Capacity of Multi-Channel Wireless Networks. Random (c, f) Assignment

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1 Capaity of Multi-Channel Wireless Networks with Random (, f Assignment ABSTRACT Vartika Bhandari Dept. of Computer Siene, and Coordinated Siene Laboratory University of Illinois at Urbana-Champaign vbhandar@uiu.edu With the availability of multiple unliensed spetral bands, and potential ost-based limitations on the apabilities of individual nodes, it is inreasingly relevant to study the performane of multihannel wireless networks with hannel swithing onstraints. To this effet, some onstraint models have been reently proposed, and onnetivity and apaity results have been formulated for networks of randomly deployed single-interfae nodes subjet to these onstraints. One of these onstraint models is termed random (, f assignment, wherein eah node is pre-assigned a random subset of f hannels out of (eah having bandwidth W, and may only swith on these. Previous results for this model established bounds on network apaity, and proved that when = O(logn, the perflow apaity is O(W nlogn and Ω(W f nlogn (where = ( f ( f...( f f+ e. In this paper we present a lower bound onstrution that mathes the previous upper bound. This establishes the apaity as Θ(W nlogn. The surprising impliation of this result is that when f = Ω(, random (, f assignment yields apaity of the same order as attainable via unonstrained swithing. The routing/sheduling proedure used by us to ahieve apaity requires synhronized route-onstrution for all flows in the network, leading to the open question of whether it is possible to ahieve apaity using asynhronous proedures. Categories and Subjet Desriptors C.2. [Computer Communiation Networks]: Network Arhiteture and Design Wireless ommuniation General Terms Performane, Theory This researh is supported in part by NSF grant CNS , US Army Researh Offie grant W9NF , and a Vodafone Graduate Fellowship. Permission to make digital or hard opies of all or part of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for profit or ommerial advantage and that opies bear this notie and the full itation on the first page. To opy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speifi permission and/or a fee. MobiHo 07, September 9 4, 2007, Montréal, Québe, Canada. Copyright 2007 ACM /07/ $5.00. Nitin H. Vaidya Dept. of Eletrial and Computer Eng., and Coordinated Siene Laboratory University of Illinois at Urbana-Champaign nhv@uiu.edu Keywords Wireless Networks, Capaity, Multiple Channels, Swithing Constraints, Random (, f Assignment. INTRODUCTION There has been muh reent interest in exploiting the availability of multiple hannels in wireless networks. The transport apaity of suh networks has also been studied under various assumptions on availability/apability of radio-interfaes. It was shown in [7] that for a single-hannel single-interfae senario, in a randomly deployed network, per-flow apaity sales W as Θ( nlogn bits/s under a Protool Model of interferene, and that if the available bandwidth W is split into hannels, with eah node having a dediated interfae per hannel, the results remain the same. While many existing standards, e.g., IEEE 802.a, 802.b, allow for multiple hannels, nodes are typially hardwareonstrained and have muh fewer interfaes. This issue was studied in [9], under a model where nodes were apable of swithing their interfae(s to any hannel. It was shown that given available hannels of bandwidth W eah, and m interfaes per node, apaity depends solely on the ratio m. For a random network, and the Protool Model, three apaity regions were established. Most relevant to our work is the region m = O(logn, where they W showed that apaity is the same as for m =, i.e., Θ( nlogn bits/s per-flow. In [3], a ase was made for the need to study the performane of multi-hannel networks in situations where there are onstraints on hannel swithing. This study was motivated on the basis of future low-ost transeiver designs involving limited tunability, as well as ognitive radio networks. As more spetrum beomes freely available for unliensed use, ost onerns are very likely to lead to situations where individual nodes an operate only over a muh smaller spetral range, and may possess heterogeneous apabilities. Thus it is quite relevant to study the impat of swithing onstraints, and attempt to quantify it. Some onstraint models were proposed in [3] to apture some expeted onstraints, and two suh models were analyzed, viz., adjaent (, f assignment and random (, f assignment. The impat of restrited swithing was quantified by the parameter f (where f is the number of hannels an individual node may swith to. Results were presented for the regime = O(logn. It was established that per-flow apaity is Θ(W f nlogn for adjaent (, f assignment. For random (, f assignment, an upper bound of O(W nlogn and a lower bound of Ω(W f nlogn were established.

2 In this paper, we establish that the per-flow apaity with random (, f assignment (under the Protool Model of interferene for the regime = O(logn (2 f is Θ(W nlogn by presenting a apaity-ahieving lower bound onstrution. It an be shown that e. Thus, the somewhat surprising impliation of this result is that when f = Ω(, random (, f assignment yields apaity of the same order as attainable via unonstrained swithing. Hene -swithability is suffiient to make order-optimal use of all hannels. Interestingly, our apaity ahieving routing/sheduling proedure requires that all routes be omputed in lok-step. This leaves open the question of whether apaity an be ahieved via asynhronous routing/sheduling proedures. 2. NOTATION AND TERMINOLOGY Throughout this paper, we use the following standard asymptoti notation [4]: f(n = O(g(n means that,n o, suh that f(n g(n for n > N o f(n f(n = o(g(n means that lim n g(n = 0 f(n = ω(g(n means that g(n = o( f(n f(n = Ω(g(n means that g(n = O( f(n f(n = Θ(g(nmeans that, 2,N o, suh that g(n f(n 2 g(n for n > N o When f(n=o(g(n, any funtion h(n=o( f(n is also O(g(n. We often refer to suh a situation as h(n = O( f(n = O(g(n. As in [7], we say that the per flow network throughput is λ(n if eah flow in the network an be guaranteed a throughput of at least λ(n with probability, as n. Whenever we use log without expliitly speifying the base, we imply the natural logarithm. 