Arbitrary Throughput Versus Complexity Tradeoffs in Wireless Networks using Graph Partitioning

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1 University of Pennsyvania SchoaryCommons Departmenta Papers (ESE) Department of Eectrica & Systems Engineering November 2006 Arbitrary Throughput Versus Compexity Tradeoffs in Wireess Networks using Graph Partitioning Saikat Ray University of Pennsyvania, Saswati Sarkar University of Pennsyvania, Foow this and additiona works at: Recommended Citation Saikat Ray and Saswati Sarkar, "Arbitrary Throughput Versus Compexity Tradeoffs in Wireess Networks using Graph Partitioning",. November This paper is posted at SchoaryCommons. For more information, pease contact

2 Arbitrary Throughput Versus Compexity Tradeoffs in Wireess Networks using Graph Partitioning Abstract Severa poicies have recenty been proposed for attaining the maximum throughput region, or a guaranteed fraction thereof, through dynamic ink scheduing. Among these poicies, the ones that attain the maximum throughput region require a computation time which is inear in the network size, and the ones that require constant or ogarithmic computation time attain ony certain fractions of the maximum throughput region. In contrast, in this paper we propose poicies that can attain any desirabe fraction of the maximum throughput region and require a computation time that is independent of the network size. First, using a combination of graph partitioning techniques and yapunov arguments, we propose a simpe poicy for tree topoogies under the primary interference mode that requires each ink to exchange ony 1 bit information with its adjacent inks and approximates the maximum throughput region using a computation time that depends ony on the maximum degree of nodes and the approximation factor. We subsequenty deveop a framework for attaining arbitrary cose approximations for the maximum throughput region in arbitrary networks and interference modes and use this framework to obtain any desired tradeoff between throughput guarantees and computation times for a arge cass of networks and interference modes. Specificay, given any ε > 0, the maximum throughput region can be approximated in these networks using a computation time that depends ony on the maximum node degree and ε. Keywords wireess networks, scheduing, medium access contro, throughput region This working paper is avaiabe at SchoaryCommons:

3 1 Arbitrary Throughput Versus Compexity Tradeoffs in Wireess Networks using Graph Partitioning Saikat Ray and Saswati Sarkar Department of Eectrica and Systems Engineering University of Pennsyvania Abstract Severa poicies have recenty been proposed for attaining the maximum throughput region, or a guaranteed fraction thereof, through dynamic ink scheduing. Among these poicies, the ones that attain the maximum throughput region require a computation time which is inear in the network size, and the ones that require constant or ogarithmic computation time attain ony certain fractions of the maximum throughput region. In contrast, in this paper we propose poicies that can attain any desirabe fraction of the maximum throughput region and require a computation time that is independent of the network size. First, using a combination of graph partitioning techniques and yapunov arguments, we propose a simpe poicy for tree topoogies under the primary interference mode that requires each ink to exchange ony 1 bit information with its adjacent inks and approximates the maximum throughput region using a computation time that depends ony on the maximum degree of nodes and the approximation factor. We subsequenty deveop a framework for attaining arbitrary cose approximations for the maximum throughput region in arbitrary networks and interference modes and use this framework to obtain any desired tradeoff between throughput guarantees and computation times for a arge cass of networks and interference modes. Specificay, given any ɛ > 0, the maximum throughput region can be approximated in these networks using a computation time that depends ony on the maximum node degree and ɛ. I. INTRODUCTION Attaining the maximum throughput region, or a guaranteed fraction thereof, through dynamic ink scheduing, is a key design goa in mutihop wireess networks. The scheduing probem invoves determination of which inks transmit packets at any given time. Appropriate scheduing of inks is key towards attaining throughput guarantees as the success of transmission in any given ink depends on which other inks transmit packets simutaneousy. The transmission schedues can not be pre-computed, and needs to be determined at every transmission epoch, as the congestion eves in the nodes and the transmission conditions in the wireess medium vary with time, and the statistics of these tempora variations are oftentimes not known a priori. Thus, the time required to determine which inks woud transmit at any transmission epoch is a key performance metric for any dynamic scheduing poicy. The contribution of this paper is to characterize tradeoffs between throughput guarantees and computation times for scheduing poicies for different casses of wireess networks. Owing to the ack of a centra controer, at every transmission epoch each ink needs to determine whether it woud transmit based on its own state and the information it acquires about the states of other nodes. The throughput guarantees usuay improve with increase in the information each ink (or rather a node which is the source of the ink) acquires about the states of other inks. The time required for each ink to decide whether to transmit at any given time depends on the time required (a) to exchange messages with other inks to earn their states and (b) to perform the computations required to arrive at an appropriate decision based on the information acquired. We refer to the tota time required in both parts as the schedue computation time, or rather the computation time. The time required in each part increases with increase in the amount of information a ink acquires about the states of other inks. Thus, an important question is how much information a ink shoud acquire about the states of other inks. The scheduing poicies that have been widey investigated can be cassified in two broad casses based on the above quaifier: the poicies that require each ink to know some attribute that depends on the states of (a) a inks in the network [14], [15] and (b) ony the inks that interfere with it [2], [9], [10],

