Scheduling wireless links by graph multicoloring in the physical interference model
aa r X i v : . [ c s . N I] A p r Scheduling Wireless Links by GraphMulticoloring in the Physical Interference Model
Fabio R. J. Vieira ∗ Jos´e F. de RezendeValmir C. BarbosaPrograma de Engenharia de Sistemas e Computa¸c˜ao, COPPEUniversidade Federal do Rio de JaneiroCaixa Postal 6851121941-972 Rio de Janeiro - RJ, Brazil
Abstract
Scheduling wireless links for simultaneous activation in such a way thatall transmissions are successfully decoded at the receivers and moreovernetwork capacity is maximized is a computationally hard problem. Usu-ally it is tackled by heuristics whose output is a sequence of time slots inwhich every link appears in exactly one time slot. Such approaches can beinterpreted as the coloring of a graph’s vertices so that every vertex getsexactly one color. Here we introduce a new approach that can be viewedas assigning multiple colors to each vertex, so that, in the resulting sched-ule, every link may appear more than once (though the same number oftimes for all links). We report on extensive computational experiments,under the physical interference model, revealing substantial gains for avariety of randomly generated networks.
Keywords:
Wireless mesh networks, Link scheduling, Physical interfer-ence model, Graph coloring, Graph multicoloring. ∗ Corresponding author ([email protected]). Introduction
Let L be a set of wireless links, each link i ∈ L characterized by a sendernode s i and a receiver node r i . Depending on the spatial disposition of suchnodes, activating more than one link simultaneously creates interference thatmay hamper the receivers’ ability to decode what they receive. In the physicalinterference model [1], the chief quantity governing receiver r i ’s ability to decodewhat it receives from s i when all links of a set S containing link i are active isthe signal-to-interference-and-noise ratio (SINR), given bySINR( r i , S ) = P/d αs i r i N + P j ∈ S \{ i } P/d αs j r i , (1)where P is a sender’s transmission power (assumed the same for all senders), N is the noise floor, d ab is the Euclidean distance between nodes a and b , and α > S of L is feasible if no two of its members share a node (incase | S | >
1) and moreover SINR( r i , S ) ≥ β for all i ∈ S , where β is a parameterrelated to a receiver’s decoding capabilities (assumed the same for all receivers)and is chosen so that β > L for activation [2–15] orby combining link scheduling with other techniques [16–22]. All these strate-gies revolve around formulations as NP-hard optimization problems, so all relyon some form of heuristic procedure drawing inspiration from various sources,some merely intuitive, others more formally grounded on graph-theoretic no-tions. Often the problem is formulated in a spatial time-division multiple access(STDMA) framework, that is, assuming essentially that time is divided intotime slots, each one accommodating a certain number of simultaneous link ac-tivations. In this case, the problem is to find T feasible subsets of L , heredenoted by S , S , . . . , S T , minimizing T while ensuring that every link appearsin exactly one of the T subsets.There is a sense in which this formulation can be interpreted in the contextof coloring a graph’s vertices. Specifically, if we regard the links to be scheduledas vertices in a graph, and furthermore say that no two vertices of a group areneighbors of each other if the corresponding set of links is feasible, then theschedule given by the sequence S , S , . . . , S T of feasible link sets establishesa coloring of the graph’s vertices with T colors in which all vertices in S k getcolor k . This interpretation suggests a generalization of the above formulationthat requires every link to appear not in exactly one of the T subsets but inany number q of subsets, provided this number is the same for all links. Inthis generalized formulation, the goal is no longer to minimize T , but ratherto find the values of T and q that minimize the ratio T /q . Returning to thevertex-coloring interpretation, now a vertex receives q (out of T ) distinct colors,each relative to the time slot in which the corresponding link is scheduled to beactivated (i.e., q of the subsets S , S , . . . , S T ).2he potential advantages of this multicoloring-based formulation are tanta-lizing. If the original formulation leads to a number T of slots while the newone leads to T ′ > T slots for some q >