3. NETWORK MODEL We onsider a network of n single-interfae nodes deployed uniformly at random over a unit torus. Eah node is the soure of exatly one flow. As in [7], eah soure S selets a destination by first fixing on a point D uniformly at random, and then piking the node D (other than itself, that is losest to D. The total bandwidth (data-rate available is W, and it is divided into hannels of equal bandwidth W, where = O(logn. We assume that 2, as = implies that f = is the only possibility, whih yields the degenerate single-hannel ase. We also assume 2 f. A justifiation for not allowing f = for 2 is given in [3], [], where it was shown that for the random (, f model (and also the adjaent (, f model desribed in [3], f = and 2 leads to zero apaity, as some flow will get no throughput w.h.p. 4. RELATED WORK It was shown by Gupta and Kumar [7] that, for a single-hannel single-interfae senario, the per flow apaity in a random network sales as Θ( nlogn W bits/s. The throughput-delay trade-off was studied in [6], and it was shown that the optimal trade-off is given by D(n = Θ(nT(n where D(n is delay, and T(n is throughput. In the multi-hannel ontext, an interesting senario arises when the number of interfaes m at eah node may be smaller than the number of available hannels. This issue was analyzed in [9] and it was shown that the apaity results are a funtion of the hannel-to-interfae ratio m. It was also shown that in the random network ase, there are three distint apaity regions: when m = W O(logn, the per-flow apaity is nlogn, when m = Ω(logn and ( ( also O n loglogn 2 logn, the per flow apaity is Θ(W m n, and when m = Ω (n ( loglogn logn 2, the per-flow apaity is Θ( Wmloglogn logn. Connetivity and apaity of multi-hannel wireless networks with hannel swithing onstraints were onsidered in [3]. Results were presented for two speifi onstraint models, viz., adjaent (, f assignment and random (, f assignment. It was shown that when = O(logn, apaity with adjaent (, f assignment sales as Θ(W f nlogn. For random (, f assignment, it was shown that apaity is O(W nlogn and Ω(W f nlogn. In this paper, we show that the apaity for this model is atually Θ(W nlogn. 5. RANDOM (, f ASSIGNMENT In this setion we briefly desribe the random (, f assignment model first introdued in [3, ], and summarize some already proven results that will be useful in proving the lower bound on apaity. In this assignment model, a node is assigned a subset of f hannels uniformly at random from the set of all possible hannel subsets of size f. Thus the probability that two nodes share at least one hannel is given by = ( f ( f...( f f+. The proofs of the following are available in [], and also [2]: LEMMA. For 2 and 2 f, the following holds: min{,2 f } ( f f LEMMA 2. min{ f,2 f } 2 5. Suffiient Condition for Connetivity It was stated and proved in [3] that, for random (, f assignment, if πr 2 (n = 800πlogn n, then the network is onneted w.h.p. We summarize the proof idea here, to provide important ontext for the results in this paper. The unit torus is divided into square ells of area a(n = 00logn n. It an be shown that there are at least 50logn nodes in eah ell w.h.p. r(n is set to 2logn 8a(n. Within eah ell, nodes are hosen uniformly at random, and set apart as transition failitators. At least 48logn nodes remain in eah ell, and they at as bakbone andidates. Consider any node in any given ell. The probability that it an ommuniate to any other random node in its range is. Then the probability that in some adjaent ell, there is no bakbone andidate node with whih it an ommuniate is less than ( 48logn =. Applying union bounds over all 8 e 48logn n adjaent ells of a node, and all 48 n nodes, the probability that at least one node is unable to ommuniate with any bakbone andidate 8 node in at least one of its adjaent ells is at most. n Assoiated with eah node x, there is a set of nodesb(x 47 alled the bakbone for x. B(x is onstituted as follows: Cells already overed by the bakbone are referred to as filled ells. x is by default a member ofb(x, and its ell is the first filled ell. From eah adjaent ell, amongst all bakbone andidate nodes sharing at least one ommon hannel with x, one is hosen uniformly at random is

3 added to B(x. Thereafter, from eah ell bordering a filled ell, of all nodes sharing at least one ommon hannel with some node already inb(x, one is hosen uniformly at random, and is added to B(x; the ell gets added to the set of filled ells. This proess ontinues, till all ells are filled. Based on previous arguments, B(x eventually overs all ells with probability at least 8 n 47. For any node-pair x and y, ifb(x B(y φ the two are obviously onneted. Suppose the two bakbones are disjoint. Then x and y are still onneted if there exists a ell where the member of B(x (let us all it q x an ommuniate with the member ofb(y in that ell (let us all it q y, either diretly, or through a third node z. q x and q y an ommuniate diretly with probability if they share a ommon hannel. Thus the ase to handle is that where no ell has q x and q y sharing a hannel. If they do not share a ommon hannel, onsider the event that there exists a third node amongst the transition failitators in the ell through whom they an ommuniate. Thus, the overall probability an be lower-bounded by obtaining for one ell the probability of q x and q y ommuniating via a third node z, given they have no ommon hannel, onsidering that eah ell has at least 2logn possibilities for z, and treating it as independent aross ells. This is elaborated further. Consider a third node z amongst the transition failitators in the same ell as q x and q y. Consider a situation where z enumerates its f hannels in some uniformly random order, and then inspets the first two hannels, heking whether the first one is ommon with q x, and heking whether the seond one is ommon with q y. This probability is ( f ( f > through z with probability p z > 2 = Ω( 2. Thus q x and q y an ommuniate. There are 2logn log 2 n possibilities for z within that ell, and all the possible z nodes have i.i.d hannel assignments. Thus, the probability that q x and q y annot ommuniate through any z in the ell is at most ( p z 2logn, and the probability they an do so is p xy ( p z 2logn. Thereafter appliation of the union bound over all ells, and all node pairs suffies to prove the result. 