Joint Roting and Schedling in Mlti-hop Wireless Netorks ith Directional Antennas Partha Dtta IBM Research India parthdt@in.ibm.com Viek Mhatre Motorola Inc. iekmhatre@motorola.com Debmalya Panigrahi CSAIL, MIT debmalya@mit.ed Rajee Rastogi Yahoo Research India rrastogi@yahoo-inc.com Abstract Long-distance mlti-hop ireless netorks hae been sed in recent years to proide connectiity to rral areas. The salient featres of sch netorks inclde TDMA channel access, nodes ith mltiple radios, and point-to-point longdistance ireless links established sing high-gain directional antennas monted on high toers. It has been demonstrated preiosly that in sch netork architectres, nodes can transmit concrrently on mltiple radios, as ell as receie concrrently on mltiple radios. Hoeer, concrrent transmission on one radio, and reception on another radio cases interference. Under this schedling constraint, gien a set of sorce-destination demand rates, e consider the problem of satisfying the maximm fraction of each demand (also called the maximm concrrent flo problem). We gie a noel joint roting and schedling scheme for this problem, based on linear programming and graph coloring. We analyze or algorithm theoretically and proe that at least 50% of a satisfiable set of demands is satisfied by or algorithm for most practical netorks (ith maximm node degree at most 5). I. INTRODUCTION Standardization of the air interface for IEEE 802.11 and 802.16 has reslted in rapid prodction and deployment of WiFi and WiMAX based mesh netorks. Sch mlti-hop mesh netorks hae seeral applications inclding ireless backhaling for 3G celllar netorks, and lo cost netorking for rral connectiity [1], [2], [3], [4]. In sch netorks, one typically makes a distinction beteen long-distance links and mch shorter local access links [5]. In this paper, e are concerned only abot long-distance links hich set p the backbone of the netork. These links are set p by highgain directional antennas monted on top of tall toers to achiee Fresnel clearance aboe obstrctions [6], [7]. Typically, a sbset of netork nodes (called gateays) hae direct connectiity to the Internet. The remaining nodes (called mesh nodes) se mlti-hopping to access the Internet throgh the gateay nodes. The focs of this paper is the design of joint roting and schedling algorithms for throghpt maximization in sch mlti-hop ireless mesh netorks. We assme that nodes are eqipped ith mltiple radios and directional antennas. Directional antennas are helpfl in mitigating interference and improing spatial rese [8], [9]. A node commnicates ith its neighboring nodes sing line-of-sight point-to-point ireless links. Each antenna is This ork as done hen the athors ere at Bell Labs Research India. connected to a single radio at the node. For sch a setting, it has been experimentally demonstrated on commercial off-theshelf ireless eqipment [8], [10] that either (i) concrrent transmission on all the radios of a node, or (ii) concrrent reception on all the radios of a node is possible. This is becase the directional antennas effectiely isolate the signals of all the incoming links. Hoeer, if one radio is transmitting dring a gien time slot, and another radio on the same node is receiing concrrently, then de to radiation leakage and the proximity of the to antennas, the transmitted signal can dron the receied signal. Hence transmission on one radio, and concrrent reception on another radio of the same node is not permitted. We refer to this operation of concrrent transmission or concrrent reception on all the radios of a node as SynOP (synchronos operation). For the aboe netork model, e consider the problem of joint roting and schedling to best meet a gien set of sorce-destination demands. Kodialam and Nandagopal hae stdied the joint roting and schedling problem for the omni-directional antenna setting ith single radio nodes in [11], and mlti-radio nodes in [12]. In [13], Narlikar et al. consider the mlti-radio setting ith directional antennas, and propose heristic algorithms for joint roting and schedling. We sho that the roting-schedling problem that arises in mlti-radio directional antenna setting is fndamentally different from the mlti-radio omni-directional antenna scenario