1, the latter schedule is preferable to theformer, even though it requires more time slots, provided only that T ′ /q < T (or qT > T ′ ). To see that this is so, first note that the longer schedule promotesan overall number of link activations given by q | L | in T ′ time slots. In order forthe shorter schedule to achieve this same number of activations, it would haveto be repeated q times in a row, taking up qT > T ′ time slots.The possibility of multicoloring-based link scheduling in the physical inter-ference model seems to have been overlooked so far, despite the recent demon-stration of its success in the protocol-based interference model [22]. Here weintroduce a heuristic framework to obtain multicoloring-based schedules fromthe single-color schedules produced by any rank-based heuristic (i.e., one thatdecides the time slot in which to activate a given link based on how it ranks rela-tive to the others with respect to some criterion). We use two iconic single-colorheuristics (GreedyPhysical [3], for its simplicity, and ApproxLogN [8, 15], forits role in establishing new bounds on network capacity), as well as a third onethat we introduce in response to improvement opportunities that we perceivedin the former two. Incidentally, the latter heuristic, called MaxCRank, is foundto perform best both as a stand-alone, single-color strategy and as a base forthe multicoloring scheme. Rank-based heuristics for single-color scheduling are usually monotonic, in thesense that first S is determined, then S out of the set R of links that remainto be scheduled, then S out of a smaller R , and so on, until R becomes empty.Choosing a link to add to the current S k depends on the feasibility of theresulting set and also on a ranking criterion that is specific to each heuristic.The ranking criterion establishes the order in which the links in R are to beconsidered for inclusion in S k .The following is the general outline of such a heuristic.1. Let k := 1, S k := ∅ , and R := L . Order R according to the rankingcriterion.2. If a link i ∈ R exists such that S k ∪ { i } is feasible, then move the top-ranking such i from R to S k and go to Step 3. If none exists, then let k := k + 1, S k := ∅ , and go to Step 2.3. If R = ∅ , then reorder R according to the ranking criterion and go toStep 2.4. Let T := k and output S , S , . . . , S T .Steps 1–4 amount to scanning the set R of unscheduled links and moving to thecurrent S k (in Step 2) the top-ranking link i ∈ R whose inclusion in S k preserves3easibility. Whenever such a move does occur, an opportunity is presented for R to be reordered (in Step 3) according to the ranking criterion.It is easy to see that both GreedyPhysical and ApproxLogN can be cast inthis sequence of steps in a straightforward manner. The ranking criterion forGreedyPhysical is nonincreasing and refers, for link i , to the number of linksin L with which i can never share a time slot; that is, links j ∈ L such that { i, j } is infeasible. It is then an immutable ranking criterion and consequentlythe reordering in Step 3 is moot. As for ApproxLogN, its ranking criterion isnondecreasing, but now refers to the Euclidean distance between the sender andthe receiver in each link. This criterion, too, is fixed and as such renders thereordering in Step 3 once again moot. We now introduce a new heuristic that can also be viewed as instantiatingSteps 1–4, but with a ranking criterion that is both more stringent than thetwo just described and also inherently dynamic, thus justifying the reorderingin Step 3. We call it MaxCRank to highlight its core principle, which is tomaximize the number of links in R that still have a chance of joining the current S k (i.e., remain “Candidates”) once a decision is made on which one of them,say i , is to be moved from R to S k . The corresponding ranking criterion isnondecreasing and refers to the number of links j ∈ R \ { i } for which S k ∪ { i, j } is infeasible. The link sets S , S , . . . , S T output by Steps 1–4 of Section 2 promote a numberof link activations given by | L | , one activation per link. If this schedule were tobe repeated q times in a row for some q >