6. SUMMARY OF OUR RESULTS In the rest of this paper, we desribe a onstrution that ahieves a per-flow throughput of Ω(W nlogn for = O(logn. In light of the upper bound of O(W nlogn proved in [3], this establishes the apaity for random (, f assignment as Θ(W nlogn in the regime = O(logn. It is easy to see the following: = ( f ( f...( f f + ( f f e Hene: f = Ω( = = Ω(. To illustrate, if we set f =, e > 2. In light of Eqn. (2, our result implies that f = Ω( suffies for ahieving apaity of the same order as the unonstrained swithing ase [9]. For f =, the previously established lower bound of Ω(W f nlogn, would have yielded a apaity degradation by a fator of the order of 4 ompared to the unonstrained swithing ase. In general, one may see that the apaity may diverge from the previous lower bound when f 0, but f. Fig. is a numerial plot (obtained by setting to 0 4, and vary- (2 Probability of sharing at least one hannel Communiation Probability with Constrained Swithing Max. Prob. with Adjaent (, f Assignment Random (, f Assignment Figure : Comparison of probability of sharing a hannel ing f from 2 to depiting how the probability ompares with the probability p max 2 f ad j = min{ f+,}. Reall that is the probability that two nodes share at least one hannel in random (, f assignment, and p max ad j is the upper bound on the probability that two nodes share at least one hannel in adjaent (, f assignment [3]. It is quite remarkable that though both models allow nodes to swith between a subset of f hannels, the additional degrees of freedom obtained via the random assignment model lead to a muh quiker onvergene of toward. The results in [3] established that onnetivity was the dominant onstraint determining apaity for adjaent (, f assignment in the = O(logn regime. The lower bound in this paper for random (, f assignment mathes the upper bound imposed by the onnetivity onstraint (see [3, ]. Thus, the quik onvergene of to leads to a quiker onvergene of apaity towards that attainable via unonstrained swithing. It is to be noted that the lower bound of [3] was obtained using a muh simpler onstrution than the one desribed in this paper. Thus the two onstrutions represent an interesting trade-off in apaity versus sheduling/routing omplexity. 7. SOME USEFUL RESULTS THEOREM. (Chernoff Upper Tail Bound [0] Let X,...,X n be independent Poisson trials, where Pr[X i = ] = p i. Let X = n X i. Then, for 0 < β : i= f/ Pr[X (+βe[x]] exp( β2 E[X] (3 3 LEMMA 3. For all 0 x : ( x e x. LEMMA 4. Suppose we are given a unit torus with n points (or nodes loated uniformly at random, and the region is subdivided into axis-parallel square ells of area a(n eah. If a(n = 00α(nlogn n, α(n 00logn n, then eah ell has at least (00α(n 50logn, and at most (00α(n+50logn points (or nodes, with high probability. LEMMA 5. Suppose we are given a unit torus with n points (or nodes loated uniformly at random. Let us onsider the set of all irles of radius R and area A(n = πr 2 on the unit torus. If A(n = 00α(nlogn n, α(n 00logn n, then eah irle has at least (00α(n 50logn, and at most (00α(n+50logn points (or nodes, with high probability.

4 LEMMA 6. If n pairs of points (P i,q i are hosen uniformly at random in the unit area network, the resultant set of straight-line formed by eah pair L i = P i Q i satisfies the ondition that no ell has more than n a(n lines passing through it. THEOREM 2. (Hall s Marriage Theorem [8], [] Given a set S, lett = {T,T 2,...T n } be a finite system of subsets ofs. ThenT possesses a system of distint representatives if and only if for eah k in,2,..,n, any seletion of k of the setst i will ontain between them at least k elements ofs. Alternatively stated: for alla T, the following is true: A A LEMMA 7. The number of subsets of size k hosen from a set of m elements is given by ( m ( k me k. k THEOREM 3. (Integrality Theorem [4] If the apaity funtion of a network flow graph takes on only integral values, then the maximum flow x produed by the Ford-Fulkerson method has the property that x is integer-valued. Moreover, for all verties u and v, the value of x(u,v is an integer. 8. LOWER BOUND ON CAPACITY A lower bound of Ω(W f nlogn for apaity with random (, f assignment was proved in [3, ]. From Lemma, it follows that f f nlogn = Ω( nlogn f. Thus for f < 00, = Ω(, and the onstrution presented in [3] (details in [] is asymptotially optimal. nlogn nlogn Thus, we propose to use this onstrution for f < 00 to ahieve apaity. We now present a onstrution that ahieves Ω(W nlogn when f 00 (thus neessarily 00. Traffi-model related results. We first state some results for the traffi model of [7] (whih is also used in this paper. For proofs, please see [2]. LEMMA 8. The number of flows for whih any node is the destination is O(logn w.h.p. LEMMA 9. For large n, at least one node is the destination for Ω(logn flows with a probability at least e ( e ( δ, where δ > 0 is an arbitrarily small onstant. Subdivision of network region into ells. We use a square ell onstrution (similar to that used in [6], and subsequently in [9], [3]. The surfae of the unit torus is divided into square ells of area a(n eah, and the transmission range is set to 8a(n, thereby ensuring that any node in a given ell is within range of any other node in any adjoining ell. Sine we utilize the Protool Model [7], a node C an potentially interfere with an ongoing transmission from node A to node B, only if BC (+ r(n. Thus, a transmission in a given ell an only be affeted by transmissions in other ells within a distane (2+ r(n from some point in that ell. Sine is independent of n, the number of ells that interfere with a given ell is only some onstant (say β. We hoose a(n= 250max{logn,} n = Θ( logn n (sine =O(logn. Then the following holds: LEMMA 0. Eah ell has at least 4na(n 5 = 200max{logn,} and at most 6na(n 5 = 300max{logn,} nodes w.h.p. PROOF. The proof has been omitted due to spae onstraints. Many of the