4 [12], [16]. We refer to the two casses as INFORMATION(N) and INFORMATION(1) poicies respectivey, where N is the number of inks in the network. By this nomencature, INFORMATION(k) is the cass of poicies that require each ink to earn the states of their k-hop interferers. A semina resut has estabished that poicies in INFORMATION(N) cass can attain the maximum possibe throughput region in arbitrary wireess networks using O(N) computation time per scheduing decision [14]. Recenty, it has been shown that a poicy in INFORMATION(1) cass can attain a guaranteed fraction of the maximum throughput region using O( G ogn) computation time per scheduing decision where G is the maximum degree, or the maximum number of neighbors of any given node, in the network [2]. The contribution of this paper is to show that in certain important casses of wireess networks, for appropriate seection of k between 1 and N, poicies can be designed in INFORMATION(k) cass so as to obtain arbitrary cose approximations for the maximum throughput region, whie requiring a computation time that depends ony on G and the desired approximation factor and is otherwise independent of the size of the network. We first consider the primary interference mode which mandates that any set of inks can be simutaneousy schedued provided they do not have any common node. Under this interference mode, when the network topoogy is a tree, given any positive constant ɛ, we obtain a distributed scheduing poicy in INFORMATION(1) cass that (a) approximates the throughput region within a factor of 1 ɛ and (b) requires a computation time of O( G /ɛ) (Section IV). We next present a genera framework for designing INFORMATION(k) poicies for approximating the throughput region arbitrariy cosey for arbitrary networks and interference modes (Section V). We subsequenty use this framework for obtaining arbitrary tradeoffs between throughput guarantees and computation times for arge casses of networks, e.g., graphs with imited cycicity and primary interference modes (Section V-B), geometric graphs (Section V- C) and quasi-geometric graphs (Section V-D) under both primary and secondary interference modes. For exampe, for the specia case where nodes are embedded in a pane and two inks interfere if and ony if at east one end-node of one ink is within a given distance D of an end-node of the other ink (i.e., geometric graphs and secondary interference mode), given any positive constant ɛ, we obtain a distributed scheduing poicy in INFORMATION(O( 2 G /ɛ2 )) cass that (a) approximates the throughput region within a factor of 1 ɛ and (b) requires a computation time of ( 2 G /ɛ2 ) O(1/ɛ2). The throughput and computation time guarantees hod in a cases even when sessions traverse mutipe inks (Section VI). Under the primary interference constraints in tree topoogies existing poicies attain (a) the maximum throughput region using a computation time of Θ(N) [14] (b) 2/3 of the maximum throughput region using a computation time of Θ( G (ogn) 2 ) [12] and (c) 1/2 of the maximum throughput region using a computation time of Θ( G ) [9]. For geometric graphs and secondary interference mode, existing poicies attain (a) the maximum throughput region using a computation time of Θ(N) [14] (b) 1/8 of the maximum throughput region using a computation time of Θ( G ogn) [2] and (c) 1/ G of the maximum throughput region using a computation time of Θ( G ) [9]. Our poicies therefore attain arbitrary desired tradeoffs between the best known guarantees for throughput and computation times. Specificay, for networks with bounded degree, our poicies approximate the throughput region within any constant factor using a computation time which depends ony on the approximation factor and does not depend on the network size, whereas existing agorithms that require constant computation time attain an approximation guarantee of at most 1/2 and 1/8 for the above cases respectivey. For networks with degrees O(ogN) (which happens in severa topoogies), our poicies approximate the throughput region within any constant factor using poy-ogarithmic computation time, whereas existing agorithms that use poy-ogarithmic computation time attain an approximation factor of at most 2/3 and 1/8 for the above cases respectivey. We now briefy describe the design of the proposed poicies, and provide the intuition behind the performance guarantees. The proposed poicies partition the network in a coection of components - the size of the components depend ony on G and ɛ. The inks that originate in a component but interfere with those in another component are shut down i.e., not schedued. Thus, the inks schedued in each component wi not interfere with those schedued in other components irrespective of the scheduing poicy in each component. Hence, the scheduing in different components can now be determined in parae. Thus, the time required to compute the overa schedue now depends ony on the size of each 2

5 3 component and is therefore determined ony by G and ɛ. We now describe how the inks in each component are schedued. The weight of each ink is the number of packets waiting for transmission in the ink, the weight of a set of inks is the sum of the weights of the inks in the set, and a set of inks in which no two inks interfere with each other is referred to as an independent set of inks. In each component the set of inks are schedued such that they constitute the maximum weighted independent set of inks in the component. When different partitioning schemes are used at different times and the size of the components in each partition is arge enough, each ink is shut down ony a sma fraction of time. Thus, the inks seected as above, constitute an independent set whose weight is at east (1 ɛ) that of the weight of the maximum weighted independent set of inks in the entire network. The throughput guarantee now foows from the existing resut that a poicy that schedues an independent set of inks whose weight is at east 1 ɛ that of the weight of the maximum weighted independent set of inks attains 1 ɛ fraction of the throughput region [10]. II. RELATED LITERATURE The probem of maximizing the throughput region in wireess networks, or attaining a guaranteed fraction thereof, has received significant attention. Tassiuas et a. have characterized the maximum throughput region and provided a poicy that attains this throughput region in an arbitrary wireess network [15]. This poicy schedues the maximum weighted independent set of inks in each sot, and hence requires Ω(e N ) computation time uness P = NP. Later, Tassiuas [14] provided randomized scheduing schemes that attain the maximum achievabe throughput region, which can be impemented in fuy distributed manner using gossip based agorithms [4]. In each sot, this poicy randomy seects an independent set of inks, compares its weight with the weight of the set of inks schedued in the previous sot and schedues the set that has the arger weight. This poicy requires Θ(N) computation time. A these poicies are in the INFORMATION(N) cass. Recenty, provabe throughput guarantees have been obtained with some poicies in INFORMATION(1) cass. Dai et. a. [3], Lin et a [10] and Wu et. a. [16] proved that a simpe greedy scheduing scheme, maxima matching, attains haf the maximum throughput region for the primary interference mode; the computation time for maxima matching is Θ(ogN). Chaporkar et. a. [2] proved that maxima matching can be generaized to attain guaranteed fraction of the maximum throughput region for arbitrary interference modes, whie retaining the ogarithmic computation time. Sarkar et. a. [12] proved that for primary interference mode and tree graphs, a queue ength dependent maxima matching attains 2/3 of the throughput region whie using Θ ( G og 2 (N) ) computation time. Lin et. a. [9] proved that a random access scheme, where inks access the medium with a probabiity that depends on their and their interferers queue engths, attains 1/2 and 1/ G the throughput region for arbitrary networks under primary interference mode and secondary interference modes respectivey, whie requiring a O( G ) computation time. Our contribution is to introduce the cass of INFORMATION(k) poicies and prove that for appropriate choices of k, poicies can be designed in the INFORMATION(k) cass so as to obtain arbitrary tradeoffs between the best throughput guarantees and the computation times obtained so far. The design of our poicies rey on the use of graph partitioning techniques. Hunt et. a. [7], Kuhn et. a. [8], Nieberg et. a [11] and Sharma et. a. [13] have devised graph partitioning techniques for obtaining arbitrary cose approximations of maximum weighted independent sets in poynomia growth bounded graphs. A graph is said to be poynomia growth bounded if the maximum number of pairwise independent nodes in any r-neighborhood of a node can be upper-bounded by a poynomia in r. Many of the graphs we consider, e.g., trees, are not poynomia growth bounded. Even in the poynomia growth bounded graphs we consider, i.e., geometric graphs, existing resuts [7], [11], [13] approximate maximum weighted independent sets ( within a factor ) of 1 ɛ using poicies in INFORMATION (N) cass which have computation times of Θ N + f(ɛ) G where f(ɛ) is a function of ɛ that increases with decrease in ɛ. Thus seecting the inks using these approximation techniques require centra contro and Θ(N) time for

