Distrib. Comput.DOI 10.1007/s00446-014-0222-7
Nearly optimal bounds for distributed wireless schedulingin the SINR model
Magnús M. Halldórsson · Pradipta Mitra
Received: 4 June 2013 / Accepted: 16 May 2014© Springer-Verlag Berlin Heidelberg 2014
Abstract We study the wireless scheduling problem in theSINR model. More specifically, given a set of n links, eacha sender–receiver pair, we wish to partition (or schedule)the links into the minimum number of slots, each satisfyinginterference constraints allowing simultaneous transmission.In the basic problem, all senders transmit with the same uni-form power. We analyze a randomized distributed schedulingalgorithm proposed by Kesselheim and Vöcking, and showthat it achieves O(log n)-approximation, an improvement ofa logarithmic factor. This matches the best ratio known forcentralized algorithms and holds in arbitrary metric spaceand for every length-monotone and sublinear power assign-ment. We also show that every distributed algorithm usesΩ(log n) slots to schedule certain instances that require onlytwo slots, which implies that the best possible absolute per-formance guarantee is logarithmic.
Keywords Wireless · Scheduling · SINR model
1 Introduction
Given a set of n wireless links, each a sender–receiver pair,what is the minimum number of slots needed to schedule allthe links, given interference constraints? This is the canonicalproblem of scheduling wireless communication, which westudy here in a distributed setting.
Supported by Grants 90032021 and 120032011 from the IcelandicResearch Fund. Preliminary version appeared in ICALP 2011.
M. M. Halldórsson (B) · P. MitraICE-TCS, School of Computer Science, Reykjavik University,Reykjavík, Icelande-mail: [email protected]
P. Mitrae-mail: [email protected]
In a wireless network, simultaneous transmissions on thesame channel interfere with each other. Algorithmic ques-tions for wireless networks depend crucially on the modelof interference considered. In this work, we use the physi-cal, a.k.a. SINR, model of interference, defined in Sect. 2. It isknown to capture reality more faithfully than the graph-basedmodels most common in the theory literature, as shown the-oretically as well as experimentally [21,23]. Early work onscheduling in the SINR model focused on heuristics and/ornon-algorithmic average-case analysis (e.g. [11]). In seminalwork, Moscibroda and Wattenhofer [22] proposed the prob-lem of scheduling an arbitrary set of links. Numerous workson various problems in the SINR setting have appeared since.
The scheduling problem has primarily been studied in acentralized setting. In many realistic scenarios, however, itis imperative that a distributed solution be found, since acentralized controller may not exist, and individual nodesin the link may not be aware of the overall topology of thenetwork. For the scheduling problem, the only rigorous resultpreviously known is due to Kesselheim and Vöcking [20],who show that a simple and natural distributed algorithmprovides an O(log2 n)-approximation.
In this work, we adopt the algorithm of Kesselheim andVöcking, but provide an improved analysis of an O(log n)-approximation. This matches the best upper bound knownfor centralized algorithms. Moreover, we show this to bebest possible for distributed algorithms that use no externalcommunication infrastructure.
2 Preliminaries and contributions
Given is a set L = {l1, l2, . . . , ln} of links, where each linklv represents a communication request from a sender sv toa receiver rv . The distance between two points x and y is
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denoted by d(x, y). The asymmetric distance from link lv tolink lw is the distance from v’s sender to w’s receiver, denotedby dvw = d(sv, rw). Let �v = d(rv, sv) denote the length oflink lv .
Let Pv denote the power assigned to link lv , or, in otherwords, sv transmits with power Pv . We adopt the SINR model(a.k.a., physical model) of interference, in which a node rv
successfully receives a message from a sender sv if and onlyif the following condition holds:
Pv/�αv∑
lw∈S\{lv} Pw/dαwv + N
≥ β, (1)
where N ≥ 0 is a universal constant denoting the ambi-ent noise, α > 0 denotes the path loss exponent, β > 0denotes the minimum SINR (signal-to-interference-noise-ratio) required for a message to be successfully received,and S is the set of concurrently scheduled links in the sameslot.We say that S is SINR-feasible (or simply feasible) if (1)is satisfied for each link in S.
A power assignment P is length-monotone if Pv ≥ Pw
whenever �v ≥ �w and sub-linear if Pv
�αv≤ Pw
�αw
whenever �v ≥�w [20]. Two widely used power assignments in this class arethe uniform power assignment, where every link transmitswith the same power; and the linear power assignment, wherePv is proportional to �α
v . A third one, mean power [6,12], hasalso proved to be versatile.
Given a set of links L , the scheduling problem is to finda partition of L of minimum size such that each subset inthe partition is feasible. The size of the partition equals theminimum number of (time or frequency) slots required toschedule all links. We will call this number the schedulingnumber of L , and denote it by χ(L) (or χ when clear fromcontext).
Distributed algorithms. A communication infrastructure forrunning distributed algorithms is generally assumed to existin traditional distributed settings. The current setting, whichabstracts the MAC layer in networks, is different, as the goalactually is to construct such an infrastructure. Thus, our algo-rithm will work with very little global knowledge and mini-mal external input.
Communication is only available over the channel. Algo-rithms operate in synchronous rounds with the senders eithertransmitting or listening in each round. When transmission issuccessful, the sender stops transmitting. This necessitates anacknowledgment from the receiver, so that the sender knowswhen his message has been heard. These acknowledgmentsare sent over the same channel as the message; thus, thereare no side-channels for control messages.
We assume that nodes have a rough estimate of the net-work size n and (senders of) links are assigned a fixed length-monotone, sublinear power function. The power assignment
indirectly requires each link to know its length as well asthe values of the path loss constant α and the technologicalparameters β and N . No information of locations is needed.
We note that the assumptions are particularly minimalwhen using uniform power. The algorithm then needs noknowledge of distances, the path loss constant α, nor the tech-nological parameters β and N . Only the polynomial boundon the number n of nodes is needed.
