where memberships are revealed one by one. Finally, we point out that these performance bounds represent the worst-case scenario and therefore can be quite loose on average. In order to better understand the performance of these algorithms in an average sense, we will evaluate them using simulations in the next section. Before proceeding, we describe how the TDMA frame construction as discussed earlier can be used for the case T > 1. Note that when T = 1, only one slot of the TDMA frame is active at any time. For T > 1, one can simply activate the schedules of T consecutive slots in a sliding-window fashion. For example, suppose T = 2 and the total frame length is 5. Then we can activate the schedules corresponding to the following pairs of slots: (1, 2), (2, 3), (3, 4), (4, 5), (5, 1). This will be a feasible schedule and it can be seen that it achieves twice the throughput of the T = 1 case. VI. EVALUATIONS In this section, we evaluate the performance of our approximation algorithms using simulations. These algorithms are implemented following the procedure described in the previous section. Given a set of K multicast groups, we first construct the baseline TDMA frame as described in Sec. V-A. As noted before, this may lead to potential group splitting that requires additional relay slots for the completion of multicast transmissions. For each algorithm, these additional slots are calculated as discussed in Secs. V-B and V-C. If the total length of the resulting frame is L, then the algorithm yields a uniform multicast throughput of 1/L. In the following, we assume that T = 1 noting that the same TDMA frame structure can be used for T > 1 to achieve throughput of T/L. We compare the performance of our approximation algorithms with the single-hop broadcast strategy. As noted before, in the fully connected topology this strategy models the behavior of the multicast algorithms proposed in prior works. Thus, our performance comparison can be viewed as comparing against any such algorithm. A. Performance when Conflict Graph is a Clique We first consider the case where the group membership is such that the resulting conflict graph becomes a clique. Examples of such groups are given in the proof of Theorem 3 where it is also shown that the maximum possible uniform multicast throughput under the class of single-hop broadcast algorithms is given by 1/K for such scenarios. We first use the construction outlined in Theorem 3 to generate K groups for different values of K in {5, 10,..., 50}. Recall that this results in d max = 2 for each K. Further, N = K(K 1) 2 and all nodes have the same participation degree. In Fig. 4, we compare the multicast throughput achieved by the approximation algorithms with that under the singlehop broadcast algorithm for different K. While the singlehop broadcast throughput decreases as 1/K as expected, it is remarkable that the throughput under both approximation algorithms remains constant (independent of K). The exact value of this throughput is 1/3. Intuitively, this follows by noticing that for this group construction, both algorithms are always able to schedule all relay transmissions using a single additional slot. In Fig. 4, we also plot the theoretical upper bound 1/d max = 1/2 and the respective lower bounds for the two algorithms: 1/d 2 max = 1/4 for Algorithm 1 and 1/4d max = 1/8 for Algorithm 2. Note that the single-hop broadcast algorithm has a trivial lower bound of 0. This plot illustrates our theoretical results that the performance ratio of these algorithms is independent of K while that of single-hop broadcast algorithms is unbounded. Next, we start with the same group membership as before but with probability 0.1, each node is assigned to be a member of the other groups in addition to its original groups. The

8 Multicast Throughput Algorithm 1 Algorithm 2 Single-Hop Broadcast Algorithm 1 Lower Bound Algorithm 2 Lower Bound Upper Bound Multicast Throughput Algorithm 1 Algorithm 2 Single-Hop Broadcast Algorithm 1 Lower Bound Algorithm 2 Lower Bound Upper Bound Number of Groups K Number of Groups K Fig. 4. Performance comparison for the multicast group construction in Theorem 3. Also shown are the theoretical upper and lower bounds. Fig. 5. Performance comparison after adding group memberships. motivation for doing this it to make the groups less homogeneous as compared to the original membership. Note that since we are potentially increasing the participation degrees of nodes, the resulting d max can become greater than 2. We repeat the simulations on the modified groups thus formed and Fig. 5 shows the results for a particular run. First, note that the performance of the single-hop broadcast algorithm does not change since the new conflict graph remains a clique for each K. However, the multicast throughput achieved by the approximation algorithms is no longer constant and tends to decrease with K. But as shown in Fig. 5, the upper bound on the multicast throughput is also decreasing with K. This can be explained by noticing that as K increases, the number of nodes N = K(K 1) 2 also grows and d max is likely to increase due to more nodes choosing additional group memberships. The two approximation algorithms outperform single-hop broadcast except for small K. B. Performance under Random Group Membership Now we consider general multicast group constructions without imposing any specific structure as in Sec. VI-A. Specifically, we fix the parameters K, N, d max and simulate the algorithms over 100 different multicast groups instances generated by uniform random assignment of nodes to groups subject to the following conditions: (1) each node belongs to at least one group, (2) each group has at least two nodes, and (3) the maximum node participation degree equals d max. We measure the average multicast throughput for each algorithm across these 100 iterations. Note that implementing the optimal single-hop broadcast would require solving an NP-hard problem (as shown in Theorem 2). Instead, we use a heuristic where maximal