Throughput Optimizing Localized Link Scheduling for Multihop Wireless Networks under Physical Interference Model

Throughput Optimizing Localized LinkScheduling for Multihop Wireless Networks

under Physical Interference ModelYaqin Zhou, Xiang-Yang Li, Senior Member, IEEE, Min Liu, Member, IEEE, Xufei Mao, Member, IEEE,

Shaojie Tang, Member, IEEE, and Zhongcheng Li, Member, IEEE

Abstract—We study throughput-optimum localized link scheduling in wireless networks. The majority of results on link schedulingassume binary interference models that simplify interference constraints in actual wireless communication. While the physicalinterference model reflects the physical reality more precisely, the problem becomes notoriously harder under the physicalinterference model. There have been just a few existing results on link scheduling under the physical interference model, and evenfewer on more practical distributed or localized scheduling. In this paper, we tackle the challenges of localized link scheduling posedby the complex physical interference constraints. By integrating the partition and shifting strategies into the pick-and-comparescheme, we present a class of localized scheduling algorithms with provable throughput guarantee subject to physical interferenceconstraints. The algorithm in the oblivious power setting is the first localized algorithm that achieves at least a constant fractionof the optimal capacity region subject to physical interference constraints. The algorithm in the uniform power setting is the firstlocalized algorithm with a logarithmic approximation ratio to the optimal solution. Our extensive simulation results demonstrateperformance efficiency of our algorithms.

Index Terms—Localized link scheduling, physical interference model, maximum weighted independent set of links (MWISL), capacityregion

Ç

1 INTRODUCTION

AS a fundamental problem in wireless networks, linkscheduling is crucial to improve network perfor-

mances through maximizing throughput and fairness. Ithas recently regained much interest from networkingresearch community because of wide deployment ofmultihop wireless networks, e.g., wireless sensor networksfor monitoring physical or environment [2], [3] throughcollection of sensing data [4]. Generally, link schedulinginvolves determination of which links should transmit atwhat times, what modulation and coding schemes to use,and at what transmission power levels should communi-cation take place [5]. In addition to its great significance inwireless networks, developing an efficient schedulingalgorithm is extremely difficult due to the intrinsicallycomplex interference among simultaneously transmittinglinks in the network.

The link scheduling problem has been studied withdifferent optimization objectives, e.g., throughput-optimum scheduling, minimum length scheduling. Ourstudy mainly focuses on maximizing throughput in multi-hop wireless networks. Taking queue length of every linkas its weight, it is well known that a throughput-optimumscheduling policy that tries to find a maximum weightedindependent set of links is generally NP-hard in wirelessnetworks [5].

Despite of numerous results gained for the problem [6],[5], [7], [8], [9], [10], most assume simple binary interfer-ence models, e.g., hop-based, range-based, and protocolinterference models [11]. Under this category of interfer-ence model, a set of links are conflict-free if they arepairwise conflict-free. Conflict of transmissions on twodistinct links is predetermined independently of theconcurrent transmissions of other links. Thus, interferencerelationships based on these models can be represented byconflict graphs, and we can leverage classic graph-theoreticaltools for solutions. However, in actual wireless communi-cation, interference constraints among concurrent transmis-sions are not local and pairwise, but global and additive.Conflict of distinct transmissions is determined by thecumulative interference from all concurrent transmissions,which is often depicted by the physical interference model,e.g., the Signal-to-Interference-plus-Noise Ratio (SINR)interference model. The global and additive nature of thephysical interference model drives previous traditionaltechniques based on conflict graphs inapplicable or trivial.Consequently, designing and analyzing scheduling algo-rithms under the physical interference model becomesespecially challenging.

. Y. Zhou, M. Liu, and Z. Li are with the Research Center of NetworkTechnology, Institute of Computing Technology, Chinese Academy ofSciences, Beijing, China.

. X.-Y. Li is with TNLIST, School of Software, Tsinghua University,Beijing, China, and also with the Department of Computer Science,Illinois Institute of Technology, Chicago, IL USA.

. X. Mao is with TNLIST, School of Software, Tsinghua University,Beijing, China.

. S. Tang is with the Department of Computer and Information Science(Research) at Temple University, Philadelphia, PA USA.