Affectance. We will use the notion of affectance, introducedin [9,17] and refined in [20] to the thresholded form usedhere. The affectance aP
w (v) on link lv from another link lw,with a given power assignment P , is the interference of lwon lv relative to the power received, or
aPw (v) = min
{
1, cv
Pw/dαwv
Pv/�αv
}
, (2)
where cv = β/(1− βN�αv /Pv) depends only on model con-
stants and on the length of lv .We will drop P and assume it to be an arbitrary length-
monotone sub-linear power strategy, unless otherwise stated.Let av(v) = 0. For a set S of links and a link lv , letaS(v) = ∑
lw∈S aw(v), referred to as in-affectance, andav(S) = ∑
lw∈S av(w), the out-affectance from lv . For setsS and R, aR(S) =∑
lv∈R∑
lu∈S av(u). Using such notation,(1) can be rewritten as
aS(v) ≤ 1 , (3)
whenever |S| > 2.
2.1 Related work
In the centralized setting, scheduling results have closely fol-lowed results on the related capacity problem, where onewants to find the maximum subset of L that can be trans-mitted in a single slot). Goussevskaia et al. [10] showed theproblem to be NP-hard for the case of uniform power onthe plane and gave O(log Δ)-approximation result (on theplane), where Δ denotes the ratio between the maximumand minimum length of a link. Same bound was shown byAndrews and Dinitz [1] but in comparison with optimumthat is allowed to choose arbitrary power. Constant factorapproximation was obtained for uniform power, also on theplane, by Goussevskaia et al. [9], which was generalized toall length-monotone, sublinear power assignments and arbi-trary metrics space by Halldórsson and Mitra [14]. Kessel-heim [18] gave a constant-factor approximation for the jointproblem of selecting links and assigning them feasible power(see also earlier work of Chafekar et al. [4]).
All the results lead to equivalent bounds for the centralizedscheduling problem with O(log n)-factor overhead. In par-
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SINR model
ticular, O(log n)-approximation holds for scheduling withlength-monotone, sublinear power [14] and with arbitrarypower control [18]. Also, the problem remains NP-hard [10].For the results in terms of Δ on the plane [1,10], this overheadcan be avoided (see, e.g., [12]). Scheduling with arbitrarypower control can also be approximated within a factor ofO(log n log log Δ) when the algorithm uses mean power. Forlinear power on the plane, an algorithm using O(χ + log2 n)
slots for instances with optimal schedule length χ was givenby Fanghänel et al. [7]; this can be improved to a constant fac-tor [26]. A bi-directional version was studied by Fanghänel etal. [6] and further treated in [12,14], and the joint multi-hopscheduling and routing was treated by Chafekar et al. [4].
In the distributed setting, the capacity problem was treatedwith no-regret learning, first by Dinitz [5], and later result-ing in a O(1)-approximation algorithm for uniform powerof Ásgeirsson and Mitra [2]. However, these game-theoreticalgorithms take time polynomial in n to converge, and thuscan be viewed more appropriately as determining capacityinstead of realizing it in “real time”.
For distributed scheduling, the only work that we areaware of is the groundbreaking paper of Kesselheim andVöcking [20], who give a distributed O(log2 n)-approximat-ion algorithm for the scheduling problem with fixed length-monotone and sublinear power assignment. Our resultsconstitute a Ω(log n)-factor improvement. Kesselheim andVöcking also extend their results to multi-hop scheduling,with the same approximation factor, for which our improve-ments do not apply, and to routing, with an extra logarithmicfactor.
A versatile measure introduced in [20] is the maximumaverage affectance A of a link set L , defined as
A(L) := maxR⊆L
avgl∈RaR(l) = maxR⊆L
aR(R)
|R| .
They then show two results that combined yield theO(log2 n)-approximation factor. On the one hand, they showthat A(L) = O(χ(L) log n). On the other hand, they presenta natural algorithm (which we also use in this work) thatschedules links in O(A(L) log n) slots. We show that bothof these bounds are tight. Thus, it is not possible to obtainimproved approximation using the measure A.
Following the original publication of this work, the resultshave been applied to distributed connectivity and aggrega-tion [3,15]. A different approach for distributed capacity wasproposed by Pei and Kumar [24], with complexity that isa function of the link lengths. In a recent follow-up work,Halldórsson et al. [13] have shown that A(L) = O(χ) forall sublinear, length-monotone power assignments other thanuniform power.
2.2 Our contributions
We achieve the following results:
Theorem 1 There is a randomized distributed O(log n)-approximation algorithm for the scheduling problem, in arbi-trary metric space and for all length-monotone sublinearpower assignments.
Theorem 2 For every n, there is an instance Ln of linkson the real line that can be scheduled in two slots butfor which every distributed algorithm uses Ω(log n)-slots(w.h.p). Thus, Θ(log n) is the best absolute approximationfactor for a distributed scheduling algorithm.
As in [20], our upper bound results hold in arbitrary dis-tance metrics (and do not require the common assumptionthat α > 2). We also show that the results hold independent ofthe ambient noise term N , extending [20]. The lower boundresult necessarily holds independent of power assignmentstrategy and for all positive values of the technical constantsα, β and N .
One of our main technical insights is to devise a differentmeasure that involves median rather than average affectance.The measure Λ = Λ(L) is given by
Λ(L) := maxR⊆L
median(A(R)),
where A(R) = {aR(l) : l ∈ R} is the multi-set of in-affectance values of links in the subset R, and median(X)
denotes the median of a multi-set X . Since we only insist thathalf of the given subset R of links have affectance boundedby Λ, the value of Λ may be much smaller than A. Indeed,we show that Λ = O(χ) and that the algorithm schedulesall links in time O(Λ log n), achieving the claimed approxi-mation factor.
The other main technical contribution of the paper is theintroduction of the concept of anti-feasibility. A set S of linksis anti-feasible 1 if av(S) ≤ 2, for every lv in S; i.e., if the out-going affectance from each link is small. A set is bi-feasibleif it is both feasible and anti-feasible. We observe in thispaper that every feasible set contains a large bi-feasible setand that certain analyses are easier on bi-feasible sets. Thishas proved useful in later works, e.g., in giving simplifiedanalysis of capacity approximation algorithms [16,19].
In the next section, we give the improved analysis of aO(log n)-factor for distributed scheduling, via the measureΛ; the treatment of acknowledgments is given in Subsect.3.2. We show in Sect. 4 that this logarithmic factor is bestpossible, and give a construction in Sect. 5 that shows thatthis result cannot be obtained in terms of the measure A.