independent sets of the multicast groups are scheduled in a greedy fashion until all groups have been scheduled. In Fig. 6 we plot the average multicast throughput vs. K for the case where d max = 4 and N = K(K 1) 2. First, note that since we have fixed d max = 4, the upper bound is a constant equal to 1/d max = 1/4. Also, for this d max, the lower bounds of both approximation algorithms are the same. Next, we note that even in the absence of any specific structure, the throughput performance of both algorithms does not decrease with K like the greedy single-hop broadcast algorithm. However, Algorithm 1 outperforms Algorithm 2 for all K even though Algorithm 2 has a superior performance ratio bound theoretically. Finally, for smaller values of K, the greedy single-hop broadcast algorithm has the better performance. In Fig. 7 we repeat this for the case where d max = 4 and N = 2K. One big difference from Fig. 6 is that greedy single-hop broadcast significantly outperforms Algorithm 2. We believe this can be explained by noting that when N = 2K, the resulting conflict graphs are relatively sparse. In that case, it becomes feasible to schedule multiple groups concurrently and splitting groups and performing multi-stage relaying like Algorithm 2 may not be a good strategy. These results suggest that, given a network instance, one should test all three algorithms and then pick the best one. Note that both Algorithm 1 and 2 are easy to implement. Solving for the optimal single-hop broadcast algorithm is NP-hard, but a greedy maximal independent set heuristic can be used. This strategy ensures that the performance ratio remains bounded at all times while the actual performance can be significantly better than the theoretical lower bounds that many be loose. VII. CONCLUSIONS In this paper, we studied the problem of maximizing the multicast throughput across multiple sessions in dense MC- MR wireless networks where the multicast group memberships can be arbitrary. Our focus was on a simple network topology where all nodes are within the transmission range of each other. While simple, this scenario is important due to its prevalence in real-world settings. Further, the problem that we investigate has not been studied before in this setting. To address this challenging combinatorial problem, we first derived an upper bound on the multicast capacity. Then we designed two polynomial time approximation algorithms that are guaranteed to achieve a constant factor of thi bounds under arbitrary multicast group memberships. These algorithms are based on a novel idea of group splitting. Our focus in this work was on the transmitter-limited regime as discussed in Sec. III. It would be interesting to perform a similar analysis for the channel-limited case where the effective upper bound is µ C K. One specific sub-case that admits a simple solution is when C T. Here, the upper bound can be achieved by activating one group at a time and assigning all C channels to it. The case when C > T needs further investigation. Also of interest are extensions of this work to consider networks where not all nodes are in each other s transmission range. We believe that the results presented in this work can serve as a building block for designing algorithms for such cases.

9 Multicast Throughput Algorithm 1 Algorithm 2 Greedy Single-Hop Broadcast Algorithm 1 & 2 Lower Bound Upper Bound Multicast Throughput Algorithm 1 Algorithm 2 Greedy Single-Hop Broadcast Algorithm 1 & 2 Lower Bound Upper Bound Number of Groups K Number of Groups K Fig. 6. Comparison under random membership when N = K(K 1)/2. APPENDIX NP-HARDNESS Proof: We show that the decision version of a general instance of the bin packing problem can be reduced to the decision version of an instance of the uniform multicast throughput maximization problem when T = 1. Assume that in the bin packing problem, we are given K objects of integral sizes S 1, S 2,..., S K respectively. Also, we are given a collection of identical bins of size N each. We assume that S k N for all k {1,..., K}. Further, we assume that there exists an integer d max > 1 such that K k=1 S k = Nd max. If K k=1 S k < Nd max, we can create another object of size S K+1 such that K+1 k=1 S k = Nd max with S K+1 N. The decision version of the bin packing problem is to determine if there is a feasible bin packing of these objects that uses only d max bins. We will now reduce it to the decision version of an instance of the uniform multicast throughput maximization problem. Specifically, consider a network with N nodes and K multicast groups with T = 1 transceiver per node. Associate a multicast group G k to each object k of the bin packing problem where the size of G k is S k. Next, assign nodes to these groups as follows. Start with G 1 and assign nodes 1, 2,..., S 1 to be its members. Then take G 2 and assign nodes (S 1 mod N) + 1, ((S 1 + 1) mod N) + 1,..., ((S 1 + S 2 1) mod N) + 1 to be its members. In general, for group G k+1, the following nodes are assigned to be its members: (σ k mod N) + 1, ((σ k + 1) mod N) + 1,..., ((σ k + S k+1 1) mod N) + 1 where σ k = k i=1 S i. This assignment ensures that the size of G k is S k. Further, it also ensures that each node has a participation degree of d max. Using Lemma 1, the maximum possible uniform multicast throughput for this network is given by µ = T/d max = 1/d max. The decision version of the uniform multicast throughput maximization problem is to determine is a throughput of 1/d max is achievable. Now note that all objects in the bin packing problem can be packed using exactly d max bins iff a throughput of 1/d max can be achieved in the corresponding throughput maximization problem. To see this, note that the only way a uniform multicast throughput of 1/d max can be achieved is if all groups can be scheduled using exactly d max slots. Further, such a schedule cannot split any group since that would require additional slots for completing the multicast transmissions. 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