Manuscript received 21 May 2013; revised 9 Aug. 2013; accepted 9 Aug.2013. Date of publication 20 Aug. 2013; date of current version 17 Sept. 2014.Recommended for acceptance by H. Frey.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference the Digital Object Identifier below.Digital Object Identifier no. 10.1109/TPDS.2013.210

1045-9219 � 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 25, NO. 10, OCTOBER 20142708

Some recent research results [12], [11], [13], [14], [15],[16], [17], [18], [19] have addressed some related chal-lenges. To the best of our knowledge, however, all of these,with throughput maximization or other optimizationobjectives such as a minimum length schedule [11], justfocus on centralized implementation. Distributed or evenlocalized scheduling under the physical interference modelis more demanding out of practical relevance.

Though [20], [21], [22] consider distributed implemen-tation of centralized algorithms, they require globalpropagation of messages and [20], [21] fail to provide aneffective localized scheduling algorithm with satisfactorytheoretical guarantee and. A scheduling algorithm withouttheoretical guarantee may cause arbitrarily bad throughputperformance, and global propagation of messages through-out network is inefficient in terms of time complexity [23],especially for large-scale networks. This motivates us todevelop localized link scheduling algorithms with prov-able throughput performance. Here by a localized sched-uling algorithm we mean that each node only needsinformation (i.e., queue length and link status) withinconstant distance to make scheduling decisions, while adistributed algorithm may inexplicitly need informationfar away. Since just local information is available for eachnode to collaborate on schedulings with globally coupledinterference constraint, it poses significant challenges todesigning efficient localized scheduling algorithms withtheoretical throughput guarantee.

In this paper we tackle these challenges of practicallocalized scheduling for throughput maximization underphysical interference with the commonly-used obliviousand uniform power assignment. We prove that ouralgorithms can respectively achieve a constant andOðlog jV jÞ fraction of capacity region for the oblivious anduniform power setting, where jV j is the number of nodes.Our extensive simulation shows that our proposed algo-rithms outperform simple heuristics.

The remainder of the paper is organized as follows. Wedefine the network model and problems in Section 2,propose our localized algorithms for the oblivious anduniform power assignment respectively in Sections 3 and 4,and evaluate their performance in Section 5. We reviewrelated works in Section 6, and conclude our work inSection 7.

2 MODELS AND ASSUMPTIONS

2.1 Network Communication ModelWe model a wireless network by a two-tuple ðV; EÞ, whereV denotes the set of nodes and E denotes the set of links.Each directed link l ¼ ðu; vÞ 2 E represents a communica-tion request from a sender u to a receiver v. Let klk or kuvkdenote the length of link l. We assume each node knows itsown location and the topology of the network.

2.2 Interference ModelUnder the physical interference model, a feasible schedule isdefined as an independent set of links (ISL), each satisfying

SINRuv ¼D Pu � � � kuvk��P

w2Tu Pw � � � kwvk�� þ � � �;

where � denotes the ambient noise, � denotes certainthreshold, and Tu denotes the set of simultaneous trans-mitters with u. It assumes path gain � � kuvk�� 1, wherethe constant � 9 2 is path-loss exponent, and � is thereference loss factor.

We consider the following two transmission powersettings.

1. Oblivious power setting: a sender u transmits to areceiver v always at the power Pu ¼ c � kuvk� where cand � are both constant satisfying c 9 0; 0 G � � �.

2. Uniform power setting: all links always transmit at thesame power Pu ¼ P . Under the uniform power case,we further assume that all links have a lengthadequately less than the maximum transmission

radiusffiffiffiffiffi�P��

�

qas links with length almost the same as the

maximum transmission radius are vulnerable to fail.

We use Pl and Pu alternatively to denote the transmittingpower of link l ¼ ðu; vÞ. The distance between u and vsatisfies r � kuvk � R, where r and R respectively denotesthe shortest link length and the longest link length. Wesuppose that r and R are known by each node.