1 For a technical reason we use a different constant here than for fea-sibility; the signal-strengthening result of [17] implies that this onlyaffects constants in the approximation factors.
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3 O(log n)-Approximate distributed schedulingalgorithm
The algorithm from [20], listed below as Distributed, is anatural backoff scheme, in the tradition of ALOHA [25].It is run synchronously, but independently, on each senderof a link. The algorithm, and all the results in this section,work for an arbitrary fixed sublinear length-monotone powerassignment.
Algorithm 1 Distributed1: k ← 02: loop3: q = 1
4·2k
4: for 4q c1 ln n slots do
5: transmit with i.i.d. probability q6: if successful (and acknowledged) then7: halt8: end if9: end for10: k ← k + 111: end loop
The algorithm is mostly self-descriptive. The constant c1
is to be chosen to satisfy the high probability bound desired.One point to note is that Line 6 necessitates some sort ofacknowledgment mechanism for the distributed algorithmto stop. For simplicity, we will defer the issue of acknowl-edgments to Sect. 3.2 and simply assume their existence fornow. Theorem 3 below implies our main positive result. LetΛ = Λ(L).
Theorem 3 If all links of a set L of n links run Distrib-uted, then L is fully scheduled in O(Λ log n) slots, with highprobability.
Intuitively, when the probability q is set right, a large frac-tion of the links transmit successfully, in expectation. This isargued in Lemma 1. What remains are then more pedestriantasks of showing that the large expectation results in goodconcentration and that the search for the right value of q isnot too expensive.
Lemma 1 Consider a subset R ⊆ L of links and a par-ticular time slot t in which each sender of R transmits withprobability q ≤ 1
2Λ. Then, the expected number of successful
transmissions is at least q·|R|4 .
Proof Define M = MΛ(R) = {lu ∈ R : aR(u) ≤ Λ}. Bythe definition of Λ,
|M | ≥ |R|/2 . (4)
Thus, it suffices then to show that at least q|M |/2 transmis-sions in slot t are successful in expectation.
Intuitively, the success probability of a link is proportionalto its in-affectance. The links in M are the ones with low in-affectance, so as long as the transmission probability q isless than 1/(2Λ), they will succeed with probability 1/2 iftransmitting.
For lu ∈ R, let Tu = Tu(t) be the indicator random vari-able that link lu transmits, and let Su = Su(t) be the indica-tor random variable that lu succeeds. We shall make use ofa few elementary facts about probabilities. For a (Bernoulli)indicator random variable X , E(X) = Pr(X). For randomvariables X1, X2, . . ., it holds by the linearity of expectationthat
∑i E(Xi ) = E
(∑i Xi
). And, for a random variable X
that assumes non-negative values, P(X ≥ 1) ≤ E(X).Armed with these facts, we can now bound the probability
that a transmitting link lu ∈ M is unsuccessful:
P(Su = 0|Tu = 1) = P
⎛
⎝∑
lv∈R
av(u)Tv > 1
⎞
⎠
≤ E
⎛
⎝∑
lv∈R
av(u)Tv
⎞
⎠
=∑
lv∈R
av(u)E(Tv)
= q · aR(u)
≤ q ·Λ ,
where the first equality uses (3), the second one uses thelinearity of expectation, and the last inequality uses the def-inition of M and that lu ∈ M .
Thus, when q ≤ 12Λ
,
P(Su = 0|Tu = 1) ≤ 1/2 ,
which allows us to bound the probability of link lu transmit-ting in the time slot by
E(Su) = P(Su = 1)
= P(Tu = 1)P(Su = 1|Tu = 1) (5)
= q(1− P(Su = 0|Tu = 1))
≥ q/2 .
The expected number of successful links in the time slotis then at least
E
⎛
⎝∑
lu∈R
Su
⎞
⎠ =∑
lu∈R
E(Su)
≥∑
lu∈M
E(Su) ≥ |M | · q/2 ≥ |R| · q/4,
using (4) and (5). This implies the lemma. ��Proof (of Theorem 3) Given Lemma 1, the theorem followsessentially from the arguments in Theorems 2 and 3 of [20].
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SINR model
Let q = 2−(1+lg Λ), i.e., the unique power of two satisfying1
4Λ< q ≤ 1
2Λ.
We first bound the probability that not all links are sched-uled during the iteration of the outer loop when q in Line 1equals q .
Let t be the first time slot where q ≤ q . Let nt be therandom variable indicating the number of links that did notsuccessfully transmit in the first t time slots.
Lemma 1 implies that for any given value s and time slott ≥ t ,
E(nt |nt−1 = s) ≤ s − q
4s,
and thus
E(nt ) ≤∞∑
s=0
P(nt−1 = s) ·(
1− q
4
)
s=(
1− q
4
)
E(nt−1).
Noting that n0 = n, this yields that
E(nt ) ≤(
1− q
4
)t
n.
Now, after t ′ := t + 4c1 ln nq time slots, the expected number
of requests remaining is
E(nt ′) = E(nt+4c1 ln n/q)
≤(
1− q
4
)4c1 ln n/q
· E(nt )
≤(
1
e
)c1 ln n
n
= n1−c1 .
By Markov’s inequality,
P(nt ′ = 0) = P(nt ′ ≥ 1) ≤ E(nt ′) ≤ n1−c1 .
Thus, with high probability all the links are scheduled whileq ≥ q .
Finally, to bound the total running time of the algorithm,we sum up the spent for values of q smaller than q , boundingt0. This is a geometric series given by
t0 =lg(1/q)∑
i=2
8c1 ln n
2−i
= 8c1 ln nlg(1/q)∑
i=2
2i
≤ 8c1 ln n · 2lg(1/q)+1
= 8c1 ln n · 2
q≤ 64c1Λ ln n,
establishing the time complexity.
3.1 Bounding the measure
We need the following lemma to get a handle on affectances.Recall that we assumed that the implicit power assignmentis length-monotone and sublinear.
Lemma 2 (Lemma 7, [20]) Let L be a feasible set and lu ∈ Lbe link with �u ≤ �v for all lv ∈ L. Then, aL(u) = O(1).