considered in [11], [12]. Frthermore, in contrast to [13], e proide a proably near-optimal scheme for joint roting and schedling for this scenario. Or main contribtions are: Formlation of the problem as an (exponential-sized) linear program (LP) ith joint roting and schedling constraints. A noel schedling algorithm based on directed edge coloring in a mlti-graph. An algorithm for joint roting and schedling based on soling a modified (polynomial-sized) ersion of the aboe LP, and then sing the schedling algorithm to prodce a alid schedle for the LP soltion. Analysis of the aboe algorithm shoing that for a large class of netork graphs (maximm node degree of 9), at least 40% of the maximm concrrently satisfiable demand for each sorce-destination pair is satisfied by or algorithm. For netork graphs ith maximm node
2 degree of 5, hich encompass most practical mesh netork deployments, or soltion has a proable performance of at least 50% of the optimm. Althogh e focs on only a TDMA model, or reslts can be easily extended to a mix TDMA-FDMA model, as ell as the OFDMA model of IEEE 802.16. II. SYSTEM MODEL AND PROBLEM FORMULATION Consider a mesh netork graph G =(V,E) here V is the set of mesh nodes, and E V V is the set of directed links. A link is formed beteen to nodes by aligning the directional antennas on each of the nodes. Reslts in [8] sho ho to dimension the transmit poers of radios of different nodes so that desired SINR (Signal to Interference and Noise) reqirements are met despite side lobe leakage, and concrrent reception on mltiple directional links at a node is possible. 1 In the netork graph, the same physical link is modeled sing to opposite directed links, and e say that a directed link is actie hen the nderlying physical link is actie in that direction. Throghot this paper links refer to directed links. Let C(e) be the capacity of link e. Since the nodes are static, and the links are line-of-sight, e assme that the link capacities are fixed, i.e., there are no shado-fading time ariations. Hoeer, note that slo time ariations in link capacities can be easily incorporated in or frameork by adapting the roting and schedling policies oer longer time scales. We are gien a set of M demands (rates), here the k th demand reqires a rate of r k from sorce s k V to destination d k V. If all the traffic is only beteen the mesh nodes and the Internet, and if the gateay nodes are assmed to be directly connected to the Internet throgh a high speed connection (a typical mesh netork architectre), then the aboe model can be easily modified as follos. We add a node (corresponding to the Internet) to the netork graph, and connect it to all the gateays sing links of infinite capacity. We make the folloing assmptions abot the capabilities of the nodes, and the interference constraints: Each node has a dedicated radio and a corresponding directional antenna for commnicating ith each of its neighbors. All the radios operate on a common channel. Transmissions are synchronized in time (TDMA operation), and in each slot, a sbset of links in the netork are actiated. Dring each time slot, a node can concrrently transmit on all its otgoing links, or concrrently receie on all its incoming links. Hoeer, concrrent transmission on one link and reception on another link ithin the same slot at a node is not alloed. 1 Note than, hen omni-directional antennas are employed, it is not possible to dimension the transmit poers for concrrent reception. Conseqently, ith omni-directional antennas, for a gien time slot/channel, only a single radio can be actiely receiing, hile ith directional antennas all the radios can concrrently receie packets. This important difference beteen omnidirectional and directional antennas reslts in to fndamentally different schedling problems. The last constraint arises becase of side-lobe radiation leakage from the directional antennas. De to this leakage, a transmission on a radio can completely dron a reception on a nearby radio. Ths in a slot, the set of nodes can be bipartitioned into a set of transmitters and a set of receiers. This immediately implies that the set of actie links in each slot forms a directed bipartite sbgraph from the transmitters to the receiers. 