2.3 Traffic Models and SchedulingThe maximum throughput scheduling is often studied inthe following models. It assumes time-slotted wirelesssystems, and single-hop flows with stationary stochasticpacket arrival process at an average arrival rate �l. Thevector AðtÞ ¼ fAlðtÞg denotes the number of packetsarriving at each link in time slot t. Every packet arrivalprocess AlðtÞ is assumed to be i.i.d over time. We alsoassume that all packet arrival processes AlðtÞ havebounded second moments and they are bound by Amax,i.e., AlðtÞ � Amax; 8l 2 E. Let a vector f0; 1gjEj denote aschedule SðtÞ at each time slot t, where SlðtÞ ¼ 1 if link l isactive in time slot t and SlðtÞ ¼ 0 otherwise. Packetsdeparture transmitters of activated links at the end oftime slots. Then, the queue length vector QðtÞ ¼ fQlðtÞgevolves as Qðtþ 1Þ ¼ maxf0; QðtÞ � SðtÞg þAðtþ 1Þ whereQlðtÞ is queue length (weight or backlog) of link l.

The throughput performance of link scheduling algo-rithms is measured by a set of supportable arrival ratevectors, named capacity region or throughput capacity. Thatis, a scheduling policy is stable, if for any arrival rate vectorin its capacity region [10], it satifies limt!1 E½QðtÞ� G1.

Though the policy of finding a MWISL to scheduleregarding to the underlying interference models isthroughput-optimal [24], finding a MWISL itself is NP-hardgenerally [25]. Thus we have to rely on approximationor heuristic methods to develop suboptimal schedulingalgorithms running in polynomial time. A suboptimalscheduling policy can achieve a fraction of the optimalcapacity region depicted by efficiency ratio � [5].

A suboptimal scheduling policy with efficiency ratio �must find a �-approximation scheduling at every time slot tto achieve � times of the optimal capacity region [26]. Itremains difficult to achieve in a decentralized manner. Thepick-and-compare approach proposed in [6] enables thatwe just need to find a �-approximation scheduling with a

ZHOU ET AL.: THROUGHPUT OPTIMIZING LOCALIZED LINK SCHEDULING FOR WIRELESS NETWORKS 2709

constant positive probability. The basic pick-and-compare[6] works as follows. At every time slot, it generates afeasible schedule that has a constant probability ofachieving the optimal capacity region. If the weight ofthis new solution is greater than the current solution, itreplaces the current one. Using this approach achieves theoptimal capacity region. The proposition below furtherextends this approach to suboptimal cases.

Proposition 1 ([8]). Given any � 2 ð0; 1�, suppose that an algo-rithm has a probability at least 9 0 of generating anindependent set XðtÞ of links with weight at least � times theweight of the optimal. Then, capacity � � L can be achieved byswitching links to the new independent set when its weight islarger than the previous one(otherwise, previous set of linkswill be kept for scheduling). The algorithm should generatethe new scheduling SðtÞ from the old scheduling Sðt� 1Þ andcurrent queue length QðtÞ.

3 THE ALGORITHM IN THE OBLIVIOUSPOWER SETTING

In this section, we focus on the design of localizedscheduling algorithm for the oblivious power setting.

3.1 Basic IdeaThe basis of our idea is to create a set of disjoint local linksets where scheduling can be done independently withoutviolating the global interference constraint. The decouplingof the global interference constraint is based on the fact thatdistance dominates the interference between two distinctlinks. That is, if a transmitting link is placed a certaindistance away from all the other transmitting links, thetotal interference it receives may get bounded. Based onthis, then we employ the partition strategies to divide thenetwork graph into disjoint local areas such that each localarea is separated away by a certain distance to enableindependent local computation of schedulings inside every

local area. Links lying outside local areas will keep silent toensure separation of local areas. As links lying outside localareas cannot remain unscheduled all the time or it willinduce network instability, we use the shifting strategy tochange partitions at every time slot to make sure that everybacklogged link will be scheduled. These locally computedscheduling link sets compose a new global schedule XðtÞ atevery time slot t.

As the pick-and-compare scheme, we choose a moreweighted schedule, denoted as SðtÞ, between a newly gen-erated schedule XðtÞ and the last-time schedule Sðt� 1Þusing QðtÞ. Meanwhile, by Proposition 1, if we guaranteeThat

P SðtÞ �QðtÞ � �S�ðtÞ �QðtÞð Þ � (1)

for some constant � 9 0; 9 0, the queue length vector QðtÞwill eventually converge to a stable state.