6 4 computing each schedue. We propose a poicy in the INFORMATION(O( 2 G /ɛ2 )) cass that computes each schedue in O( 2 G /ɛ2 ) time using a simper partitioning technique, and sti attains desired approximation guarantees for the maximum throughput region. The partitioning technique used in [8] however requires time for computing each schedue which does not depend on N as we, but this technique approximates a maximum weighted independent set arbitrariy cosey ony when the weights are a equa. Since different inks have different queue engths in a network, this partitioning technique does not provide throughput guarantees. Finay, Brzezinski et. a. have recenty used graph partitioning techniques for providing throughput guarantees using Θ(N) scheduing schemes for networks with mutipe channes [1]. Their goa is to divide the graph in subgraphs such that different subgraphs are assigned different channes, and a greedy maxima weight scheduing, which requires Θ(N) computation time, maximizes the throughput region in each subgraph. Driven by different goas, we use different partitioning schemes. O(1/ɛ2 ) G III. SYSTEM MODEL We consider scheduing at the MAC ayer in a wireess network. We assume that time is sotted. The topoogy in a wireess network can be modeed as a graph G = (V, E), where V and E respectivey denote the sets of nodes and inks. A ink exists from a node u to another node v if and ony if both u and v can receive each others signas. Each session represents a tripet (i, u, v) where i is the identifier associated with the session and u and v are source and destinations of the session. At the MAC ayer, each session traverses ony one ink, but mutipe sessions may traverse a ink. We consider a network with N sessions. We now introduce terminoogies that we use throughout the paper. Some of these are we-known in graph theory; we mention these for competeness. A node i is a neighbor of a node j, if there exists a ink from i to j, i.e., (i, j) E. Two inks (sessions) are adjacent to each other if they have common nodes. By definition, a ink is adjacent to itsef. The degree of a node u is the number of inks in E originating from or ending at u. The maximum degree in G, G, is the maximum degree of any node in G. A ink i interferes with ink j if j can not successfuy transmit a packet when i is transmitting. A subset of inks is said to be independent if if no ink in the subset interferes with another ink in the subset. Let X be the coection of independent sets of inks. We now describe the packet arriva process. We assume that at most α max 1 packets arrive for any session in any sot. Let  i (t) be the number of packets that session i generates in sot t. We assume that a packet arriving in a sot arrives at the end of the sot, and may not be transmitted in the sot. The arriva process {Âi(t)} is independent and identicay distributed for a t. A scheduing poicy is an agorithm that decides in each sot the subset of sessions that woud transmit packets in the sot. Ceary, a subset of sessions can transmit packets in any sot if no two sessions in the subset traverse the same ink and the inks the sessions traverse constitute an independent set X, i.e., if X X. Every packet has ength 1 sot. Thus, if a session is schedued in a sot, it transmits a packet in the sot. Let ˆDi (t) be the number of packets that session i transmits in sot t, i = 1,...,..., N. Now, ˆDi (t) {0, 1} and depends on the scheduing poicy. Let ˆQi (t) be the queue ength before the arrivas and the transmissions in sot t. Then ˆQ i (t + 1) = ˆQ i (t) + Âi(t) ˆD i (t). Definition 1: The network is said to be stabe if there exists a finite rea number Γ such that with probabiity 1, T 1 im sup ˆQ i (t)/t Γ, i = 1,..., N. (1) T t=0 We consider a virtua-queue Q associated with ink that contains a packets waiting for transmission for a sessions that traverse. Note that the virtua queue in a ink may contain packets of sessions

7 5 L (0) L (1) L (0) L (1) (a) Tree topoogies (b) Topoogies with imited cycicity Fig. 1. The figures demonstrate the edge sets L (0), L (1) under the primary interference mode for (a) a tree and (b) topoogy with imited cycicity. L (0) L (0) GRID 0 GRID 0 GRID 1 GRID 1 D D (a) Geometric graphs under primary interference (b) Geometric graphs under secondary interference Fig. 2. The figures demonstrate the edge sets L (0), L (1) for a geometric graph under (a) primary and (b) secondary interference modes. traversing in both directions. Let A (t) and D (t) respectivey denote the number of arrivas and departures in sot t in virtua queue Q. Ceary, the arriva process {A (t)} is independent and identicay distributed for a t. Let EA (t) = λ. The arriva rate of ink i is λ i, i = 1,..., E. The arriva rate vector λ is an E dimensiona vector whose components are the arriva rates. Now, Q (t + 1) = Q (t) + A (t) D (t). Aso, (1) hods if and ony if im sup T 1 T t=0 Q i(t)/t is finite. The throughput region Λ π of a scheduing poicy π is the set of arriva rate vectors λ for which the network is stabe under π. An arriva rate vector λ is said to be feasibe if it is in the throughput region of some scheduing poicy. The maximum throughput region Λ is the set of feasibe arriva rate vectors. A scheduing poicy π is said to approximate the maximum throughput region within a factor 1 ɛ if for each arriva rate vector λ Λ, (1 ɛ) λ Λ π. IV. INFORMATION(1) POLICY FOR APPROXIMATING THE MAXIMUM THROUGHPUT REGION ARBITRARILY CLOSELY IN TREE TOPOLOGIES We assume that G is a tree and consider the primary interference mode. Under this interference mode, two inks interfere if and ony if they have a common end-point. A matching is a set of inks such that no two inks in the set are adjacent to each other. Thus, a vaid schedue in a sot is a matching in the basic graph G. Thus, X is the set of a matchings in G. This interference mode is encountered in networks ike Buetooth where each node has a singe transceiver and a unique frequency in its neighborhood. We now describe the scheduing poicy which we refer to as TREE-PARTITION-MATCHING (k), and abbreviate as TPM(k). Here, k is a parameter which determines the throughput region and the computation time of the poicy. We first introduce the foowing notations. The eve of a node in a tree is its distance from the root of the tree. A ink = (u, v) is the parent of a ink = (v, w) if the eve of v exceeds that of w, and then