We now prove the following complementary result. It canbe contrasted with Lemma 9 of [20], which without the anti-feasibility condition can only give av(L) = O(log n). Thesecond part of the lemma essentially follows Lemma 11 of [2](which had the unnecessary assumption that L is feasible).
We first need the following result.
Lemma 3 ([12]) Let lu, lv be links with min(au(v), av(u)) ≤1/q. Then, duv · dvu ≥ q2/α · �u�v .
Lemma 4 Let L be an anti-feasible set with length-monotoneand sublinear power and let lv ∈ L be a link with �v ≤ �u,for every lu ∈ L. Then, av(L) = O(1).
Proof We first use a variation of the signal strengtheningtechnique of [17], given as Theorem 7 in the Appendix. Thisallows us to decompose the set L into 4 · 3α2 sets, whereeach set S satisfies aw(S) ≤ 1
3α , for all lw ∈ S. We shall provethe claim for S; the claim will then hold for L by summingover the 4 · 3α2 sets.
Let lu = (su, ru) (lw = (sw, rw)) be a link in Swhose sender (receiver) is closest to sv , i.e., d(sv, su) =minlx∈S d(sv, sx ) (d(sv, rw) = minlx∈S d(sv, rx )), respec-tively. Let h := d(sv, su).
First, suppose for the sake of contradiction that there is alink lx ∈ S, lx = lw, such that d(sv, rx ) < h/2. Then, by thedefinition of lw,
d(sv, rw) ≤ d(sv, rx ) <h
2. (6)
Also, by the definition of lu ,
d(sv, sx ) ≥ d(sv, su) = h (7)
and
d(sv, sw) ≥ d(sv, su) = h . (8)
Thus, by the triangular inequality and bounds (8) and (6),
�w ≥ d(sv, sw)− d(sv, rw) >h
2, (9)
and similarly by bound (7) and the supposition,
�x ≥ d(sv, sx )− d(sv, rx ) >h
2. (10)
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M. M. Halldórsson, P. Mitra
On the other hand, by the triangular inequality, bound (6) andthe supposition,
d(rw, rx ) ≤ d(rw, sv)+ d(sv, rx ) <h
2+ h
2= h. (11)
Applying the triangular inequality and bounds (11), (9)and (10), we get that
dwx · dxw ≤ (�w + d(rw, rx ))(�x + d(rw, rx ))
< (�w + h)(�x + h)
< 9�w�x ,
which contradicts Lemma 3 (given the strong feasibilityproperty of S). Thus, our supposition must be false; namely,for all links lx in S, lx = lw, it holds that
d(sv, rx ) ≥ h
2. (12)
Let lx be a link in S, lx = lw. We have that
dux = d(su, rx )
≤ d(su, sv)+ d(sv, rx ) (Triangular ineq.)= h + d(sv, rx ) (Def. of h)
≤ 3d(sv, rx ) (Inequality(12))
= 3dvx
. (13)
We observe that Pv ≤ Pu holds by length-monotonicity.Also, note that since the maximum affectance between
links in S is 13α , the thresholding in the affectance definition
does not take effect, implying that au(x) = cxPudα
ux
�αx
Px. Thus,
using (13) and that Pv ≤ Pu ,
av(x) ≤ cxPv
dαvx
�αx
Px≤ cx
3α Pu
dαux
�αx
Px= 3αau(x) . (14)
Finally, summing over all links in S,
av(L) = av(w)+∑
lx∈S\{lw}av(x)
≤ 1+∑
lx∈S\{lw}av(x)
≤ 1+ 3α∑
lx∈S\{lw}au(x)
≤ 1+ 3α · aS(x)
≤ 1+ 3α · 2 ,
using the definition of affectance (2), (14), and anti-feasibility,respectively. The lemma follows. ��
We can now derive the needed bound on the measure.
Theorem 4 Let L be a set of links. Then, Λ(L) = O(χ(L)).
Proof Let χ = χ(L) and let R be an arbitrary subset R ⊆ L .To prove the theorem, it suffices to show that at least half ofthe links in R have in-affectance at most O(χ(L)). Con-sider a partition of R into χ feasible subsets S1, S2, . . . , Sχ ,and define S′i = {lv ∈ Si : av(Si ) ≤ 3}. Let R′ = ∪i S′i .We observe that R′ contains at least two thirds of the linksin R. ��
Claim
|R′| ≥ 2|R|3
. (15)
Proof Let i ∈ {1, 2, . . . , χ}. Since Si is feasible, it followsfrom (3) that aSi (v) ≤ 1, for every link lv ∈ Si . Let Si =Si\S′i . Now,
aSi(Si ) ≤ aSi (Si ) =
∑
lv∈Si
aSi (v) ≤∑
lv∈Si
1 = |Si | .
But, aSi(Si ) =∑
lv∈Siav(Si ) > 3 · |Si |, by the definition
of Si . Thus, |Si | ≤ aSi(Si )/3 < |Si |/3, and |S′i | = |Si | −
|Si | > 2|Si |/3. ��
We next bound affectances from R to R′. Let c2 (c3) be theconstant implicit in the big-oh notation in Lemma 2 (Lemma4), respectively.
Claim aR(R′) ≤ (c2 + c3)|R| · χ.
Proof We first observe that for every i, j ,
aS j (S′i ) =∑
lu∈S′i
∑
lv∈S j�v≥�u
av(u)+∑
lu∈S′i
∑
lv∈S j�v<�u
av(u)
≤∑
lu∈S′i
c2 +∑
lv∈S j
∑
lu∈S′i�v<�u
av(u)
≤ c2|S′i | +∑
lv∈S j
c3
≤ c2|Si | + c3|S j |, (16)
using Lemma 2 and rearrangement in the first inequality, andLemma 4 in the second. Summing up over the feasible setsin R and R′, we have that
aR(R′) =∑χi, j=1 aS j (S′i )
≤∑χi, j=1
(c2|Si | + c3|S j |
)(By(16))
=∑χi, j=1(c2 + c3)|Si | (By symmetry)
= (c2 + c3)χ∑χ
i=1 |Si |= (c2 + c3)χ |R| (Defn. of Si ),
establishing the claim. ��
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SINR model
We continue with the proof of the theorem. From the claim,it follows that the average in-affectance aR(l ′) over the linksl ′ ∈ R′ is at most
aR(R′)|R′| ≤
(c3 + c4)|R| · χ|R′| ≤ μ := 3(c2 + c3)
2χ,
applying (15). By Markov’s inequality, at least three fourthsof the links in R′ have in-affectance from R at most fourtimes the average, μ.