2 In the rest of this paper, nless stated otherise, a bipartite graph refers to a directed bipartite graph. A schedle of length T specifies for each slot 1 to T, the set of links that are actie in that slot. In or setting, a schedle is alid if the set of links that are actie in any time-slot forms a directed bipartite graph. In other ords, a alid schedle is a seqence of bipartite sbgraphs. Note that the bipartite sbgraphs need not be disjoint and the same link may appear in mltiple time-slots, either as part of different directed bipartite sbgraphs or in the same sbgraph hich is repeated in mltiple time-slots. The tilization of a link e in the schedle is then defined as the ratio of the nmber of slots in hich e is actie to the length of the schedle. The effectie capacity of link e in a schedle is therefore the prodct of its tilization and its total capacity C(e). A roting specifies the amont of flo on eery link for eery sorce-destination demand. The link flo ector of a roting specifies the total flo load (de to all sorcedestination demands) on each link. We say that a roting can be schedled if there is a alid schedle sch that the effectie capacity of eery link in the schedle is at least as mch as the total flo load on the link. Note that some rotings may not hae a feasible schedle. The concrrent flo ale of a roting is the largest λ sch that the roting satisfies at least λ fraction of eery sorce-destination demand. For a graph G and a set of sorce-destination demands, the maximm concrrent flo (or max flo) is defined as the maximm λ sch that there is a roting that has concrrent flo ale of λ. Then, or joint roting and schedling problem is the folloing: Gien a netork graph and a set of sorcedestination demands, among all rotings that can be schedled in or model, find a roting (and its corresponding schedle) that has the maximm concrrent flo. An LP formlation. Let B be the set of all directed bipartite sbgraphs of the gien graph G =(V,E). By simple conting argments, B =2 n 2 here n = V. Ths, any feasible actiation of links in a time slot is one of the sbgraphs in B, and any schedle can be represented as a eighted combination of elements B i B. Here the eight of a particlar directed bipartite sbgraph B i is the fraction of time-slots in hich B i is actie. For a schedle π, let B1, B2,..., B B be the associated eights. Let 1 e,b be the indicator fnction hich is 1 if link e is present in the directed bipartite sbgraph B B. 2 A directed bipartite graph diides the set of nodes into to sets, the transmitter set and the receier set, sch that, each edge in the graph originates from a node in the transmitter set, and terminates at a node in the receier set.
3 Maximize λ (1) Sbject to: M = f(e, k), e E (2) k=1 λr k e N in () e N ot (s k ) f(e, k) e N in (s k ) f(e, k), k =1...M (3) f(e, k) = f(e, k), e N ot () s k,d k, k =1...M (4) f(e, k) 0, e E, k =1...M (5) B C(e) Bi 1 e,bi, e E (6) i=1 B Bi =1, Bi 0, B i B (7) i=1 Fig. 1. LP for joint schedling-roting problem Let N in () represent the set of incoming links at node, and N ot () represent the set of otgoing links at node. As mentioned in Section II, e are gien a demand ector of M flos, sch that, demand k consists of sorce node s k and destination node d k ith a desired rate r k.letf(e, k) denote the load on link e de to demand k, and let S and D denote the set of sorce and destination nodes for these demands. Then the maximm concrrent flo problem in or model can be formlated as the LP in Figre 1. In the aboe LP, Eq. (2)-(5) are the reglar flo constraints, hile Eq. (6)- (7) are the schedling constraints. The LP finds the optimm eights Bi for all the bipartite graphs, and the corresponding link flo ales. While soling the aboe problem gies the optimal max-flo soltion, its comptational complexity is ery high. This is becase the nmber of bipartite sbgraphs in B is exponential in the nmber of nodes, and hence the LP has exponential nmber of ariables Bi. We therefore look for approximation algorithms for the aboe problem. Or general plan in designing the approximation algorithms is as follos. Consider a particlar roting ith a link flo ector f : E R + 0 representing the total flo load on each directed link in E. The corresponding roting can