8 6 is a chid of. Links (u, v 1 ), (u, v 2 ),... are sibings of each other; different priorities are associated with different sibings such that between any two sibings one is oder and the other is younger. Let J = { E : is a parent or oder sibing of }. For j = 0,..., k 1, et L (j) be the set of inks (u, v) such that eves of u and v are j and j + 1 moduo k (Figure 1(a)). A forma description of TPM(k) foows. TREE-PARTITION-MATCHING (k) In sot t, every ink seects an integer in the range [0,... k 1]; each integer is seected with probabiity 1/k and a inks seect the same integer. Let i(t) be the integer seected in sot t. A ink contends if and ony if (a) its virtua-queue has packets to transmit, and (b) E \ L (i(t)). A ink schedues itsef if and ony if (a) it contends and (b) inks in J do not schedue themseves. When a ink is schedued, the head of ine packet in the corresponding virtua queue is served. Note that TPM (k) beongs in the INFORMATION (1) cass irrespective of the vaue of k, and is simpe to impement since each ink ony needs to inform its adjacent inks about whether its virtua queue is empty or non-empty. We now evauate the computation time for TPM(k). Note that in any sot the inks that contend constitute a forest such that those in a tree of the forest do not interfere with those in a different tree of the forest. Thus, the scheduing in different components can be determined in parae. The maximum ength of a path in any tree in the forest is k. Each ink that contends decides whether to schedue itsef immediatey after it knows the decisions of its parents and oder sibings that contend. Thus, each ink waits for the scheduing decision of at most k G inks. Thus, the overa computation time is O(k G ). Theorem 1: If λ Int(Λ), then (1 1/k) λ Λ TPM(k). We first outine the intuition behind Theorem 1. First, intuitivey a scheduing poicy π that schedues a ink if and ony if (a) it has a packet to transmit and (b) inks in J do not schedue themseves, maximizes the throughput region in a tree. This is because whenever a ink has a packet to transmit, π schedues either or a ink in J ; the optimum poicy aso schedues at most one ink in J {} in each sot. Ceary, the computation time for π is O(d G ) where d is the depth of the tree, and d is O( E ). Now, by preventing the contention of a subset L (i(t)) of inks in each sot t, TPM (k) partitions the graph in a forest where the depth of each tree is at most k, and uses the above scheduing poicy in each tree of the forest. This reduces the computation time of TPM (k) to O(k G ). The choice of L (0),..., L (k 1), and different seections of i(t) {0,..., k 1} in each sot t ensures that a ink contends with probabiity 1 1/k in each sot t; this in turn ensures that the maximum throughput region reduces ony by a factor of 1 1/k. Proof: The resut ceary hods if k = 1. Thus, we assume that k > 1. The arriva rate vector is (1 1/k) λ where λ Int(Λ). Since λ Λ and X constitutes of a matchings of the inks, ( J {} λ < 1 P 1 max ) [6], [15]. Let δ = min J {} λ, 1. Ceary, δ > 0. Consider a ink = (u, v) where eve of v 2 E max λ exceeds u; then χ denotes the sum of the eve of u and the number of oder sibings of. Observe that the queue engths of the virtua queues constitute a Markov chain. We consider a yapunov function V ( Q) = δ χ Q δ χ Q ( We prove that E Q = J Q. ( ) ( ) V Q (t + 1) V Q(t) Q(t) = Q ) < 1 for a sufficienty arge Q, where V ( Q). Then, from Foster s theorem (Theorem in [5]) the Markov chain representing the queue ength process Q (t) is positive recurrent. Aso, E (Q (t)) < for each under the steady state P K 1 t=0 distribution for the above Markov chain. Thus, im Q (t) K <. The resut foows. K

9 7 V ( ) Q(t + 1) V ( Q(t) ) = δ χ (Q (t + 1) Q (t)) (Q (t + 1) + Q (t)) + 2 δ χ Q (t + 1) J Q (t + 1) 2 δ χ Q (t) J Q (t) 2 δ χ (A (t) D (t)) Q (t) + 2 δ χ (A (t) D (t)) δ χ Q (t) J (A (t) D (t)) +2 δ χ (A (t) D (t)) Q (t) + 2 δ χ (A (t) D (t)) (A (t) D (t)) J J 2 δ χ Q (t) (A (t) D (t)) + δ χ χ A (t) + 4N 2 αmax 2 J {} : J 2 δ χ Q (t) (A (t) D (t)) + δ A (t) + 4N 2 αmax. 2 (2) J {} : J The ast inequaity foows since 0 < δ 1, χ < χ if J. From (2), ( ( ) ( ) E V Q (t + 1) V Q(t) Q(t) = Q ) (2/k) k 1 δ χ E Q (t) (A (t) D (t)) + δ A (t) Q(t) = Q, i(t) = m m=0 J {} : J +4N 2 α 2 max (2/k) δ χ Q k(1 1/k) J {} (since L (j) for ony one j {0,..., k 1} λ (k 1) + k(1 1/k)δ : J and D (t) = 1 for some J {} uness Q (t) = 0 or L (i(t)) ) 2(1 1/k) δ χ Q λ 1 + δ λ + 4N 2 αmax 2 J {} 2(1 1/k) E max λ δ δ χ Q : J λ + 4N 2 α 2 max < 1 for sufficienty arge Q (since δ > 0 and k > 1). The resut foows. Thus, TPM ( 1/ɛ ) attains a throughput region that is at east 1 ɛ times that of the maximum throughput region. The computation time for TPM ( 1/ɛ ) is O( G /ɛ). V. INFORMATION (k) POLICIES FOR APPROXIMATING THE MAXIMUM THROUGHPUT REGION ARBITRARILY CLOSELY FOR ARBITRARY NETWORKS AND INTERFERENCE MODELS We first provide a genera framework for approximating the maximum throughput region arbitrariy cosey in arbitrary networks and interference modes using poicies in INFORMATION (k) cass (Section V- A). Subsequenty, we eucidate the utiity of the framework in severa important casses of networks