It follows that
|M4μ(R)| = |{l ∈ R : aR(l) ≤ 4μ}| ≥ 3|R′|4≥ |R|
2,
applying (15) in the second inequality. That is, at least halfof the links in R have in-affectance at most 4μ. Hence, themedian in-affectance of links in R is bounded above by 4μ:
Median(A(R)) ≤ 4μ.
Since this holds for every given R, the theorem follows. ��
3.2 Acknowledgments
In the preceding exposition, we ignored the issue of sendingacknowledgments from receivers to senders. We can treatacknowledgments in a fashion similar to Kesselheim andVöcking [20]. We outline their approach briefly, but directthe reader to their paper for the details.
A special slot for acknowledgments is inserted betweenthe time slots used by Algorithm 1. A node that success-fully received a packet will transmit an acknowledgment withprobability p = 1/8. The power P∗v used for the acknowledg-ment on link lv is chosen to be proportional to P∗v ∼ �α/Pv
(using the right scaling factor).Kesselheim and Vöcking showthat at least half of these acknowledgments are successfulin expectation. That implies that we can modify Lemma 1to claim that the expected number of successfully acknowl-edged transmissions is at least p · q|R|/4 = q|R|/32, losingonly a constant factor. The rest of the arguments are thenidentical.
The only catch is the assumption in [20] that there areno weak links in the instance; a link lv is said to be weakiff cv > Cβ, for an appropriately chosen constant C (whosevalue affects the choice of p). We show here how to extend theapproach to deal with weak links. For simplicity of exposi-tion, we illustrate it for the case of uniform power and assumethat weak links satisfy cv > 3 max(β, 1). For non-uniformpower assignment, we can obtain the same bounds up to con-stant factors by showing that the power used by any two weaklinks differ by at most a constant factor since their lengthsdiffer by at most a constant factor.
The original transmissions, using Algorithm 1, areunchanged, but we allocate a separate time slot for theacknowledgments of weak links. Each receiver of a success-
fully transmitting weak link sends an acknowledgment in thattime slot with probability p′ (to be chosen).
The key observation in the following lemma is that weaklinks must be spatially well-separated. This implies that dif-ferences between the positions of the sender and receiver ofa link are minor, allowing us to relate the success probabilityfor an acknowledgment in terms of the observed success ofthe original transmission.
Lemma 5 Assume the use of uniform power. Suppose recep-tions are acknowledged with probability p′ ≤ 1
2(1+2 ln 3·α)α.
Let lv be a weak link that transmits successfully in a giventime slot t of Algorithm 1. Then, a given transmitting linkreceives acknowledgment with probability at least p′/2.
Proof Let lu be any other weak link that successfully trans-mitted at time t . Since both were successful,
au(v) = cv
Pu
dαuv
�αv
Pu= cv
�αv
dαuv
≤ 1. (17)
Thus, from (17) and the assumption that cv ≥ 3, we get that
dαuv ≥ cv�
αv ≥ 3�α
v , and dαvu ≥ 3�α
u . (18)
By the triangular inequality, duv ≤ dvu + �u + �v , which by(18) implies that
duv
(
1− 1
31/α
)
≤ duv − �v ≤ dvu + �u ≤(
1+ 1
31/α
)
dvu .
Some algebra and the inequality ex − 1 ≥ x (for any valuex) gives that(
1+ 131/α
)
(1− 1
31/α
) ≤ 1+ 2
e(ln 3)/α − 1≤ 1+ 2α
ln 3.
Combining the two displayed formulas above gives that
dαuv ≤
(
1+ 2α
ln 3
)α
dαvu . (19)
Now, let l∗v = (s∗v , r∗v ) = (rv, sv) be the dual link of lv ,with the roles of sender and receiver reversed. A transmissionon lv is acknowledged on l∗v . Let l∗u be another dual linkattempting to transmit an acknowledgment. The affectanceof l∗u on l∗v (under uniform power) is denoted and given by
au∗(v∗) = min
(
1, cv
(�v
dvu
)α)
.
Since the receptions on the original links lu and lv were simul-
taneously satisfied, we know that au(v) = cv
(�v
duv
)α
, i.e.,
thresholding did not occur. We use that along with (19) tobound au∗(v∗) by
123
M. M. Halldórsson, P. Mitra
au∗(v∗) ≤ cv
(�v
dvu
)α
≤(
1+ 2α
ln 3
)α
cv
(�v
duv
)α
=(
1+ 2α
ln 3
)α
au(v). (20)
Let S be the set of weak links that successfully transmittedin slot t and let S∗ be the set of the corresponding dual links.
Let T ∗u be the indicator random variable that the dual linkl∗v transmits an acknowledgment, and recall that P(T ∗u ) = p′.The in-affectance of a dual link l∗v due to other acknowledg-ment transmissions is a random variable Xv given by
Xv =∑
l∗u∈S∗T ∗u · au∗(v
∗) .
A transmitting dual link l∗v is successful iff Xv ≤ 1. Theexpected in-affectance of a link l∗v that transmits an acknowl-edgment is bounded by
E(Xv) = E
⎛
⎝∑
l∗u∈S∗T ∗v · au∗(v
∗)
⎞
⎠
=∑
l∗u∈S∗E(T ∗v ) · au∗(v
∗)
= p′∑
l∗u∈S∗au∗(v
∗)
≤ p′(
1+ 2α
ln 3
)α ∑
l∗u∈S∗au(v)
= p′(
1+ 2α
ln 3
)α
aS(v), (21)
by the linearity of expectation, the expectation of Bernoullivariables, (20), and the one-to-one correspondence betweenlinks in S and S∗. Applying the assumed upper bound on p′to (21), we find that
E(Xv) ≤ aS(v)
2≤ 1
2,
by the feasibility of S. By Markov’s inequality, the proba-bility that a link receives less than twice the expected in-affectance is at least 1/2, i.e., a dual link that does attemptto transmit an acknowledgment has at least 50% chance ofsuccess. Namely,
P(Xv ≤ 1) = 1− P(Xv > 1) ≥ 1− E(Xv) ≥ 1
2.