1, the total number of link activationswould grow by a factor of q and so would the number of time slots used. Thatthe same growth law should apply both to how many links are activated andto how many time slots elapse indicates that the most basic scheduling unit is S , S , . . . , S T itself, not any number of repetitions thereof.However, activating the links in S the second time around does not neces-sarily have to be restricted to time slot T + 1. Instead, it may be possible totake advantage of some room left in previous time slots for at least one of thelinks in S . With this type of precaution in mind, advancing link activations insuch a manner might result in a sequence of link sets S , S , . . . , S T ′ containingexactly q activations of every link in L for some q > T < T ′ < qT .Clearly, in this case the most basic scheduling unit would be S , S , . . . , S T ′ ,not S , S , . . . , S T any more. Not only this, but the new basic scheduling unitwould be preferable to the previous one, since a total of q | L | link activationswould be attainable in fewer time slots ( T ′ rather than qT ).A heuristic to find the greatest q > T ′ < qT , if any exists, issimply to wrap Steps 1–4 in an outer loop that iterates along with q = 1 , , . . . ApproxLogN replaces the requirement of feasibility in Step 2 by conditions that are suf-ficient for it to be satisfied. This is done to make sure that certain algorithmic performanceguarantees hold, but that is of no concern to us here. S k from being reset to ∅ any later than the first time it isconsidered. At the end of each iteration, say the q th, the value of T ′ is updated(to the number of time slots elapsed since the beginning) and the ratio T ′ /q is computed. The iterations continue while this ratio is strictly decreasing. Atthe end of the first iteration we get T ′ = T , but successful further iterationswill produce a sequence of strictly decreasing T ′ /q values. A new quantity ofinterest is then the gain G incurred by the resulting heuristic, that is, the ratioof T to the last T ′ /q , hence G = qT /T ′ . The least possible value of G , of course,is G = 1, which corresponds to the case in which the iterations fail already for q = 2. We give results for two families of randomly generated networks, henceforthreferred to as type-I and type-II networks. As will become apparent, type-Inetworks are more realistic. We use type-II networks as well because they wereused in the performance evaluation of ApproxLogN [8, 15] and thus provide amore direct basis for comparison. A network’s number of nodes is henceforthdenoted by n .A type-I network is generated by first placing all n nodes inside a squareof side ℓ uniformly at random. A node’s neighbors are then determined as afunction of the value of d s i r i for which SINR( r i , { i } ) = β . Denoting such adistance by ρ yields ρ = ( P/βN ) /α , so a node’s neighbor set is the set of nodesto which the Euclidean distance does not surpass ρ . Any two nodes that areneighbors of each other become a link in L , sender and receiver being decideduniformly at random (so that a node may, e.g., be the sender in a link and thereceiver in another). For fixed n , increasing ℓ causes the number of links, | L | ,to decrease precipitously, though in the heavy-tailed manner of an approximatepower law (Fig. 1). It also causes the network’s number of connected componentsto increase from about 1 to nearly n (a component per node) through a sharptransition in between (Fig. 2).In a type-II network, the number n of nodes is necessarily even. Of these, n/ n/ ℓ and then,for each receiver, placing the corresponding sender inside a circle of radius ρ centered at it, also uniformly at random. A type-II network has n/ ℓ affects interference only. We give results for all three single-color heuristics mentioned in Section 2,namely GreedyPhysical, ApproxLogN, and MaxCRank, and also for their multicoloring-based versions, obtained as explained in Section 3. These results are given as5 | L | ℓ (m) Figure 1: Average number of links in type-I networks as a function of the squareside ℓ for n = 100. All data points are averages over 1 000 network instances.Additional relevant parameters are P = 300 mW, N ≈ × − W (for abandwidth of 20 MHz at room temperature), α = 4, and β = 25 dB. N u m b e r o f c onn ec t e d c o m pon e n t s ℓ (m) Figure 2: Average number of connected components in type-I networks as afunction of the square side ℓ for n = 100. All data points are averages over1 000 network instances. Additional relevant parameters are P = 300 mW, N ≈ × − W (for a bandwidth of 20 MHz at room temperature), α = 4,and β = 25 dB. 6 / | L | in the former case (the normalized schedule length, since | L | is a clearupper bound on T ), and as the gain G in the latter.The data in Fig. 3 refer to type-I networks and as such are given as a func-tion of the square side ℓ . The number of nodes is fixed throughout (at n = 100),so the networks get sparser (fewer links, more connected components) as ℓ isincreased. In the single-color cases (panel (a) of the figure), all three heuris-tics start out with T = | L | for the very dense networks (very small ℓ ), butsmaller densities quickly reduce interference so that T falls significantly below | L | . MaxCRank is the best performer throughout, followed by GreedyPhysicaland ApproxLogN. As for the heuristics’ multicoloring-based versions (panel (b)),there is practically no gain for the densest networks, but again this is reversedas interference abates with increasing ℓ . MaxCRank is still the top performerand ApproxLogN the bottom one (in fact, the only of the three heuristics forwhich G = 1 is sometimes attained).The results for type-II networks, given in Fig. 4, are presented as a function of | L | = n/