intermediate results in the rest of this paper assume that the high-probability event of Lemma 0 holds. We also state the following fats: f (4 For large n, sine = O(logn, and 2 f : f(n = O(na(n = f(n = O( n a(n (5 f(n = O( = f(n = O( n a(n (6 a(n Some properties of SD D routing. Reall that we use the traffi model of [7], where eah soure S first hooses a pseudodestination D, and then selets the node D nearest to it as the atual destination. In [7], the route SD D was followed, whereby the flow traversed ells interseted by the straight line SD, and then took an extra last hop if required. In our ase, it may not always suffie to use SD D routing (we elaborate on this later. However, this is still an important omponent of our routing proedure, and so we state and prove the following lemmas (similar results were stated in [] for SD D routing: LEMMA. Given only straight-line SD routing (no additional last-hop, the number of flows that enter any ell on their i-th hop is at most 5na(n 4 w.h.p., for any i. PROOF. Let us onsider the straight-line part SD of an SD D route. Thus all the n SD lines are i.i.d. Denote by Xi k the indiator variable whih is if the flow k enters a elld on its i-th hop. Then, as observed in [6] (proof of Lemma 3, for i.i.d. straight lines, the Xi k s are identially distributed, and Xk i and Xj l are independent for k l. However for a given flow k, at most one of the Xi k s an be as a flow only traverses a ell one. Then Pr[Xi k = ] = a(n = 250max{logn,} n. Let X i = n Xi k. Then E[X i] = na(n. Also, for a given i, the Xi k s k= are independent [6]. Then by appliation of the Chernoff bound from Thereom (with β = 4 : Pr[X i 5E[X i] ] exp( E[X i] 4 48 Pr[X i 250max{logn,} ] 4 exp( 250max{logn,} 48 < n 5 2 The maximum value that i an take is = 2n a(n 250max{logn,} < n. Also the number of ells is a(n n. Then by appliation of union bound over all i, and all ellsd, the probability that X i 5E[X i] 4 is less than, and thus the number of flows that enter any ell on any hop is less than 5na(n 4 = 250max{logn,} 4 n 3 with probability at least. Resultantly, sine X n 3 i is an integer, we an say that it is at most 5na(n 4 w.h.p. (7

5 LEMMA 2. The number of flows for whih any single node is the destination is O(na(n w.h.p. PROOF. The proof has been omitted due to spae onstraints. LEMMA 3. If a node is the destination of some flow, then that flow s pseudo-destination must lie within either the same ell, or an adjaent ell w.h.p. PROOF. The proof has been omitted due to spae onstraints. LEMMA 4. The number of SD D routes that traverse any ell is O(n a(n w.h.p. PROOF. The proof for this lemma is largely based on a proof in [6]. It has been omitted due to spae onstraints. Please see [2]. Having stated and proved these lemmas, we now establish some properties of the spatial distribution of hannels, and thereafter desribe our sheduling/routing proedure further: DEFINITION. We define a term M u where M u = 9 f na(n 25 = 90 f max{logn,}. Then the following holds: LEMMA 5. If there are at least 200max{logn,} nodes in every ell, of whih we hoose 80max{logn,} nodes uniformly at random as andidates to examine, then, in eah ell, amongst those 80max{logn,} andidate nodes, at least f 4 hannels have at least M u nodes apable of swithing on them, w.h.p. PROOF. The proof has been omitted due to spae onstraints. Similar to the onstrution for onnetivity from [3] that we briefly summarized in Setion 5., we will onstrut a bakbone for eah node. However, sine our onern is not merely onnetivity but also apaity, these bakbones need to be onstruted arefully, to ensure that no bottleneks are formed. Conditioning on Lemma 0, there are at least 200max{logn,} nodes in eah ell w.h.p. Initially, from eah ell, we hoose 80max{logn,} nodes uniformly at random as bakbone andidates. The remaining nodes (whih are at least 20max{logn,} in number are deemed transition failitators. DEFINITION 2. (Proper Channel A hannel i is deemed proper in elld if it ours in at least M u bakbone andidate nodes ind. LEMMA 6. For eah ell of the network, the following is true w.h.p.: if the number of proper hannels in the ell is, then f PROOF. The proof follows from Lemma 0 and Lemma 5. Besides, we an also show the following: LEMMA 7. Consider any elld. LetW i be the set of all nodes in the 8 adjaent ellsd(k, k 8, that are apable of swithing on hannel i. This an be viewed as a speial variant of the Coupon Colletor s problem [0], where there are different types of oupons, and eah box has a random subset of f different oupons. Some other somewhat different variants having multiple oupons per box have been onsidered in work on oding, e.g., [5]. For a set of nodesb, definec(b = { j j proper ind and u B apable of swithing on j}. If f 00, the following holds w.h.p.: hannels i, B W i suh that B = f na(n : C(B This is true for all ellsd. PROOF. The proof has been omitted due to spae onstraints. 8. Routing and hannel assignment Partial Bakbones. As mentioned earlier, the routing strategy is based on a per-node bakbone struture similar to that used to prove the suffiient ondition for onnetivity in [3]. However, instead of onstruting a full bakbone for eah node, only a partial bakboneb p (x is onstruted for eah node x.b p (x only overs those ells whih are traversed by flows for whih x is either soure or destination. A flow first proeeds along the route on the soure bakbone and will then attempt to swith onto the destination bakbone. We shall explain the bakbone onstrution proedure in detail later. First we show how a flow an be routed along these bakbones from its soure to its destination. LEMMA 8. Suppose a flow has soure x and destination y. Thus it is initially onb p (x and finally needs to be onb p (y. Then after having traversed 2 distint ells (hops (reall that 2 f and = O(logn, it will have found an opportunity to make the transition w.h.p. If the routes of eah of the n flows get to traverse at least 2 distint ells (note that eah individual route needs to traverse at least so many distint ells; two different flows may have ommon ells on their respetive routes, then all n flows are able to transition w.h.p. PROOF. Consider a flow traversing a sequene of ells D,D 2,... Then