Acknowledgment is received by a sender if: a) the receiverattempts to send it (T ∗v = 1), and b) the affectance is lowenough for success (Xv < 1). The probability of the receiptof acknowledgment is then
P(T ∗v = 1 ∧ Xv ≤ 1) = P(T ∗v = 1) · P(Xv ≤ 1) ≥ p′
2,
using the independence of transmissions across links. Hence,the lemma. ��
4 Ω(log n)-Factor lower bound for distributedscheduling
In this section, we prove Theorem 2. Namely, we constructa set of 2n unit length links on the line that can be scheduledin two slots while no distributed algorithm can schedule theset in less than Ω(log n) slots.
We assume that all senders start at the same time in thesame state and use the same (randomized) algorithm. Notethat the algorithm in Sect. 3 operates under these assump-tions. We allow N , α, and β to be arbitrary positive values. LetPmax denote the maximum power, which we assume withoutloss of generality suffices comfortably to transmit a messagea unit distance, i.e., that
Pmax
N≥ 2β. (22)
We start with a gadget F with two identical links of length1, in a yin-yang position, i.e., with the sender of one linkin the same position as the receiver of the other. In casewe are uncomfortable having nodes share the exact sameposition, it suffices that the nodes be separated by at most(Pmax/(β Pmin))1/α , when there is a lower bound Pmin onthe power that can be used.
The gadget construction ensures that a transmission on alink is successful only if the other link in the gadget does nottransmit. This holds independent of the power used on theselinks. Also, the bound (22) ensures that for every link lv ,
cv ≤ 2β . (23)
The combined construction consists of n such gadgetsFi , i = 1, 2, . . . n, placed on the line as follows. Define
z := max
(2Pmax
βN, 4βn
)1/α
. (24)
The sender of one link and the receiver of the other link inFi are placed at point i(z+ 1) and the other two nodes of Fi
are placed at i(z + 1)+ 1. This completes the construction.
Claim When using uniform power (Pmax ), the affectancefrom links of other gadgets is negligible.
To see this, consider a link lu in some gadget Fi and letlv be a link in a different gadget. The distance between thelinks is at least dvu ≥ z. By (23) and (24),
av(u) ≤ cv
�αu
dαvu≤ 2β
1
zα≤ 1
2n.
There are 2n−2 links in other gadgets, i.e., in Fi := ∪ j =i Fj .Therefore,
aF (u) ≤ (2n − 2)av(u) < 1,
123
SINR model
as claimed.Thus, the behavior of links in other gadgets is immaterial
to the success of a link. This also implies that the schedulingnumber of this set of links is 2. Note that since the construc-tion uses equi-length links, the only possible oblivious powerassignment is the uniform one.
We also observe that nodes cannot glean any informationfrom other gadgets (even if it is hard to see how that infor-mation could be used).
Claim Links cannot receive transmissions from other gad-gets.
Proof Consider links lu and lv in different gadgets. Since thedistance between the links is at least z, and lu can transmitwith power at most Pmax , the signal-to-noise-ratio of thetransmission from lu at the receiver rv of lv (even in theabsence of interference) is at most
Pmax/zα
N≤ β
2< β .
Hence, by (24), rv cannot decode transmissions from lu . ��To prove the lower bound, we first argue that the two links
in a given gadget must transmit with the same probability ineach round (until one is successful). We say that gadget Fi
is active at time t if neither link of Fi has succeeded by timet − 1, and denote the event by Ai (t). Let Tu(t) denote theindicator random variable that link lu transmits at time t .
Lemma 6 Let Fi be a gadget and t ≥ 0 be a time. Thetransmission probabilities of the two links in Fi at time t areidentical and independent, conditioned on Fi being active attime t.
Proof Let lu and lv be the links in gadget Fi . Let Tu = Tu(t)and Tv = Tv(t), for short. By symmetry, the distributionsof Tu and Tv are identical, thus we need only to prove theirindependence.
The history of a link at time t is the binary vector of previ-ous transmission attempts. If Fi was active at time t , then byour previous observation the history vectors of the two linksin the gadgets are identical.
We can model the randomness used by the algorithmsas an i.i.d. random choice over a set F of functions. Eachf ∈ F is a function that takes a history of past transmis-sions and receptions over previous slots, and returns a binarytransmission decision. Note that if Ai (t) occurs then the his-tories of lu and lv over the previous t − 1 slots are identical.The different histories that can result in Ai (t) occurring aredisjoint; thus, it is enough to prove independence for a fixedhistory H . Let fu and fv denote the functions chosen by luand lv , and allow them also to represent the event that theyget chosen. Once again, by symmetry, there is some F ′ ⊆ Fsuch that H happens iff fu ∈ F ′ and fv ∈ F ′. We will use
the Iverson bracket [X ] to denote the value 1 if X is true and0 otherwise.
Then, for fixed Boolean outcomes a and b,
P(Tu = a, Tv = b | H)
=∑
fu∈F ′, fv∈F ′P( fu fv)[ fu(H) = a][ fv(H) = b]
=∑
fu∈F ′, fv∈F ′P( fu)P( fv)[ fu(H) = a][ fv(H) = b]
=∑
fu∈F ′P( fu)[ fu(H) = a] ·
∑
fv∈F ′P( fv)[ fv(H) = b]
= P(Tu = a|H)P(Tv = b|H) ,
thereby proving independence. We have used that P( fu fv) =P( fu)P( fv) in the second equality, which follows from thefact that fu and fv are chosen a priori and independently.
��We now show that it takes logarithmic rounds, in expecta-
tion, for all n gadgets to finish. Intuitively, the best option foreach link is to transmit with probability 1/2, and thus at mosthalf of the active gadgets become inactive in any given round,requiring lg n rounds for all gadgets to become inactive.
Theorem 5 Let Zn be a random variable whose value is thesmallest time t at which none of the gadgets are active. Then,E(Zn) = Ω(log n).
Proof Consider gadget Fi . By Lemma 6, both links in thegadgets use the same transmission probability while the gad-get is active. Let pt denote that transmission probability attime t . Then,
P(Ai (t + 1)|Ai (t)) = p2t + (1− pt )
2 ,
which is minimized for pt = 12 .