10 8 and interference modes (Section V-B, V-C, V-D). We consider both primary and secondary interference modes. For the primary interference mode, we generaize the throughput and computation time guarantees presented in the previous section to graphs with imited cycicity (Section V-B) and geometric and quasigeometric graphs (Section V-C.1). For the secondary interference mode, we obtain simiar resuts for geometric (Section V-C.2) and quasi-geometric graphs (Section V-D). In Section V-E, we discuss how these poicies can be impemented. A. Genera Framework We consider an arbitrary network and an interference mode as described in Section III. We consider a poicy π(k) that consists of k subsets of inks L (0),..., L (k 1) such that the inks in a component of G (j) = (V, E \ L (j) ) do not interfere with those in other components of G (j). In every sot t, every ink seects an integer in the range [0,... k 1]; each integer is seected with probabiity 1/k and a inks seect the same integer. In any sot t, the weight of a ink is the number of packets waiting for transmission in the virtua queue associated with the ink, and the inks that constitute a maximum weighted independent set in the interference graph of any component of G (i(t)) are schedued. Without oss of generaity, inks with zero weight are not schedued. When a ink is schedued, the virtua queue associated with transmits a packet. Note that π(k) is competey specified once L (0),..., L (k 1) are specified. We now describe when π(k) approximates the maximum throughput region within an approximation factor that depends ony on k. We first introduce the foowing definition. Definition 2: A coection of subsets E 1,..., E q of E is said to be c-approximate if for (a) any given E -dimensiona vector of non-negative rea numbers W = (W 1,..., W E ) and (b) any coection of subsets of E, X 1,... X q such that X i X and X i E i q W c max W. i=1 X i X We now present the key technica emma that aows us to obtain desired throughput guarantees. Lemma 1: Let L (0),..., L (k 1) be c-approximate. Then, ) ( E Q i (t)d i (t) Q(t) = Q i (1 c/k) max Q i (t). We first provide the intuition behind the resut. Now, the weight of the inks schedued by π(k) differs from the maximum weight of any schedue in the sot by at most the weight of the maximum weight independent set among inks that do not contend in the sot. Now, if L (0),..., L (k 1) are c approximate, the expected weight of the maximum weight independent set in L (j) for j = 0... k 1 turns out to be at most c/k times that of the weight of the maximum weight independent set in the sot. Thus, the expected weight of the schedued inks is at east (1 c/k) times that of the weight of the maximum weight of any schedue in the sot. Proof: Let i(t) be the integer seected by inks in sot t, and B(t) = arg max X L (i(t)) Q (t). X

11 Now, i Q i(t)d i (t) (max Q i(t) ) i B(t) Q i(t). Now, E Q (t) Q(t) = Q k 1 ( = P i(t) = j Q(t) = Q ) E Q (t) Q(t) = Q, i(t) = j B(t) j=0 j=0 B(t) B(t) k 1 = (1/k) E Q (t) Q(t) = Q, i(t) = j k 1 = (1/k) (c/k) max max j=0 X L (j) X Q (t) Q i (t) (since L (0),..., L (k 1) are c approximate). ( Thus, E i Q i(t)d i (t) Q(t) = Q ) (1 c/k) max Q i(t). Lemma 2: Let L (0),..., L (k 1) be c-approximate. Then, if λ Int(Λ) and k > c, (1 c/k) λ Λ π. We first provide the intuition behind the above resut. When L (0),..., L (k 1) are c-approximate, from emma 1 it foows that π(k) schedues inks such that the expected weight of the schedued inks in any sot is at east (1 c/k) times that of the maximum weight independent set of inks in the sot. The throughput guarantee now foows using yapunov arguments simiar to those in [10], [15]. Refer to appendix A for the proof. Once we prove that the coection L (0),..., L (k 1) is c-approximate, Lemma 2 aows us to approximate the maximum throughput region within a factor of 1 ɛ for any ɛ > 0 using π(k) for k = c/ɛ. In the next subsections we wi prove that in arge casses of networks the coection L (0),..., L (k 1) can be seected so as to render it c-approximate for different constant factors c. Note that different components in each G (j) can schedue the inks in parae as the inks in different components do not interfere. Thus, π(k) can be impemented provided in each sot and in each component either one, or a inks, know the weights of a inks in the component. In either case, π(k) is in INFORMATION(ˆk) cass where ˆk is the maximum diameter of the interference graph of any component of G (j) for any j {0,..., k 1}. The maximum diameter is upper bounded by the number of inks in any component of G (j) for any j {0,..., k 1}. The computation time for π(k) wi again be determined by the maximum size (number of inks or number of nodes or both) of a component in G (j) for j {0,..., k 1}. We wi show that for a arge cass of networks, the size of each component and therefore the overa computation time depends ony on G and k. 9 B. Graphs with Limited Cycicity Using the above genera framework, we generaize the tradeoffs between throughput and computation times to networks with imited cycicity. Specificay, we assume that there exists a constant H such that the maximum ength of a cyce in G is upper bounded by H +1. We sti consider the primary interference mode. We now describe L (0),..., L (k 1) for the scheduing poicy which we refer to as H-LIMITED-CYCLICITY- PARTITION-MATCHING (k), and abbreviate as H-LCPM(k). Consider a spanning tree T of G. For H- LCPM (k), for j = 0,..., k 1, L (j) E is the set of inks (u, v) such that the eves of u and v in T are (a) ess than or equa to jh moduo kh and (b) greater than jh moduo kh respectivey (Figure 1(b)). Intuitivey, for H-LIMITED-CYCLICITY-PARTITION-MATCHING (k), when i(t) = j, eves jh, jh + kh, jh + 2kH,... partition the graph, and L (j) consists of the inks that cross these eves. The set of edges of a graph corresponds to the set of vertices in the interference graph. There exists an edge between two vertices u and v in the interference graph if at east one of the corresponding edges interferes with the other.