be schedled in or setting proided there exists a schedle hich can achiee a tilization of at least /C(e) for each link e E. Wefirst derie a set of necessary conditions for schedling a particlar link flo ector. In other ords, if there exists a schedle that achiees the reqired tilization, then these conditions hae to be satisfied. The optimal flo ector gien by the soltion to the LP mst therefore satisfy the necessary conditions. Then, e derie a set of sfficient conditions for the same problem, i.e. if these sfficient conditions are satisfied, then the link flo ector can be schedled. Or sfficient conditions are based on a schedling algorithm that can schedle any link flo ector that satisfies these conditions. Adding the sfficient conditions to the LP formlation possibly changes the optimal ale of λ. Hoeer, by comparing the necessary and sfficient conditions, e bond the decrease in the optimal LP ale hen the sfficient conditions are added to the LP. The sfficient conditions replace the exponential nmber of constraints inoling Bi in the LP and make the LP tractable. We sole the LP to obtain a flo ector hich can no be schedled since it satisfies the sfficient conditions. The ratio beteen the necessary and sfficient conditions gies the approximation bonds on or algorithm. We may note that the oerall plan of or algorithm is similar to [11], hoeer or necessary and sfficient conditions as ell as or schedling algorithms are ery different. III. NECESSARY CONDITION To schedle a link flo ector f corresponding to a roting, each link e mst hae a tilization of at least /C(e). Since a node cannot actiate an incoming and an otgoing link simltaneosly, the sm of tilization of an incoming link and that of an otgoing link (at the same node) mst add p to at most 1. Theorem 1: If a link flo ector f can be schedled, then C(e ) 1, V, e N in(), e N ot (). (8) The aboe necessary condition at a node depends only on the most heaily loaded pair of incoming and otgoing links. This is becase in or model e are alloed SynOP at a node (synchronos reception on all the links, and synchronos transmission on all the links). In sharp contrast, the model considered in [11] has omni-directional antennas, and therefore only allos one link to be actie per channel, at a node. Ths their necessary condition depends on the sm of the loads on all links of a node. IV. SUFFICIENT CONDITIONS USING EDGE COLORING OF A DIRECTED MULTIGRAPH In this section, e gie a ne schedling algorithm and an associated set of sfficiency conditions. We first introdce some terminology that e se throghot this section. A mltigraph is a graph in hich there cold be mltiple parallel edges beteen to nodes. In the rest of the paper, nless explicitly specified, by mlti-graph, e refer to a directed mlti-graph. The nderlying simple (directed) graph of a mlti-graph is prodced by replacing each mlti-edge by a single (directed) edge. The mltiplicity of an edge in the nderlying simple graph is defined as the nmber of copies of the edge that are present in the mlti-graph. Any to different copies of the same simple edge in the mlti-graph are said to be parallel edges. Note that an edge from to is not a parallel edge of an edge from to. Sch edges are referred to as anti-parallel or opposite edges. For any directed graph (either a mlti-graph or a simple graph), the corresponding ndirected graph is prodced by replacing each set of parallel and anti-parallel edges (i.e. all edges beteen to ertices and in both directions) by a single ndirected edge. Let τ be the dration of a time slot. Since C(e) bps is the capacity of link e, in order to carry a traffic of bps, link