Thus,
P(Ai (t + 1)|Ai (t)) ≥ 1
2. (25)
Note that P(Ai (0)) = 1 and for every t > 0, Ai (1) ∩Ai (2) ∩ · · · ∩ Ai (t) = Ai (t). Then, for every t ′ ≥ 0,
P(Ai (t′)) = P(Ai (0))
t ′∏
t=2
P(Ai (t)| ∩ j<t Ai ( j))
= 1 ·t ′∏
t=2
P(Ai (t)|Ai (t − 1))
≥ 2−(t ′−1) ,
by (25). In particular, for t ′ ≤ t0 := lg n,
P(Ai (t′)) >
1
n. (26)
123
M. M. Halldórsson, P. Mitra
Let Qt ′ = ∩i Ai (t ′) be the event that none of the n gadgetsare active at time t ′. Since events of different gadgets areindependent, it holds for every t ′ ≤ t0, using (26), that
P(Qt ′) =n∏
i=1
(1− P(Ai (t′)) ≤
(
1− 1
n
)n
≤ e−1 .
Then, by definition of expectation,
E(Zn) =∞∑
t=1
Pr(Qt ) ≥ t0 · Pr(Qt0) ≥ (1− e−1) lg n .
��Note that bounding E(Zn) suffices to lower bound the
expected time before all links successfully transmit, since bydefinition a link cannot succeed as long as the correspondinggadget is active.
5 Tight bound on analysis via A
We achieved a O(log n)-approximation by avoiding the mea-sure A in our analysis. In contrast, the O(log2 n) bound in[20] is achieved by proving two separate bounds involvingA: first ALG = O(A log n), and second A = O(χ log n),where ALG is the expected time taken by the algorithm. Thetightness of the bound on ALG under any oblivious powerassignment follows from Sect. 4, as it is easy to verify thatA = Θ(1) in that construction. We give a construction belowfor which the second bound is tight. Thus, going through Ais not sufficient to obtain improved bounds, and differentanalysis is required.
Our construction uses uniform power. This is necessary,since for other oblivious power assignments A = O(χ), bythe recent results of [13]. We shall assume for the rest of thissection, for simplicity, that N = 0 and β = 1, but note thatthese assumption are not essential. We make no restrictionon α beyond it being positive.
Theorem 6 For every number n and every number t, thereis a set L of n links with χ(L) = Θ(t) and A(L) =Ω(χ(L) log(n/t)) under uniform power.
This lemma shows, perhaps surprisingly, that there canbe a huge difference between the in-affectance and the out-affectance of a link in a feasible set, thereby illustrating theneed for the bi-feasibility concept.
Lemma 7 For every n, there is a set L of n links on theline and a link l0 ∈ L, such that under uniform power, L isfeasible while a0(L) = Ω(log n).
Proof We form the set L = {l0, l1, . . . , ln−1} as follows. Thesender si of link li is positioned at coordinate d(s0, si ) =c · i1/α · 2i , where c > 1 is a constant to be determined. The
length of the link li is �i = 2i and the receiver ri is positionedat ri = si + �i = (c · i1/α + 1)2i .
Then,
a0(L) ≤n−1∑
i=1
(�i
d0i
)α
=n−1∑
i=1
(2i
(c · i1/α + 1)2i
)α
<1
(2c)α
n−1∑
i=1
1
i
= Ω(log n) .
To show feasibility, we first bound distances between linksby:
di−1,i = d0i − d(s0, si−1)
= (c · i1/α + 1)2i − c(i − 1)1/α2i−1
> c · i1/α2i−1 ,
and for m > 0,
di+m,i = d(s0, si+m)− d(s0, ri )
> c(i + m)1/α2i+m − (c · i1/α + 1)2i
> ci1/α2i+m − (c · i1/α + 1)2i+m−1
= (2c · i1/α − (c · i1/α + 1))2i+m−1
≥ (c − 1)2i+m−1 .
We then bound the in-affectance of each link by
aL(i) =∑
k,k<i
ak(i)+∑
k,k>i
ak(i)
≤ i · ai−1(i)+n−i−1∑
m=1
(�i
di+m,i
)α
≤ i ·(
�i
di−1,i
)α
+n−i∑
m=1
(2i
(c − 1) · 2i+m−1
)α
≤ i ·(
2i
c · i1/α · 2i−1
)α
+ 1
(c − 1)α
∑
m=0
(1
2α
)m
= 2
cα+ 1
(c − 1)α· 2α
2α − 1.
Thus, when c ≥ 1+(
3(
1+ 12α−1
))1/α
, it holds that aL(i) ≤1 for each link li , i.e., L is feasible. ��
We now turn to proving Theorem 6. We construct the setL that satisfies the claim of the theorem. Let L be the setfeasible under uniform power and the link l0 = (s0, r0) ∈ Lwith aL(l0) = Ω(log n), promised by Lemma 7.
Let S be an arbitrary set of n links located on the real linethat is feasible under uniform power. For instance, we can
123
SINR model
take S = L . Let U be the largest coordinate of a node inS and assume without loss of generality that all coordinatesare non-negative and U ≥ 1; thus, all nodes of links in Sare contained in [0, U ]. For instance, when S = L , thenU = (c · n1/α + 1)2n .
We now form the set L ′ obtained from L by scaling allnode positions (and therefore distances) by factor U . Then,link i in L ′ has length 2iU and has sender at position c ·i1/α · 2iU . As N = 0, scaling does not change affectances,since the scaling factor cancels out. Thus, in particular, theproperties of Lemma 7 apply also to L ′.
Let L1 denote the union of t isometric copies of S withlinks in the same position as S. Similarly, let L2 denote theunion of t isometric copies of L ′. Finally, we form the com-bined instance L = L1 ∪ L2 with a total of n = 2tn links.
Observe that all links in L ′ (and therefore also those inL2) are located between the sender and receiver of link 0 inL ′; therefore, they are between s0 and all the receivers in L ′(and those in L1). Thus, for every lv ∈ L ′ and li ∈ L1, itholds that dvi > d0i , which implies that the affectance of lvon L ′ is at least
av(L ′) > a0(L ′) = Ω(log n),
since we use uniform power and by Lemma 7. Thus, summingover all tn links in L2,
aL(L) ≥ aL2(L1) = t · aL2(L ′) = |L2|Ω(t log n),
implying that
A ≥ 1
|L|aL(L) = Ω
(
t logn
t
)
.