2, the number of links. Because ℓ is fixed throughout (at ℓ = 1 000 m),increasing | L | causes the impact of accumulated interference to be felt moreseverely. One consequence of this is that, for the single-color heuristics (panel (a)of the figure), T increases almost linearly with | L | . Another consequence, nowrelated to the multicoloring-based versions of the heuristics (panel (b)), is thatgains above 1 are increasingly hard to come by as | L | is increased. MaxCRankcontinues to be the top performer in all cases, followed by GreedyPhysical, thenby ApproxLogN. Although it may at first seem striking that ApproxLogN has performed so poorlyacross most of our experiments, it should be kept in mind that this heuristic,in all likelihood, was never meant as a serious contender for single-color linkscheduling. In fact, and as noted in Section 2, ApproxLogN approaches thechecking of feasibility rather indirectly, verifying sufficient conditions for feasi-bility to hold instead of the property itself. This is bound to prevent Approx-LogN from scheduling links for activation when they could be scheduled. Whatmust be kept in mind, then, is that the use of such indirect conditions has led toimportant performance and capacity bounds. ApproxLogN, therefore, remainsan important contribution despite its performance in more practical settings.What really is striking in our results, though, is the appearance of greater-than-1 gains practically across the board, particularly for MaxCRank or Greedy-Physical as the base, single-color heuristic. Link schedules, once determined,are meant to be used repetitively, so every link is already meant to be sched-uled for activation over and over again, indefinitely. Conceptually, what ourmulticoloring-based wrapping of single-color heuristics tries to do is to inter-twine some number of repetitions of a single-color schedule, taking up fewertime slots than the straightforward juxtaposition of the same number of rep-etitions of that schedule. By doing so, more link activations can be packed7 T / | L | (a) ApproxLogNGreedyPhysicalMaxCRank 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1000 3000 5000 7000 9000 11000 13000 G ℓ (m)(b) Figure 3: Performance of GreedyPhysical, ApproxLogN, and MaxCRank ontype-I networks. Data are given for the heuristics’ single-color versions (a) andfor their multicoloring-based versions (b). All data points are averages over1 000 network instances. Confidence intervals are less than 1% of the mean atthe 95% level, so error bars are omitted. All networks have n = 100 nodes.Additional relevant parameters are P = 300 mW, N ≈ × − W (for abandwidth of 20 MHz at room temperature), α = 4, and β = 25 dB.8 T / | L | (a) ApproxLogNGreedyPhysicalMaxCRank 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 100 3200 6400 12800 25600 G |L| (b) Figure 4: Performance of GreedyPhysical, ApproxLogN, and MaxCRank ontype-II networks. Data are given for the heuristics’ single-color versions (a) andfor their multicoloring-based versions (b). All data points are averages over1 000 network instances. Confidence intervals are less than 1% of the meanat the 95% level, so error bars are omitted. All networks have ℓ = 1 000 m.Additional relevant parameters are P = 300 mW, N ≈ × − W (for abandwidth of 20 MHz at room temperature), α = 4, and β = 25 dB.9ogether in earlier time slots. As a consequence, the basic schedule to be usedfor indefinite repetition is now one that leads to higher network capacity andpossibly higher throughput.As we mentioned earlier, multicoloring-based link scheduling of the sort wehave demonstrated has roots in the multicoloring of a graph’s vertices (as wellas edges, in many cases). As such, a rich body of material, relating both tocomputational-complexity difficulties and to workarounds in important cases,is available. Further developments should draw on such knowledge, aiming toobtain more principled, and perhaps even better performing, heuristics. Acknowledgments
The authors acknowledge partial support from CNPq, CAPES, and a FAPERJBBP grant.
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