3.2 Detailed DescriptionWe first describe the partition and shifting strategies [9], asillustrated in Fig. 1. The plane is partitioned into cells withside length d ¼ R, by horizontal lines x ¼ i and verticallines y ¼ j for all integers i and j. A vertical strip with indexi is fðx; yÞji G x � iþ 1g. Similarly, we define the horizontalstrip j. cellði; jÞ is the intersection area of a vertical stripi and a horizontal strip j. A super-subSquareði; jÞ is theset of cells: fcellðx; yÞjx 2 ½i �K þ at; ðiþ 1Þ �K þ atÞ; y 2 ½j �K þ bt; ðjþ 1Þ �K þ btÞg; and a sub-squareði; jÞ inside it isthe set of cells:fcellðx; yÞjx 2 ½i �K þ at þM; ðiþ 1Þ �K þat �MÞ; y 2 ½j �K þ bt þM; ðjþ 1Þ �K þ bt �MÞg. The cor-responding link set Yij (or Lij) consists of links with bothends inside super-subSquareði; jÞ (or sub-squareði; jÞ). Letconstant K ¼ 2M þ J , where M is a constant that wouldbe defined in Lemma 2. Let integers at; bt; 0 � at; bt GK be thehorizontal and vertical shifting respectively, then we call theresulting plane PartitionðK;at; btÞ. By separately adjustingat; bt, we can get K2 different partitions for a plane totally.

Fig. 1. The partition and shifting process. (a) Partition ðK; at; btÞ. Here the gray area is the local area the link set of which participate in computing thenew schedule; the links in the white area keep silent. (b) Partition ðK; at; btÞ. Change the partition through shifting to the right by one cell, ensuring thatlinks in the white area in previous partitions have opportunity to be scheduled.


At each time slot each nodes runs Algorithm 1 tocollaboratively compute a globally feasible scheduling. Asevery node knows the locality from which it will collectinformation, it then participates the corresponding localcomputation, and at last it sends (if it is a coordinator) orreceives (if not) the results.

Algorithm 1 Distributed Scheduling by node v under theoblivious power setting

1: state ¼White; active ¼ No; Coordinator ¼ No;2: Calculate which cell node v resides in regarding to the

current partitionðK; at; btÞ;3: if v is the closest node to the center of super-subSquare

then4: Coordinator ¼ Yes;5: end if6: if Coordinator = Yes then7: Collect QðtÞ and Sijðt� 1Þ.8: Compute X ijðtÞ in sub-squareði; jÞ by enumeration;9: if Sijðt� 1Þ �QðtÞ 9X ijðtÞ �QðtÞ then

10: SijðtÞ ¼ Sijðt� 1Þ;11: else12: SijðtÞ ¼ X ijðtÞ;13: end if14: Broadcast RESULTðSijðtÞÞ in super-subSquareði; jÞ;15: end if16: if state ¼White then17: if receive message RESULTðSijðtÞÞ then18: if v 2 SijðtÞ then19: state ¼ Red; active ¼ Yes;20: else21: state ¼ Black; active ¼ No;22: end if23: end if24: end if

Time Slot t ¼ 0Every node first decides in which cell it resides whenða0; b0Þ ¼ ð0; 0Þ; then it participates in the process ofcomputing a local scheduling Sijð0Þ for the sub-squareði; jÞit belongs to. Let the solution Sð0Þ be the union of the localsolutions Sijð0Þ for all sub-squares.

Time Slot t � 1:Every node decides in which cell it resides by a partitionstarting from ðat; btÞ. The shifting strategy for at and bt worksas follows. We let at ¼ tmodK; and bt ¼ ðbt þ 1ÞmodK ifat ¼ 0, or it keeps unchanged. Each node then participates incomputing the new local scheduling, denoted asX ijðtÞ, for itssub-squareði; jÞusing the weightQðtÞ. LetSijðt� 1Þ be the set

of links from Sðt� 1Þ(the global solution at time slot t� 1)falling in the super-subSquareði; jÞ instead of sub-squareði; jÞ.If Sijðt� 1Þ �QðtÞ 9X ijðtÞ �QðtÞ, let SijðtÞ ¼ Sijðt� 1Þ, elseSijðtÞ ¼ X ijðtÞ, the global solution is the union of SijðtÞ fromall super-subSquares.