if the representative ofb p (x (let us all it q x in D i an ommuniate (diretly or indiretly with the representative ofb p (y (let us all it q y in D i, it is possible to swith fromb p (x tob p (y. If q x and q y share a hannel this is trivial. If q x and q y do not share a hannel, we onsider the probability that the two an ommuniate via a third node from amongst the transition failitators in D i, i.e. there exists a transition failitator z suh that z shares at least one hannel with q x and one hannel with q y. In Setion 5., we summarized a proof from [3] showing that q x and q y an ommuniate through a given z with probability p z > = Ω(. Given 2 log 2 n our hoie of ell area a(n, and onditioned on the fat that eah ell has 200max{logn,} nodes (Lemma 0, of whih 80max{logn,} are deemed bakbone andidates and the rest are transition failitators, there are at least 20 max{logn,} 20logn possibilities for z within that ell. All the possible z nodes have i.i.d. hannel assignments. Thus, the probability that q x and q y annot ommuniate through any z in the ell is at most ( p z 20logn, and the probability they ommuniate through some z is p xy ( p z 20logn. Hene, the probability that this happens in none of the 2 distint ells is at most ( p xy logn f ( p z 2 < ( 20 2 logn 2 e 20logn (from Lemma 3. Applying the union bound over all n n flows, the 20 probability that all flows are able to transition is at least. n 9

6 D( D(2 D(3 D D D(8 Previous hop bakbone node prev. hop for at most 4 entering bakbones in step k D InomingD(4 bakbone links P S D(7 D(6 D(5 Figure 2: Illustration of detour routing Hene, we require eah route to have at least 2 distint hops (note that this is not a tight bound on the minimum number of required hops. Resultantly, we annot stipulate that all flows be routed along the (almost straight-line path SD D. If SD D is short, a detour may be required to ensure the minimum route-length, akin to detour-routing in the onstrutions of [3]. Suh flows are said to be detour-routed. Flow Transition Strategy. As per our strategy, a non-detourrouted flow is initially in a progress-on-soure-bakbone mode, and keeps to the soure bakbone till there are only 2 distint intermediate ells left to the destination. At this point, it enters a ready-for- transition mode, and atively seeks opportunities to make a transition to the destination bakbone along the remaining hops. One it has made the transition into the destination bakbone, it proeeds towards the destination on that bakbone along the remaining part of the route, and is thus guaranteed to reah the destination. Thus, we stipulate that the (almost straight-line SD D path be followed if the straight-line route omprises h 2 distint intermediate ells (hops. If S and D (hene also D lie lose to eah other, the hop-length of the straight line ell-to-ell path an be muh smaller. In this ase, a detour path SPD D is hosen (Fig. 2, using a irle of radius 2 r(n in a manner similar to that in the onstrutions desribed in [3, ] (onsider a irle of this radius entered around S, hoose a point P on the irle, and follow the route SPD D. A detour-routed flow is always in ready-for-transition mode. The need to perform detour routing for some soure-destination pairs does not have any substantial effet on the average hop-length of routes or the relaying load on a ell, as we show further. LEMMA 9. If the number of flows in any ell is x in ase of pure straight-line routing, it is at most x + O( n4 r 2 (n f 4 = x + O(log 6 n w.h.p. in ase of detour routing. PROOF. The proof has been omitted due to spae onstraints. LEMMA 20. The number of flows traversing any ell is O(n a(n w.h.p. even with detour routing. PROOF. The proof has been omitted due to spae onstraints. LEMMA 2. The number of flows traversing any ell in readyfor-transition mode is O(log 6 n w.h.p. PROOF. The proof has been omitted due to spae onstraints. Figure 3: Cell D and neighboring ells during bakbone onstrution Bakbone Constrution. The bakbone onstrution proedure is required to take load-balaning into aount. Thus we an desribe the proedure for onstruting the bakboneb p (x of x as follows: Given a elld, the 8 ells adjaent to elld are denoted as D( j, j 8 (Fig. 3. B p (x is onstituted as follows. Let S D b be the subset of ells that must be overed byb p (x where S omprises ells traversed by the flow for whih x is the soure, andd b omprises the ells traversed by flows for whih it may be the destination. x is by default a member ofb p (x. We onsider bakbone onstrution for the route from eah soure to its pseudo-destination below. Some routes require an additional last hop to reah the atual destination node. However, from Lemma 3, the only suh last hop routes that may enter a ell orrespond to pseudo-destinations in the 8 adjaent ells. Then applying Lemma 4 to the set of pseudo-destinations, they are only O(na(n suh pseudo-destinations, and thus only O(na(n suh last-hop flows entering the ell. Hene we an aount for them separately. Expanding bakbones tos. We first over ells ins. Reall that we are only onstruting the SD part and not onsidering the possible additional last hop at this stage. This has two sub-stages. In the first stage, we onstrut bakbones for soure nodes whose flow does not require a detour. In the seond sub-stage we onstrut bakbones for soure nodes whose flow requires a detour. Straight-line bakbones: This step proeeds in a hop-by-hop manner for all non-detourrouted flows in parallel (eah of whih has a unique soure x. Any ell ofs in whih there is already a node assigned tob p (x is alled a filled ell. Thus initially x s ell is filled. We then onsider the ell ins that is traversed next by the flow. We onsider all nodes in that ell sharing one or more ommon hannel with x. This provides a number of alternative hannels on whih the flow an enter that ell. Let h max be the maximum hop-length of any non-detour-routed SD route. Then h max = O( and the proedure has h max a(n steps. In step k, for eah soure node x whose flow has k or more hops,b p (x expands into the ell entered by x s flow on the k-th hop. Eah elld performs the following proedure: The bakbones are extended by onstruting bipartite graphs that aid load-balane.