12 10 Ceary, the components of G (j) are such that the inks in a component do not interfere with those in other components. We now evauate the computation time for H-LCPM (k). Let the set of edges in T be Ê. Note that the maximum ength of a path in T (j) = (V, Ê \ L(j) ) is kh. Thus each component in T (j) has O( kh G ) nodes. Each component of G (j) consists of severa components of T (j). Let u and v be nodes that are in different components of T (j) but the same components of G (j). Then the common ancestor of u and v in T is at a distance of at most H from both u and v in T. Thus, at most H H (j) G components of T can constitute the same component in G (j). Thus, each component in G (j) has O(H (k+1)h G ) nodes. Now, each independent set X of inks in each component of G (j) is a matching in the corresponding component of G (j). The time needed to compute a maximum weighted matching in each such component is therefore O(H 3 3(k+1)H G ). Thus, the overa computation time is O(H 3 3(k+1)H G ). If G is a bipartite graph, the overa computation time is O(H 2 2(k+1)H G ). The diameter of any component of T (j) is O(kH). Since a component of G (j) consists of at most H H G components of T (j), the diameter of any component of G (j) is O(kH 2 H G ). Thus, H-LCPM (k) beongs in INFORMATION(kH 2 H G ) cass. We now prove the foowing key resut which wi be used in obtaining throughput guarantees for H-LCPM (k). Lemma 3: L (0),..., L (k 1) is 4 approximate. Proof: Let W be an arbitrary N-dimensiona vector of non-negative rea numbers, X = arg max X W, and X 0,..., X k 1 be arbitrary subsets of inks such that X j X (i.e., X i is a matching) and X j L (j), j = 0,..., k 1. We need to prove that k 1 j=0 X j W 4 X W. Note that for any ink, W W i. S Let η (j) = X j S. Thus, ( k 1 k 1 ) W η (j) W. (3) j=0 X j X j=0 ( k 1 ) Hence, we need to show that j=0 η(j) 4 for each X. Consider = (u, v) E. Without oss of generaity, et eve of u in T be ess than or equa to that of v in T. There exists a unique j such that eve of u in T is in ((j 1)H, j H] mod kh. Note that is not adjacent to any ink in L (q) where q < (j 1) mod k or q > (j + 1) mod k. Since X j s are matchings, at most 1 ink in X j is adjacent to when j {(j 1) mod k, (j + 1) mod k}, and at most 2 inks in X j are adjacent to. Thus, ( k 1 j=0 η(j) ) 4 for each X. The resut foows. Theorem 2: If λ Int(Λ), then (1 ɛ) λ Λ H-LCPM( 4/ɛ )). Using k = 4/ɛ, c = 4, Theorem 2 foows from emmas 3 and 2. Now, H-LCPM ( 4/ɛ ) is in INFORMATION(H 2 H G /ɛ) cass and requires O(H 3 3H(1+ 4/ɛ ) G ) computation time. Thus, H-LCPM wi be usefu for sma vaues of H. Finay, Theorem 2 provides the throughput guarantees of 1-LCPM ( 4/ɛ ) for trees as we. But, 1-LCPM ( 4/ɛ ) approximates the maximum throughput region for trees within a factor of 1 ɛ using a computation time of O(1/ɛ) G, whereas TPM ( 1/ɛ ) attains the same throughput guarantee using ony O( G /ɛ). C. Geometric Graphs A graph is said to be geometric if nodes are embedded in the first quadrant of the 2-dimensiona pane, and a ink exists between nodes u and v if and ony if the distance between them is ess than a certain vaue say D. The distance D is referred to as the transmission range. We first consider the primary interference mode (Section V-C.1) and subsequenty consider the secondary interference mode (Section V-C.2).

13 11 1) Geometric Graphs with primary interference mode: We consider a geometric graph G with primary interference mode. We now describe the L (0),..., L (k 1) for the poicy GEOMETRIC-GRAPH-PARTITION- MATCHING(k) which we abbreviate as GGPM(k). We wi consider k different grids each of which consists of a series of horizonta and vertica ines. Here, L (j) is the set of inks that cross vertica or horizonta ines of grid j. We now describe how these grids are constructed. Each grid consists of horizonta and vertica ines parae to the x and y axes respectivey and the distance between any two cosest horizonta (vertica) ines is kd. Each grid is specified by its first horizonta and vertica ines. The first horizonta and vertica ines of grid j are given by y = jd and x = jd respectivey for j = 0,... k 1. Figure 2(a) eucidates the grids and the choices of L (0),..., L (k). Note that the inks in a component of G (j) do not interfere with those in other components. We first evauate the computation time for GGPM(k). The overa computation time equas the worst case computation time in a component. Let ν be the maximum number of nodes in any component of G (j) = (V, E\L (j) ) for any j. We next show that ν is O( G k 2 ). Thus, the computation time for GGPM(k) is the time required to compute a maximum weighted matching in a component with O( G k 2 ) nodes, which is O( 3 G k6 ). Lemma 4: For any j = 0,..., k 1, a component in G (j) = (V, E \ L (j) ) has O( G k 2 ) nodes. Proof: Consider some j 0,..., k 1. A component in G (j) consists of nodes in a square encosed by the cosest horizonta and vertica ines of the jth grid. The side of such a square is at most kd units. Such a square can be fied with κ = 2k 2 sma squares with sides equa to D/ 2. Ceary, κ ( 2 + 1) 2 k 2. Let I be a maxima independent set of nodes in the component, i.e., there does not exist an edge between any two nodes in I and every node in the component is either in I or has an edge to some node in I. Since the distance between any two points in any sma square is at most D, there cannot be more than one node from I present in any sma square. Therefore, I κ. Thus, the first part of the emma foows. Ceary, ν I G. Thus, ν κ G ( 2 + 1) 2 G k 2. Aso, the maximum number of inks in any component of G (j) is at most ν G which is O( 2 G k2 ). Thus, GGPM (k) is in INFORMATION(O( 2 G k2 )) cass. We now prove the foowing key resut which wi be used in obtaining throughput guarantees for GGPM(k). Lemma 5: L (0),..., L (k 1) is 20 approximate. Proof: The proof is simiar to that for emma 3. We point out the differences. We need to prove that k 1 j=0 X j W 20 X W. Reation (3) hods in this case as we. Hence, we need to show ( k 1 ) that j=0 η(j) 20 for each X. Now, note that the k grids do not share any common ine. Let SUPERGRID consist of a ines of a grids. Then SUPERGRID is a grid where the distance between any two cosest horizonta (vertica) ines is D. Ceary, η (j) = 1 for any X j X. If X \ X j, η (j) is the number of inks in X j that interferes with. Since these inks are in X j, they do not interfere with each other. Thus, η (j) 2 since at most 2 inks can be adjacent to but are not adjacent to each other. Thus, η (j) η (j) 2 for any X. Next, for each X we upper-bound the number of js in {0,..., k 1} such that η (j) > 0. Now, > 0 if either L (j) or L (j) but interferes with a ink in L (j). Note that for any, L (j) for at most 4 js in {0,..., k 1}. The observation foows from the fact that L (j) ony if both end nodes of are within a distance of D from a horizonta or vertica ine of grid j; this can happen at most 2 times for vertica ines and 2 more times for horizonta ines of SUPERGRID. Next, for any, L (j) but interferes with (i.e., is adjacent to) a ink in L (j) for at most 6 js in {0,..., k 1}. This happens ony if one of the nodes of is within D units of a horizonta or vertica ine of grid j. This can happen at most 3 times for vertica grid ines and 3 more times for horizonta grid ) ines of SUPERGRID. Thus, for each X, η (j) > 0 for 10 js in {0,..., k 1}. Hence, ( k 1 j=0 η(j) Theorem 3: If λ Int(Λ), then (1 ɛ) λ Λ GPIS( 20/ɛ, ɛ)) = 20 for each X.