On the other hand, the set L clearly has a scheduling numberat most 2t , as it is formed by 2t feasible sets. Hence, thetheorem.
6 Conclusions
We have given a distributed scheduling algorithm that isO(log n)-approximate in the scheduling model, and shownthis factor cannot be improved in general. Our lower boundconstruction, however, applies only to instances with smallscheduling number.
A similar randomized scheduling algorithm was shownby Fanghänel et al. [7] to yield an asymptotic constant-factorapproximation for the case of linear power assignment. Onekey difference is that in the case of linear power, all links havelow affectance (O(χ)), while for general sublinear length-monotone power assignments this only holds on average.
It remains an important and intriguing open questionwhether a better asymptotic approximation ratio can beobtained.
7 Affectance reduction
The following is given (with minor modification) in[8, Theorem 4.1].
Theorem 7 Let S be an anti-feasible set and p < 2 be a
value. Then, S can be partitioned into t =(
4p
)2sets S1, S2,
. . . , St , each satisfying av(Si ) ≤ p, for every lv ∈ Si .
Proof We first partition S into a sequence T1, T2, . . . of setsas follows. Order the links in S in decreasing order. For eachlink lv , assign lv to the first set Tj for which av(Tj ) ≤ p/2,i.e. the accumulated affectance of lv on the previous, longerlinks in Tj is at most p/2. Since each link lv originally hadout-affectance at most 2, then by the additivity of affectance,the number of sets used is at most 2
p/2 = 4p .
We then repeat the same approach on each of the sets Ti ,processing the links this time in increasing order. The numberof sets is again 4
p for each Ti , or 4p 2 in total. In each final
slot (set), the affectance of a link on the shorter links in thesame slot is at most p/2. In total, then, the out-affectance ofeach link is at most 2 · p/2 = p. ��Acknowledgments We thank Marijke Bodlaender for helpful discus-sions leading to the derivation of Lemma 7.
References
1. Andrews, M., Dinitz, M.: Maximizing capacity in arbitrary wirelessnetworks in the SINR model: Complexity and game theory. In:INFOCOM, pp. 1332–1340, IEEE (2009)
2. Ásgeirsson, E.I., Mitra, P.: On a game theoretic approach to capac-ity maximization in wireless networks. In: INFOCOM (2011)
3. Bodlaender, M.H., Halldórsson, M.M., Mitra, P.: Connectivity andaggregation in multihop wireless networks. In: PODC (2013)
4. Chafekar, D., Kumar, V., Marathe, M., Parthasarathy, S., Srini-vasan, A.: Cross-layer latency minimization for wireless networksusing SINR constraints. In: Mobihoc (2007)
5. Dinitz, M.: Distributed algorithms for approximating wireless net-work capacity. In: INFOCOM (2010)
6. Fanghänel, A., Kesselheim, T., Räcke, H., Vöcking, B.: Obliviousinterference scheduling. In: PODC, pp. 220–229 (2009)
7. Fanghänel, A., Kesselheim, T., Vöcking, B.: Improved algorithmsfor latency minimization in wireless networks. In: ICALP, pp. 447–458 (2009)
8. Goussevskaia, O., Halldórsson, M.M., Wattenhofer, R.: Algorithmsfor wireless capacity. IEEE/ACM Transactions on Networking(2013). To appear
9. Goussevskaia, O., Halldórsson, M.M., Wattenhofer, R., Welzl, E.:Capacity of arbitrary wireless networks. In: INFOCOM, pp. 1872–1880 (2009)
10. Goussevskaia, O., Oswald, Y.A., Wattenhofer, R.: Complexity ingeometric SINR. In: Mobihoc, pp. 100–109 (2007)
11. Gupta, P., Kumar, P.R.: The capacity of wireless networks. IEEETrans. Inf. Theory 46(2), 388–404 (2000)
12. Halldórsson, M.M.: Wireless scheduling with power control. ACMTrans. Algorithms 9(1), 7 (2012)
13. Halldórsson, M.M., Holzer, S., Mitra, P., Wattenhofer, R.: Thepower of non-uniform wireless power. In: SODA (2013)
123
M. M. Halldórsson, P. Mitra
14. Halldórsson, M.M., Mitra, P.: Wireless capacity with obliviouspower in general metrics. In: SODA (2011)
15. Halldórsson, M.M., Mitra, P.: Distributed connectivity of wirelessnetworks. In: PODC (2012)
16. Halldórsson, M.M., Mitra, P.: Wireless capacity and admission con-trol in cognitive radio. In: INFOCOM, pp. 855–863 (2012)
17. Halldórsson, M.M., Wattenhofer, R.: Wireless communication isin APX. In: ICALP, pp. 525–536 (2009)
18. Kesselheim, T.: A Constant-factor approximation for wirelesscapacity maximization with power control in the SINR model. In:SODA (2011)
19. Kesselheim, T.: Approximation algorithms for spectrum alloca-tion and power control in wireless networks. Ph.D. thesis, RWTHAachen University, Aachen, Germany (2012)
20. Kesselheim, T., Vöcking, B.: Distributed contention resolution inwireless networks. In: DISC, pp. 163–178 (2010)
21. Maheshwari, R., Jain, S., Das, S.R.: A measurement study of inter-ference modeling and scheduling in low-power wireless networks.In: SenSys, pp. 141–154 (2008)
22. Moscibroda, T., Wattenhofer, R.: The complexity of connectivityin wireless networks. In: INFOCOM (2006)
23. Moscibroda, T., Wattenhofer, R., Weber, Y.: Protocol designbeyond graph-based models. In: Hotnets (2006)
24. Pei, G., Kumar, V.A.: Distributed link scheduling under the physicalinterference model. In: INFOCOM (2012)
25. Tanenbaum, A.: Computer networks, 4th edn. Prentice Hall Pro-fessional Technical Reference (2002)
26. Tonoyan, T.: On the problem of wireless scheduling with linearpower levels. CoRR abs/1107.4981 (2011)
123