7 LEMMA 22. If f 00, then it is possible to devise a bakbone onstrution proedure, suh that, after step h max of the bakbone onstrution proedure for S (for non-detour-routed flows, eah ell has O( n a(n inoming bakbone links on a single hannel, and eah node appears on O( n a(n (soure bakbones, w.h.p. PROOF. This proof assumes that the high probability events in Lemma 0, Lemma, Lemma 6, and Lemma 7 our. We present an indutive argument. Reall that we are expanding bakbones to over ells ins. At eah step of the (indutive onstrution, we first have a hannel-alloation phase, followed by a node-alloation phase. We prove that after step k of the bakbone onstrution proedure, the following two invariants hold for all ells of the network: Invariant : Eah node is assigned at most 4 new inoming bakbone links during step k. Thus after step k, it appears in a total of O(4k = O(k bakbones. Invariant 2: No more than 5na(n new bakbone links enter the ell on a single hannel during step k. Thus, in total O( kna(n inoming bakbones (entering the ell are assigned (inoming links on a single hannel after step k. If the above two Invariants hold, then it is easy to see that after h max steps, elld will have no more than 5h maxna(n = O( n a(n bakbone links assigned to any single hannel, and no node ours on more than 4h max = O( = O( n a(n a(n bakbones (from Eqn. (6. We prove that the Invariants hold, by indution, as follows: If Invariant holds at the end of step k, then Invariant 2 ontinues to hold after the hannel-alloation phase of step k. If Invariant 2 holds after the hannel-alloation phase of step k, then Invariant will ontinue to hold after the node-alloation phase of step k, and thus both Invariants and 2 will hold at the end of step k. Base Case: Before the proedure begins, at step 0, eah node is assigned to its own bakbone, for whih it is effetively the origin (and this an be viewed as a single bakbone link inoming to this node from an imaginary super-soure. Thus after Step 0, Invariant holds trivially, and Invariant 2 is irrelevant, and thus trivially true. Indutive Step: Suppose Invariants and 2 held at the end of step k. Consider a partiular elld during step k. Let the number of proper hannels ind be. From Lemma 6, we know that f for eah ell. Eah flow that enters elld in step k has a previous hop-node in one of the 8 adjaent ells. Also note that, from Lemma 6, eah previous hop node has at least 3 f 4 of elld s proper hannels available to it as hoies (sine it an swith on f hannels of whih at most f 4 may be non-proper in elld. Channel-Alloation. Construt a bipartite graph with two sets of verties (Fig. 4; one set (all itl has a vertex orresponding to eah of the (soure bakbones that enter the elld in step k. From Lemma, it proeeds that L 5na(n 4. The other set (all itp has 5na(n 5na(n i.e., P = 5na(n. verties for eah proper hannel i in elld, Set L Set V L One vertex for eah bakbone entering ell D in step k.. 0 Set P Figure 4: Bipartite Graph for CellD in step k Channel i verties Set N(V Channel i2 verties Channel i3 verties Channel i verties Channel i verties 5na(n verties for eah proper hannel A bakbone vertex is onneted to all the verties for the hannels proper ind on whih its previous hop node an swith (and whih are therefore valid hannel hoies for entering the ell D. We show that there exists a mathing that pairs eah bakbone vertex to a unique hannel vertex, through an argument based on Hall s marriage theorem (Theorem 2. Thus, we seek to show that for allv L, N (V V, wheren (V P is the union of the neighbor-sets of all verties inv. We first note the following: 3 f 4 5na(n 3 f 4 5 f na(n 3 f na(n ( 5na(n = Consider the following two ases: 5 f na(n 3 f 4 4 ( na(n Case : V < 8. Consider any setv of bakbone verties suh that V < 8. Then, sine there are at most f 4 non-proper hannels in a ell, every previous hop node has at least 3 f 4 3 f 4 proper hannel hoies. For eah proper hannel there are 5na(n 5na(n assoiated hannel verties. Thus we ( obtain that N (V 3 f 5na(n 4 8 (from Eqn. 8. Thus N (V V. Case 2: V 8. Now onsider setsv of size at least 8. Note that sine Invariant held till end of step k, no more than 4 bakbone links were assigned to any single node in 8 D(k in step k. k= Intuitively, in order to show that N (V V for all suhv, we first state and prove the observation that if a hannel overload ondition ours, resulting in N (V < V for somev, then the (8

8 overload must also manifest itself in some hannel-aligned subset (i.e. a subset where all flows have some ommon proper hannel i available to them. Thus, to show that no overload ondition ours, it suffies to show that no overload ondition ours in any of these ritial hannel-aligned subsets, whih an be shown using Lemma 7. The argument is formalized as follows: LetV i be the set omprising all setsu i L, suh that all bakbone verties inu i have hannel i assoiated with them (i.e., all bakbone verties inu i have i available to them as a valid proper hannel hoie for enteringd. Claim (a. U U S V i : i proper ind = N (U L 8 Proof of Claim (a: We know thatu V i for some i that is proper ind. Also, sine no node an be the previous hop in step k of more flows than those assigned to it in step k, and Invariant held after step k, it proeeds that no previous hop node is ommon to more than 4 entering bakbone links. LetA be the set of distint previous hop nodes assoiated with U. Then A 4 U 4 ( 8 f na(n 4 + f na(n 2 > f na(n f 500 > 2. Observe thata thus on- (note that f na(n tains at least one subsetb satisfying B = f na(n + f na(n 4 4. Reognizing that all members ofa, and hene all members ofb, are apable of swithing on hannel i, we an invoke Lemma 7 onb, to obtain that when f 00, C(B 3 8. This yields: N (U ( ( C(B 5na(n 5na(n C(B 3 8 5na(n ( 3 8 5na(n 8 3 na(n na(n 4 L. Claim (b. Consider a setv L. Then: N (V < V = i proper ind,s i V s.t. : S i V i and S i 8 5na(n 8 Proof of Claim (b: Suppose N (V < V. Let us denote by S i V the set of all bakbone verties inv that are assoiated with hannel i (i.e., have hannel i available as a valid proper hannel hoie for entering ell D. Consider the bipartite sub-graph G V indued byv N (V, and assign all edges unit apaity. Construt the graph G V {s,t} where s is a soure node having a unit apaity edge to all verties v V, and t is a sink node, onneted to eah vertex u N(V via a unit apaity edge. We try to obtain a (s,t flow g suh that all edges (s,v are saturated. Eah vertex v V sub-divides the unit of flow reeived from s equally amongst all edges (v,u outgoing from it. Sine eah vertex ( has edges to verties of at least 3 f 4 hannels, this yields at least 3 f 5na(n 4 8 edges (see Eqn. 8. Thus eah v V 8 ontributes at most units of flow to a vertex u N (V, i.e., g(v,u 8. Hene no vertex u N(V gets more than h(u = g(v,u = 8 S i units of flow, where i is the hannel v S i orresponding to vertex u. Resultantly, if S i 8 for all hannels i that are proper in elld, this implies that h(u, and setting g(u,t = h(u yields the desired (s,t flow. Hene g is a valid flow that allows a unit of flow to pass through eah vertex (9 v V. From the Integrality Theorem (Theorem 3, we an obtain an integer-apaity flow that yields a mathing of size V. Thus, from Hall s marriage theorem (Theorem 2, N (V V (else a mathing of size V ould not have existed. This yields a ontradition. Thus there must exist a proper hannel i, ands i V suh thats i V i and S i > 8. Sine set-ardinality must ne- 8, and Eqn. essarily be an integer, it proeeds that S i (9 holds. Claim (. V L suh that V 8 : N (V V Proof of Claim (: Suppose N (V < V for some suhv. Then, from Claim (b, there exists a sets i V suh thats i V i, and S i 8. ThusS i qualifies as a set to whih Claim (a applies. Invoking Claim (a on this sets i, it follows that N (V N (S i L V. This yields a ontradition. Hene: N (V