In our algorithm, at increases from 0 to K � 1 in Ksequent time slots when bt is fixed, and bt increases by 1every K time slots. The vertical and horizontal distancebetween any two sequent sub-squares is 2M. Initially, bothat and bt are 0. Thus in the worst case it takes 2M �K timeslots for a uncovered link to lie between two sequent sub-squares in horizontal line, i.e., increase bt by 2M; andanother 2M time slots to be finally covered by a sub-square,i.e., increase at by 2M. Thus we say it will take ðK þ 1Þ2Mtime slots to get every link covered by a sub-square. Thevertical and horizontal distance between any two sequentsuper-subSquares is 0. A link crosses two vertical andhorizontal cells at most. In worst case, it requires K timeslots to increase bt by 1, and another 1 time slot to increaseat by 1, where a uncovered link finally gets covered by asuper-subSquare.

3.3 Theoretical Analysis and ProofGiven a network ðV; EÞ, supposing [Zi is a set of disjointlocal link sets inside for scheduling, where Zi 2 E andZi \ Zj ¼ if i 6¼ j, for any link l 2 Zi, if l is activated, then

Il ¼ Ilin þ Iloutwhere Il denotes cumulative interference from all otheractivated links in the network, Ilin denotes the total interfer-ence from simultaneously transmitting links inside Zi andIlout denotes the total interference from transmissions outside.

Therefore, we can do independent scheduling inside Ziwithout consideration of Ilout from concurrent transmis-sions outside Zi, if Ilout gets bounded by a constant, i.e.,

Ilin � ð1� "Þ � Ilmax; Ilout � " � Imax; 0 G " G 1; l 2 Zi:

Let Imax denote the maximum interference that the longestlinks in E can tolerant during a successful transmission,and Ilmax represent the maximum interference that anactivated link l can tolerant during a successful transmis-sion. Then we have the two Lemmas below.

Lemma 1. In the oblivious power setting, the number of inde-pendent links for a local link set Zi inside a square with asize length JR is bounded by a constant. Let OPTi be a local

MWISL of Zi, jOPTij � ðffiffi2p

JRÞ�ð1�"Þ ½1��

��r��c� � þ 1.

Proof. This proof is available in Appendix A which is avai-lable in the Computer Society Digital Library at http://doi.ieeecomputersociety.org/10.1109/TPDS.2013.210.Ì

TABLE 1Summary of Notations


Lemma 2. Under the oblivious power setting, if the Euclideandistance between any two disjoint local link sets is at leastM �R, then activated links in each local link set suffer abounded cumulative interference from all other activated linksets, i.e., for each activated link l in local link set Zi,

Ilout � " � Imax; 0 G " G 1;

where M is a constant, satisfying M � ½2�c�R��jOPTijub

ð��2Þ"Imax�

1�.

Herein jOPTijub denotes an upper bound of the size of thelocal optimal MWISL for Zi.

Proof. This proof is available in Appendix B available online.Ì

To analyze the theoretical performance of our method,we first review the following definitions.

Definition 2 (Affectness [16]). The relative interference oflink l� on l is the increase caused by l� in the inverse of theSINR at l, namely rl� ðlÞ ¼ Ill�=ðPl�kuvk

��Þ. For convenience,define rlðlÞ ¼ 0. Let cl ¼ �

1��=ðPl�kuvk��Þ be a constant thatindicates the extent to which the ambient noise approaches the

required signal at receiver tv. The affectness of link l caused by aset S of links, is the sum of relative interference of the links in Son l, scaled by cl, or aSðlÞ ¼ cl �

Pl�2S

rl� ðlÞ.

Definition 3 (p-Signal Set [16]). We define a p-signal set tobe one where the affectness of any link is at most 1=p. Clearly,any ISL is a 1-signal set.