14 12 Using k = 20/ɛ, c = 20, Theorem 3 foows from emmas 5 and 2. GGPM ( 20/ɛ ) is in INFORMATION(O( 2 G /ɛ2 )) cass and requires O( 3 G /ɛ6 ) computation time. In the next subsection, we propose a technique that computes each schedue in O( 2 G /ɛ2 ) time whie attaining a throughput region of (1 ɛ) times that of the maximum throughput region. 2) Geometric Graphs with Secondary Interference Mode: We consider a geometric graph G and the secondary interference mode. In this interference mode, a ink i interferes with ink j if one end point of j is within distance D from an end point of i. Note that if two inks interfere under the primary interference mode they aso interfere under the secondary interference mode but the converse is not true. This mode is an abstraction of bidirectiona wireess inks where a transmissions use a singe channe and overapping packets aways cause a coision. Note that an independent set of inks is no onger a matching in G. We now describe the L (0),..., L (k 1) for poicy GRAPH-PARTITION-INDEPENDENT-SET(k) which we abbreviate as GPIS(k). Just as in Section V-C.1, we consider k different grids. Now, L (j) is the set of inks that either cross or are adjacent to inks that cross vertica or horizonta ines of grid j (Figure 2(b)). Note that the inks in a component of G (j) do not interfere with those in other components. We first evauate the computation time for GPIS(k). Again, the overa computation time equas the worst case computation time in a component of G (j). The maximum size of any independent set of inks in a component is O(k 2 ) (in the proof of emma 4 I is O(k 2 ) for any I). Aso, each component of G (j) has O( 2 G k2 ) inks. Thus, in any component of G (j), the maximum weighted interference set can be computed in ( 2 G k2 ) O(k2). Thus, the computation time for GPIS(k) is ( 2 G k2 ) O(k2). Again, ike GGPM (k), GPIS (k) is in INFORMATION(O( 2 G k2 )) cass. We make the foowing observations about L (0),..., L (k 1). Let ψ = {j : L (j) }. Then, ψ 6 for any E (Observation 1). This hods because L (j) ony if at east one of the nodes of is within a distance of D from a horizonta or vertica ine of grid j, which can happen at most 3 times for vertica ines and 3 more times for horizonta ines of SUPERGRID. For any, L (j) but interferes with a ink in L (j) for at most 8 js in {0,..., k 1} (Observation 2). This happens ony if one of the nodes of is within 2D units of a horizonta or vertica ine of grid j but none of the nodes of is within a distance of D from any ine of grid j. This can happen at most 4 times for vertica grid ines and 4 more times for horizonta grid ines of SUPERGRID. We now prove the foowing key resut which wi be used in obtaining throughput guarantees for GPIS(k). Lemma 6: L (0),..., L (k 1) is 112 approximate. ( k 1 ) Proof: The proof is simiar to that for emma 5. Like in emma 5, we need to prove that j=0 η(j) 112 for each X. Now, η (j) 8 for any X as the number of inks that interfere with but do not interfere with each other is at most 8 [2]. Next, ) from observations 1 and 2, for each X, η (j) > 0 for 14 js in {0,..., k 1} Hence, 8 14 = 112 for each X. ( k 1 j=0 η(j) Theorem 4: If λ Int(Λ), then (1 ɛ) λ Λ GPIS( 112/ɛ )). Using k = 112/ɛ, c = 112, Theorem 4 foows from emmas 6 and 2. GPIS ( 112/ɛ ) is in INFORMATION(O( 2 G /ɛ2 )) cass and requires ( G /ɛ) O(1/ɛ2) computation time. We now combine the graph partitioning technique with a poicy design technique proposed by Tassiuas [14] so as to attain 1 ɛ times the maximum throughput region whie computing each schedue in ony O( 2 G /ɛ2 ) time. We denote the poicy as GRAPH-PARTITION-GRADUAL-IMPROVEMENT-INDEPENDENT- SET(k) and abbreviate it as GPGIIS (k). Note that this poicy does not beong in the genera cass of poicies π(k) described in Section V-A. In GPGIIS (k) each ink is associated with k 6 secondary virtua queues: Q (S) i, i {0,..., k 1}\ ˆψ where ˆψ is the union of ψ and max(0, 6 ψ ) arbitrary eements of {0,..., k 1} \ ψ. Whenever a packet arrives in the virtua queue Q it is routed to one of the secondary virtua queues with equa