V. Therefore, by appliation of Hall s marriage theorem (Theorem 2, eah bakbone vertex an be mathed with a unique hannel vertex, and the orresponding bakbone will be assigned to the hannel with whih this vertex is assoiated. Thus all bakbones get assigned a hannel, and (sine there are at most 5na(n hannel verties for eah proper hannel no more than 5na(n inoming bakbone links are assigned to any single hannel. While Hall s marriage theorem proves that suh a mathing exists, the mathing itself an be omputed using the Ford-Fulkerson method [4] on a flow network obtained from the bipartite graph by adding a soure with an edge to eah vertex inl, a sink to whih eah vertex inp has an edge, and assigning unit apaity to all edges. Thus Invariant 2 ontinues to hold after the hannel-alloation phase of step k. Node-Alloation. Having determined the hannel eah bakbone should use to enter elld, we need to assign a node in ell D to eah bakbone. For this, we again onstrut a bipartite graph. In this graph, the first set of verties (all itf omprise a vertex for eah bakbone entering elld in step k. The seond set (all it R omprises 4 verties for eah bakbone andidate node in ell D. A vertex x inf has an edge with a vertex y inr iff the atual bakbone andidate node assoiated with y is apable of swithing on the hannel assigned to the bakbone assoiated with vertex x in the preeding hannel-alloation phase. Eah vertex x F has degree at least 4M u, sine it is assigned to a proper hannel, whih by definition has at least M u representatives in elld, eah of whih has 4 assoiated verties inr. Also reall that M u = 9 f na(n 25. One again we seek to show that for all V F, N (V V. Consider any setv F. Sine no hannel is assigned more than 5na(n entering bakbone links in this step, the verties in V are umulatively assoiated with at least m V distint proper hannels. Sine eah 5na(n of these hannels have at least M u bakbone andidate nodes apable of swithing on them, and any one node an only swith on up to f proper hannels, this implies that the number of nodes in ell D umulatively assoiated with these m V proper hannels 5na(n is at least V M u f 5na(n 9 f na(n V 25 5 f na(n 9 V 25, and as eah node has 4 verties, it follows that N (V 4 ( 9 V V 25 > V.

9 Then invoking Hall s Marriage Theorem again, eah vertex x F an be mathed with a unique vertex y R, and the atual network node assoiated with y is deemed the bakbone representative for the bakbone orresponding to vertex x in elld (the mathing an again be omputed via the Ford-Fulkerson method. Sine there are at most 4 verties assoiated with a node, no node is assigned more than 4 inoming bakbone links in step k, and Invariant ontinues to hold after the node-alloation phase of step k. Thus we have shown that both Invariants and 2 ontinue to hold after step k. Hene after step h max (where h max 2, eah elld has a(n O( h maxna(n = O( n a(n entering bakbone links per hannel, and eah node appears on O(h max = O( = O( n a(n a(n (from Eqn. (6 soure bakbones. Detour bakbones: From Lemma 9 the number of additional flows traversing a ell due to detour routing is only O(log 6 n, and eah suh flow will at most traverse the ell twie. Thus detour flows do not pose any signifiant load-balaning issue at any ell, and we an grow the bakbones ins for these flows in any manner possible, i.e. by assigning links to any eligible node/hannel (at least one eligible node is guaranteed to exist sine, as a onsequene of Lemma 6, eah node an swith on at least 3 f 4 hannels that are proper in the next ell. Additional last hop: We now aount for the possible additional last hop that some flows may have, yielding an additional ell ins (in addition to those traversed from soure to pseudo-destination. We already argued that at most O(na(n = O( n a(n flows (from Eqn. (5 enter any ell on their additional last hop. Thus, even if their bakbone links are assigned to the same hannel/node, we would still have O( n a(n flows per node and hannel in any ell for thes stage. Expanding bakbone tod b S. In this stageb p (x expands into the ells traversed by flows for whih x is the destination. Note that by our routing strategy a flow will only attempt to swith to the destination bakbone when it enters ready-for-transition mode. From Lemma 2, the total number of flows traversing a ell in ready-for-transition mode is O(log 6 n (ounting possible repeat traversals, whih is muh smaller than O( n a(n. Thus flows on their destination bakbone do not pose any major load-balane issues, and the bakbones an be expanded into ells ofd b S by assigning links to any eligible node/hannel. 8.2 Proving load-balane within a ell We now show that no hannel or interfae bottleneks form in the network when our desribed onstrution is used. Per-Channel Load LEMMA 23. The number of flows that enter any ell on any single hannel is O( n a(n w.h.p. PROOF. A flow on routed,d 2,...,D j,d j... may enter a elld j on a hannel i if ( the flow is in progress-on-sourebakbone mode, or it is in ready-for-transition mode, but is yet to find a transition into the destination bakbone, and i is the shared hannel between the soure bakbone nodes ind j andd j, or (2 z q x q00 y Figure 5: Two additional transition links for a flow lying wholly within the ell the flow has already made a transition, and i is the shared hannel between the destination bakbone nodes ind j andd j We first onsider the flows that enter a ell in progress-on-sourebakbone mode, i.e., are proeeding on their soure bakbones. Reall that these are all non-detour-routed flows, sine detour-routed flows are always in ready-for-transition mode. Then the number of suh flows that enter any ell on a single hannel is O( n a(n from Lemma 22. We now need to aount for the fat that some of these flows may be in the ready-for-transition mode. From Lemma 2 there are O(log 6 n flows traversing any ell in ready-for-transition mode w.h.p. (reall that these inlude the detour-routed flows with their repeat traversals ounted separately, and the possible additional last D D hop. Thus regardless of whether they are still on their soure bakbone, or have already made the transition to their destination bakbone, no hannel an have more than O(log 6 n suh flows entering the ell. Hene the number of flows entering on a single hannel is O( n a(n + O(log 6 n = O( n a(n w.h.p. for eah ell of the network. LEMMA 24. The number of flows that leave any ell on any single hannel is O( n a(n w.h.p. PROOF. Note that the flows that leave the ell, must then enter one of the 8 adjaent ells on that hannel (as the orresponding bakbone link for a flow leaves the urrent ell, and enters an adjaent ell. Thus, flows leaving the ell on a hannel an be no more than 8 times the maximum number of flows entering a ell on any one hannel, whih has been established as O( n a(n in Lemma 23. Hene, the total number of flows leaving any ell on a single hannel is also O( n a(n w.h.p. LEMMA 25. The number of additional transition links sheduled on any single hannel within any ell is O(log 6 n w.h.p. PROOF. Reall the transition strategy outlined in the proof of Lemma 8, whereby the flow loates a ell along the route where the soure bakbone node q x, and destination bakbone node q y are onneted through a third node z. This yields two additional links q x z, and z q y that lie entirely within the ell (Fig. 5. Note that the number of flows performing this transition in the ell an be no more than the number of flows traversing the ell in ready-fortransition mode. From Lemma 2 there are O(log 6 n suh flows traversing any ell w.h.p. In the worst ase, we an ount 2 additional links for eah suh flow as being all assigned to one hannel. The result proeeds from this observation. Per-Node Load LEMMA 26. The number of flows that are assigned to any single node in any ell is O( n a(n w.h.p.