Lemma 3 ([16]). There is a polynomial-time protocol that takesa p-signal set and refines into a p0-signal set, for p0 9 p,increasing the number of slots by a factor of at most 4ðp0pÞ

2. Itindicates that a p-signal set can be refined into at most 4ðp0pÞ

2

p0-signal set through a polynomial-time algorithm, e.g., afirst-fit algorithm. Using the result, we have the Lemma below.

Lemma 4. The weight of X ijðtÞ has a constant approximationratio to the weight of the intersection set by the local link setLij and the global optimal MWISL S�ðtÞ.

Proof. Normally any ISL is a 1-signal set. That is, for theaffectness of a normal ISL it holds that

aSðlÞ ¼ cl �Xl�2S

rl� ðlÞ ��

1� ��=Pl� I

lmax

Pl� 1; (2)

whereas the affectness of a locally computed ISL forsub-square must satisfy that,

aSijðtÞ ðlÞ ¼ cl �X

l�2SijðtÞrl� ðlÞ �

� � ð1� "Þ1� ��=Pl

� Ilmax

Pl� 1� ": (3)

Therefore, by Lemma 3, a normal MWISL can be re<fined into 4

ð1�"Þ21

ð1�"Þ Signal link sets at most. Since the1

ð1�"Þ Signal link set returned by enumeration is most

weighted, so the locally computed link scheduling setsX ijðtÞ has a weight

W X ijðtÞ� �

� ð1� "Þ2

4W OPTijðtÞ� �

: (4)

Let S�ijðtÞ ¼ S�ðtÞ \ Lij denote the intersection by Lij andS�ðtÞ, where S�ðtÞ is the global optimal MWISL at timeslot t. It is obvious that

W S�ijðtÞ� �

�W OPTijðtÞ� �

� 4

ð1� "Þ2W X ijðtÞ� �

:

Ì

Theorem 1. SðtÞ ¼ [SijðtÞ computed by our algorithm is anindependent link set under the physical interference model inthe oblivious power setting. The weight of SðtÞ, i.e., W ðSðtÞÞ,is a constant approximation of the weight of the global optimalMWISL with probability of at least 1=K2.

Proof. The proof consists of two phases. We first provethat SðtÞ ¼ [SijðtÞ is an independent set. We next de-rive the approximation bound that SðtÞ achieves.

Phase IWe rely on induction to infer that at every time slot SðtÞis a union of disjoint activated local link sets that areseparated by at least M cells from each other. Then wehave that SðtÞ is an independent link set under thephysical interference model by Lemma 2. The followingare the details.

For any link l 2 SðtÞ, assuming l 2 SijðtÞ, the total inter<ference l suffers from all the other simultaneouslytransmitting links in SðtÞ is denoted by IlSðtÞ.At time slot 0, every local activated link set Sijð0Þ ¼X ijð0Þ is kept 2M cells away from each other, soSð0Þ is an independent set by Lemma 2.

At time slot 1, either Sijð0Þ or X ijð1Þ is chosen to be apart of Sð1Þ. For those super-subSquares whose Sijð1Þ ¼Si0j0 ð0Þ \ Yijð0Þ, their distance is kept at least 2M cellsaway. For those super-subSquares whose Sijð1Þ ¼X ijð1Þ, their distance is also kept at least 2M cellsaway. And the distance between the two kinds of link setis at least 2M � 1 cells away. So Sð1Þ is an independentset.

At some time slot t�, t� 9 1, for some super-subSquares,Sijðt�Þ consists of disjoint subsets from several differentSi0j0 ðt� � 1Þ which fall into Yijðt�Þ, i.e.,

Sijðt�Þ¼Sðt� � 1Þ\Yijðt�Þ¼[i0j0

Si0j0 ðt� � 1Þ\Yijðt�Þ� �

: (5)

For instance, as illustrated in Fig. 2, link sets fl1; l2gand fl3; l4g respectively get scheduled in two differentsuper-subSquares (i.e., super-subSquare(1, 2) andsuper-subSquareð2; 2Þ) at time slot t� � 1, the linksfl1; l2; l3; l4g are then get scheduled in the samesuper-subSquare(1, 2) at time slot t�. Let Fij

i0j0 ðt�Þ ¼Si0j0 ðt� � 1Þ \ Yijðt�Þ for brevity. Each Si0j0 ðt� � 1Þ iskept at least M cells away from each other, so iseach Fij

i0j0 .Clearly, Sðt�Þ can be divided into two separated subsets,

one formed by some subsets of Sðt� � 1Þ, the other formedby newly computed X ijðt�Þ, i.e.,