4 (a) Mlti-graph G k (b) Simple graph G k s (c) Almost simple graph G k as (d) Residal graph G k+1 Fig. 2. Eoltion of residal graph G k in the Schedling Algorithm. Edge beteen and is tight, and therefore contribtes to copies in the almost simple graph G k as, hile all other edges contribte one copy. The solid edges in Gk as are in Gk s, hile the dotted edge belongs to the set Êk as. C(e)τ e shold be schedled in at least slots eery second. We therefore associate eight (e) ith link e, (e) = 1 τ C(e), (9) and consider a mlti-graph G here link e = (, ) is replaced by (e) directed links from to. (We assme that /C(e) are rational, and e choose τ so that all the (e) are integers.) No, or schedling problem redces to allocating timeslots to each edge in the mlti-graph G, here the dration of each time-slot is τ seconds. Considering each time-slot as a color, the slot allocation problem is an edge coloring problem for a directed mlti-graph sch that, the folloing to types of interference constraints are satisfied. Local Oerlap Constraint: At eery node, the color (timeslot) allocated to an otgoing edge is different from the colors allocated to the incoming edges. This ensres that no node concrrently transmits oer one radio, and receies oer another radio. Parallel Oerlap Constraint: No to parallel edges are allocated the same color. If to parallel edges get the same color, then it implies that the corresponding link needs to be actiated tice in the same time-slot; clearly, this is not possible. A similar mlti-graph edge coloring problem as considered in [11] for ndirected mlti-graphs here the local oerlap constraints don t exist. In another related ork in [10], a similar edge coloring problem as stdied in the context of simple (directed) graphs here the parallel oerlap constraint does not exist. The athors called this problem the directed edge coloring problem or the DEC problem. In the folloing sbsection, e proide a brief oerie of the DEC algorithm proposed in [10]. DEC algorithm for simple graphs. Let G be the gien simple directed graph ith maximm ertex degree Δ. Consider the ertex coloring of the ndirected graph corresponding to G ith K Δ+1 colors. 3 Let each ertex color class (set of ertices ith the same color) form a sperertex and let N be the set of sperertices in G. First, note that each sperertex is an independent set of ertices (i.e., no pair of ertices ith 3 Recall that the ertex coloring problem on an ndirected graph aims to color all the ertices in the graph sing the minimm nmber of colors sch that no to ertices sharing an edge are assigned the same color. It is ellknon that the ertex coloring problem is NP-hard [14]. Hoeer, for or prpose a (sb-optimal) polynomial-time ertex coloring algorithm sffices: the ell-knon greedy algorithm sing Δ+1 colors here Δ is the maximm degree of a ertex in a graph. Ths e assme K Δ+1. Fig. 3. K ξ(k) K ξ(k) 2 2 6 4 3 3 9 5 4 4 10 5 5 4 20 6 Vales of ξ(k) for different ales of K the same color hae an edge connecting them). Ths, the graph indced by G on these sperertices G N =(N,E N ) has all the edges in E, except that the terminals of each edge are no longer ertices in V bt the corresponding sperertices in N. No, sppose e are gien n = ξ(k) edge colors, here ξ(k) is the smallest integer k sch that ( k k/2 ) K. Then, for each sperertex N, e assign a distinct set of n/2 edge colors. By the definition of ξ(k), ( n n/2 ) K; therefore, e hae sfficiently many edge colors for each sperertex to get a distinct edge color set. The edge color set at a sperertex represents the colors that may be sed to color an edge leaing. No, it can be easily shon that corresponding to each edge in G, there is at least one edge color hich is in the edge color set at its origin sperertex and is not in the edge color set at its destination sperertex. This color is assigned to the corresponding edge. Ths, this algorithm ses ξ(k) colors for the DEC problem. It as shon in [10] that ξ(k) log K for large ales of K. For smaller ales of K, see Figre 3. A. Mlti-DEC schedling algorithm Since or problem of edge coloring the mlti-graph generated by eights (e) in Eq. (9) is similar to the DEC problem, e call it the directed mlti-edge coloring problem or the mlti-dec problem. Formally, e are gien a mlti-graph G =(V,E), and e need to color the edges of this mltigraph ith the minimm set of colors sch that the local and the parallel oerlap constraints are satisfied. Roghly speaking, the algorithm e propose for mlti-dec splits the gien mltigraph into seeral almost simple sbgraphs, and edge colors each of these sbgraphs sing a modified DEC algorithm. We first present some definitions reqired to describe or schedling algorithm. A (directed or ndirected) mlti-graph is said to be almost simple if the maximm mltiplicity of any edge in the graph is 2. The eight of a node in a mlti-graph H =(V,F) (denoted by W (, H)) is the sm of mltiplicities of the most heaily loaded incoming and otgoing links of the node. Ths, if (e) is the mltiplicity of edge e in H, then W (, H) = max {(, )+(, )}. (,),(,) F Also, let W max (H) = max V W (, H). A directed edge (, ) is said to be tight if (, ) =W max (H). Similarly, a