10 PROOF. A node is always assigned the single flow for whih it is the soure. A node is also assigned flows for whih it is the destination, and from Lemma 8 there are at most D(n = O(logn suh flows for any node w.h.p. Besides, a node may be assigned flows that are in the ready-to-transition mode, for whih it failitates a transition (if it is a transition failitator node, or on whose destination bakbone it figures. There are O(log 6 n suh transitioning flows in a ell w.h.p. from Lemma 2. Thus a node an only have O(log 6 n suh flows assigned. We now onsider the flows in progress-on-soure-bakbone mode that do not originate in the ell. These nodes are on their sourebakbone, and from Lemma 22, eah node has at most O( n a(n suh flows assigned. Thus, the resultant number of assigned flows per node is +D(n+O(log 6 n+o( n a(n = O( n a(n. 8.3 Transmission shedule As mentioned earlier, from the Protool Model assumption, eah ell an fae interferene from at most a onstant number β of nearby ells. Thus, if we onsider the resultant ell-interferene graph (a graph with a vertex for eah ell, and an edge between two verties if the orresponding ells an interfere with eah other, it has a hromati number at most + β. Hene, we an ome up with a global shedule having +β unit time slots in eah round. In any slot, if a ell is ative, then all interfering ells are inative. The next issue is that of intra-ell sheduling. We need to shedule transmissions so as to ensure that at any time instant, there is at most one transmission on any given hannel in the ell. Besides, we also need to ensure that no node is expeted to transmit or reeive more than one paket at any time instant. We onstrut a onflit graph based on the nodes in the ative ell, and its adjaent ells (note that the hop-sender of eah flow shall lie in the ative ell, and the hop-reeiver shall lie in one of the adjaent ells, exept for transition links, for whih both lie in the ative ell as follows: we reate a separate vertex for eah flow for whih a node in the ell needs to transmit data (we ount repeat traversals or additional transition links as distint flows for the purpose of sheduling; these have been aounted for while bounding the number of flows in a ell in previous lemmas. Sine the flow has an assigned hannel on whih it operates in that partiular hop, eah vertex in the graph has an impliit assoiated hannel. Besides, eah vertex has an assoiated pair of nodes orresponding to the hop endpoints. Two verties are onneted by an edge if ( they have the same assoiated hannel, or (2 at least one of their assoiated nodes is the same. The sheduling problem thus redues to obtaining a vertex-oloring of this graph. If we have a vertex oloring, then it ensures that ( a node is never simultaneously sending/reeiving for more than one flow (2 no two flows on the same hannel are ative simultaneously. Thus, the number of neighbors of a graph vertex is upper bounded by the number of flows requiring a transmission in the ative ell on that hannel, and the number of flows assigned to the flow s two hop endpoints (both hop-sender and hop-reeiver. It an be seen from Lemma 23, Lemma 24, Lemma 25 and Lemma 26 that the degree of the onflit graph is O( n a(n O( n a(n Ω( n +O( n a(n +O(log 6 n+o( n a(n +O( n a(n = (note that O(log 6 n = O( n a(n, sine n a(n = logn. Thus the graph an be olored in O( n a(n olors. Hene, the ell-slot (whih an be assumed to be of unit time is nlogn divided into O( n a(n = O( equal length subslots, and all traversing flows get a slot for transmission. This implies that eah flow gets a Ω( nlogn fration of the time. Moreover, eah ell gets at least one slot in + β slots, where β is a onstant, and eah hannel has bandwidth W. Thus eah flow gets a throughput of at ( (W least +β Ω( nlogn = Ω(W nlogn. We thus obtain the following theorem: THEOREM 4. When = O(logn, and 2 f, the per-flow network apaity with random (, f assignment is Θ(W nlogn. 9. CONCLUSION We have established the apaity of a random network with random (, f assignment, for = O(logn, 2 f. Our result indiates that apaity is Θ(W nlogn. Thus, when f = Ω(, one an ahieve apaity of the same asymptoti order as with unonstrained swithing. There still remain some interesting open questions pertaining to the random (, f model, in terms of what is ahievable via stritly asynhronous routing/sheduling. However, the results in this paper, along with prior results in [3], have been able to demonstrate that it may be possible to ahieve good throughput harateristis even when devies are subjet to swithing onstraints. Designing pratially feasible and effiient protools for networks of devies with onstrained swithing ability is still an open and interesting problem domain. 0. REFERENCES [] V. Bhandari and N. H. Vaidya. Connetivity and Capaity of Multi-Channel Wireless Networks with Channel Swithing Constraints. Tehnial Report, CSL, UIUC, January [2] V. Bhandari and N. H. Vaidya. Capaity of Multi-Channel Wireless Networks with Random (, f Assignment. Tehnial Report, CSL, UIUC, June [3] V. Bhandari and N. H. Vaidya. Connetivity and Capaity of Multi-Channel Wireless Networks with Channel Swithing Constraints. In Proeedings of IEEE INFOCOM, Anhorage, Alaska, May [4] T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introdution to Algorithms. MIT Press, 990. [5] C. Fragouli, J. Widmer, and J.-Y. 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