15 13 probabiity. The poicy divides the time axis in frames of k sots. In the jth sot of each frame, for different inks E, the secondary virtua queues Q (S) j contend. Ony the secondary virtua queues that contend can be schedued for transmission and those that are schedued for transmission transmit their head of ine packets if they are non-empty. We now describe which contending secondary virtua queues are schedued for transmission in the jth sot of each frame. Note that Q (S) j does not exist if L (j) as then j ψ ˆψ. Thus, in the jth sot of each frame, no secondary virtua queue associated with any ink L (j) contends and at most one secondary virtua queue associated with each ink E \ L (j) contends. A ink is said to contend if one secondary virtua queue associated with it contends. Thus, for each j the inks that contend in the jth sot of each frame constitute components such that inks in different components do not interfere, and the inks in each component are a priori rank ordered in some manner. Links in the ordered ist sequentiay seect themseves with a probabiity p (0, 1) and those that interfere with the seected inks remove themseves from the ist. The weight of each contending ink is the number of packets waiting for transmission in the contending secondary virtua queue associated with it. The seected inks are schedued in each component if their tota weight exceeds the tota weight of the inks schedued in the same component in the jth sot of the previous frame; otherwise the inks schedued in the same component in the jth sot of the previous frame are schedued again. The contending secondary virtua queues associated with the schedued inks are schedued. The computation time of GPGIIS (k) is ceary O(γ) where γ is the maximum number of inks in any component of G (j) ; hence this computation time is O( 2 G k2 ). Aso, GPGIIS (k) is in INFORMATION(O( 2 G k2 )) cass. Theorem 5: If λ Int(Λ), then (1 8/k) λ Λ GPGIIS(k)). Proof: Consider a fictitious system that consists of ony the secondary virtua queues Q j for a. Let ˆΛ (j) be the maximum throughput region of this fictitious system. Then [15] Int(ˆΛ (j) ) = { λ : λ = β XI X, where β X = 1, β X 0 for each X X and β φ > 0}. X E\{:j ˆψ } X E\{:j ˆψ } Consider a scheduing poicy π that schedues secondary virtua queues that satisfy the foowing properties. 1) Q j (t) constitutes an irreducibe aperiodic markov chain. 2) In each sot t there is a positive probabiity associated with scheduing the secondary virtua queues associated with inks in X (t) where X (t) = arg max X E\:j ˆψ X Q j (t). 3) If X 0 and X 1 are the sets of inks associated with the secondary virtua queues schedued in sots t 1 and t then X 1 Q j (t) X 0 Q j (t). Then π stabiizes the fictitious system for any arriva rate vector λ Int(ˆΛ (j) ) [14], [4]. Let (1 6/k) λ be the arriva rate vector in the system where λ Int λ). Let λ (j) consist of those components of λ for which j ˆψ. From (5), λ (j) Int( Λ (j) ). We now consider the secondary virtua queues Q j for a at sots j, k + j, 2k + j,... in the actua system. Note that in the actua system these secondary virtua queues are schedued ony in these sots. We can therefore assume without oss of generaity that packets arrive in these queues ony in these sots as we whie the number of arrivas in sot mk + j is the number of arrivas in the actua system between sots ((m 1)k + j, mk + j] ([0, j]) for a positive integer m (m = 0). Note that the expected number of arrivas in secondary virtua queue Q j in sot mk + j is now k(1/(k 6))(1 6/k)λ = λ. Thus, the arriva rate vector for these secondary virtua queues is λ (j) Int( λ (j) ). Now, observe that GPGIIS(k)

16 14 satisfies properties (1) to (3) for these secondary virtua queues, since inks that contend in different components of G (j) do not interfere. Thus, for each j, the system consisting of these virtua queues are stabiized. The resut foows. Thus, for k = 6/ɛ, a poicy GPGIIS (k) in INFORMATION(O( 2 G /ɛ2 )) cass, attains a throughput region of 1 ɛ times that of the maximum throughput region using a computation time of O( 2 G /ɛ2 ). Note that GGPM (k) can be simiary modified to attain a throughput region of 1 ɛ times that of the maximum throughput region, using k = 4/ɛ and a computation time of O( 2 G /ɛ2 ). Finay, GPGIIS(k) attains substantiay better tradeoffs than GPIS(k) between throughput and computation time guarantees. But, at the same time GPGIIS(k) is ikey to have substantiay higher deay as compared to GPIS(k). This is because since, unike GPIS(k), GPGIIS(k) segregates the incoming traffic in each ink in mutipe queues and in each sot aows at most one queue in each ink to contend, when the contending queue is empty it does not schedue the ink even if the ink s interferers are not schedued and other queues in the same ink are non-empty. More importanty, unike GPIS(k), GPGIIS(k) does not schedue the queues whose expected weight is cose to that of the maximum weight independent set of queues, and instead attains stabiity by graduay improving the weight of the schedued queues. This behavior is known to significanty increase the deay, e.g., simuations have demonstrated that the poicy proposed by Tassiuas et a. [15] that schedues the maximum weight independent set in each sot has substantiay ower deay as compared to the randomized poicy proposed again by Tassiuas [14] that attains stabiity through simiar improvements as above. An interesting topic for future research is to investigate the tradeoffs between deay and computation times of scheduing poicies. D. Quasi-Geometric Graphs A graph is said to be quasi-geometric if nodes are embedded in the first quadrant of the 2-dimensiona pane, and a ink (a) exists between nodes u and v if the distance between them is ess than ιd where ι < 1 (b) may exist between nodes u and v if the distance between them is between ιd and D and (c) does not exist between nodes u and v if the distance between them is greater than or equa to D. Under primary interference constraints, as before, two inks interfere if and ony if they are adjacent. Under secondary interference constraints, two inks, interfere if and ony if (a) they are adjacent and (b) there is an edge between at east one end node of and another end node of. We first consider the secondary interference mode. Now, inks L (0),..., L (k 1) are seected as in the previous subsection, and GPGIIS (k) attains a throughput region which is 1 6/k of the maximum throughput region as before. However, each component of G (j) has O( G k 2 /ι 2 ) nodes, and O( 2 G k2 /ι 2 ) inks. Thus, the computation time for GPGIIS (k) is O( 2 G k2 /ι 2 ). Aso, GPGIIS (k) is in INFORMATION(O( 2 G k2 /ι 2 )) cass. Thus, GPGIIS ( 6/ɛ ) attains a throughput region which is 1 ɛ of the maximum throughput region, requires a computation time of O ( 2 G /(ι2 ɛ 2 )) and is in INFORMATION(O ( 2 G / (ι2 ɛ 2 ))) cass. Simiary, under the primary interference mode, a throughput region of 1 ɛ of the maximum throughput region can be attained using a poicy in INFORMATION(O( 2 G / (ι2 ɛ 2 ))) cass which requires O( 2 G /(ɛ2 ι 2 )) computation time. E. Distributed Impementation of the Scheduing Poicies We discuss two possibe distributed impementations for π(k). In one, in each sot one ink in each component determines which inks wi be schedued in the component and broadcasts the decisions in the entire component, and in another every ink does this computation. For the first each contending ink communicates its weight to the ink that computes the decisions in its component, and for the second each contending ink broadcasts its weight in its entire component. The probem with both impementations is that the size of the packets can not be bounded by any function of the network size since the queue engths exceed any given number with positive probabiity. A better soution is to have each ink broadcast the increase in its weight since the previous epoch in which the ink was in the same component (that is the same random number was seected). Now, the expectation of the magnitude of this increase is O(k)