5 ertex is said to be tight if W (, H) =W max (H). For example, the edge beteen nodes and in Fig. 2(a) is tight in mlti-graph G k, since W max (G k ) = W (, G k ) = W (, G k )=5. Let G s and G be the nderlying simple graph and corresponding ndirected graph of the mlti-graph G that is to be directed edge colored. Sppose that G can be ertex-colored sing K colors. Then, from [10], G s or any of its sbgraphs can be directed edge colored sing ξ(k) colors. We no gie a schedling algorithm that assigns colors to the edges of G. Schedling Algorithm. Case (i): ξ(k) =1: G s is a directed bipartite graph. We partition G into W max (G) directed bipartite graphs, each of hich is gien a distinct color. Case (ii): ξ(k) 2: We se the folloing iteratie procedre. Let G k be the sbgraph of G sed in the k th iteration of the algorithm. Initialize G 1 = G. In each iteration k, perform the folloing steps (each iteration ses a ne set of colors): 1) Form an almost simple graph, G k as, as follos. Initialize G k as = G k s, the nderlying simple graph of G k. If edge (, ) is tight in G k, then add another copy of (, ) to G k as (see Fig. 2(c)). Let these second copies of edges form the set Êas. k 2) Color the edges in G k s sing the DEC algorithm from [10] (at most ξ(k) ne colors). For any edge e Êk as, assign one of the ξ(k) colors (that ere sed in coloring the edges in G k s) exclding the color assigned to its parallel edge in G k s. 3) The sbgraph G k+1 sed in the next iteration is the residal graph obtained by remoing the edges colored in this iteration from G k (see Fig. 2(d)). If G k+1 has no edge then the algorithm terminates. Otherise, increment k, and go to Step 1. It can be shon that the color assignment gien by the aboe algorithm does not iolate local and parallel oerlap constraints [15]. Also e can sho the folloing theorem (proof in [15]) on the nmber of colors reqired by the aboe algorithm. Theorem 2: The mlti-dec algorithm ses W max colors hen ξ(k) =1, and ξ(k)w max /2 colors hen ξ(k) 2, here the corresponding ndirected graph of G can be ertex colored sing K colors. This theorem gies the folloing sfficiency conditions. Corollary 1: Using the Schedling Algorithm, a link flo ector f can be schedled if C(e ) min { 1, } 2 ξ(k) (10) for each pair of edges e N in () and e N ot () for each ertex. B. Joint Roting and Schedling Algorithm We exploit the sfficiency conditions of Corollary 1 in or algorithm. In the original LP formlation (Eq. (2)-(7)), e replace the schedling constraints (Eq. (6)-(7)) ith the constraints: C(e ) Y, V,e N in(),e N ot () (11) { } 2 here Y = min 1, ξ(k). Note that the aboe constraint amonts to no more than Δ 2 constraints for each node. Ths, the LP is tractable. The link flo ector in the soltion of the LP is no schedled sing the Schedling Algorithm. The approximation ratio of this algorithm is gien by the folloing theorem (fll proof in [15]). Theorem 3: The aboe LP (along ith the Schedling Algorithm) proides a 2/ξ(K)-approximation to the joint roting and schedling problem hen ξ(k) > 2, and proides an exact soltion for the cases of ξ(k) =1and ξ(k) =2. Proof Sketch: Comparing the necessary condition in Eq. (8) and the sfficient condition in Eq. (10), e note if ξ(k) =1or ξ(k) =2, the necessary and sfficient conditions are identical, and for ξ(k) > 2, their ratio is at most 2/ξ(K). Ths, any flo ector that satisfies the sfficient condition is at least 2/ξ(K) times the optimm. Recall that, K Δ+1, here Δ is a maximm degree of a node in a graph. Noting the ales of ξ(k) from Figre 3, e obsere that for a large class of practical netork deployments (maximm node degree of 9), or proposed algorithm obtains a soltion that is ithin 40% of the optimm. For more typical netork deployments (maximm node degree of 5), or soltion comes to ithin 50% of the optimm. REFERENCES [1] K. Chebrol, B. Raman, and S. Sen. 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