Sðt�Þ ¼[ij

Sijðt�Þ ¼[pq

[i0j0

Fpqi0j0 ðt�Þ

( )[ [mn;

mn6¼pq

Xmnðt�Þ

8<:

9=;:(6)


SinceSpq

Si0j0

Fpqi0j0 ðt�Þ is a subset of Sðt� � 1Þ, it is composed

by disjoint subsets with a mutual distance of M cells atleast. The distance between any distinct Xmnðt�Þ is noless than 2M cells. Then we consider the distancebetween a disjoint subset of

Spq

Si0j0

Fpqi0j0 ðt�Þ and a Xmnðt�Þ.

Since Xmnðt�Þ locates in sub-squareðm; nÞ, which is Mcells away from the border of super-subSquareðm; nÞ,the distance between a disjoint subset of

Spq

Si0j0

Fpqi0j0 ðt�Þ

and a Xmnðt�Þ is still no less than M cells. Comprehen-sively, Sðt�Þ consists of disjoint subsets which areseparated by at least M cells.

Note that a disjoint subset of Sðt�Þ does not equalizeto a Sijðt�Þ since a Si0j0 ðt� � 1Þ may be reservedcompletely in different super-subSquares at time slott�. Here we denote the disjoint subset by iðt�Þ, andSðt�Þ ¼

S iðt�Þ.

By Lemma 2, for each link l 2 iðt�Þ, where iðt�Þ comesfrom the former part of equation (6), we have IlSðt�Þ � Ilmax.Meanwhile, for each l 2 iðt�Þ, where l 2 iðt�Þ comesfrom the later part of (6), it holds that IlSðt�Þ � Ilmax. Then wehave

IlSðt�Þ � Ilmax; 8l 2 Sðt�Þ; (7)

indicating that Sðt�Þ is an independent set.Next we consider situations at time slot t� þ 1. Similarly,

Sðt� þ 1Þ composes of disjoint subsets separated by no lessthanM cells. Using the same technique as at time slot t�, wecan get that Sðt� þ 1Þ is still an independent set.

By induction we can conclude that SðtÞ is a union ofdisjoint activated subsets separated by M cells at least,thus it is an independent set. Herein we finish the firstphrase of the proof.

Phase IIWe derive the approximation ratio by the pigeonholeprinciple, Lemma 4, and Proposition 1.

Let DðtÞ denote the link set of the removed strips atPartitionðK; at; btÞ. D�ðtÞ represents a subset of S�ðtÞ,links of which fall inside DðtÞ, i.e., D�ðtÞ ¼ DðtÞ \ S�ðtÞ.Recalling that there are K2 different partitions for a planetotally. If we tried all these partitions in a single slot, eachcellði; jÞ would appear in the ‘‘removed’’ strips at most

2KM times. Then we let DiðtÞ denote the link set of theremoved strips when the ith one of the K2 partitionshappens, and let D�i ðtÞ ¼ DiðtÞ \ S�ðtÞ. Thus the weight ofall D�i ðtÞ in the K2 partitions should be 2KMW ðS�ðtÞÞ, i.e.,

XK2�1

i¼0

W D�i ðtÞ� �

� 2KMW S�ðtÞð Þ: (8)

Since actually we only experience one partition at timeslot t, by the pigeonhole principle the correspondingPartitionðK;at; btÞ has a probability at least 1=K2 to bean optimal partition, the weight of whose removed linksin D�ðtÞ is no greater than that of other partitions.Instantly we have the following with probability of 1=K2

at least,


� 1

K2�XK2�1

i¼0


� 2M

KW S�ðtÞð Þ; (9)

W [S�ijðtÞ� �

¼W S�ðtÞnD�ðtÞð Þ � 1� 2M

K

W S�ðtÞð Þ: (10)

Following Lemma 4, with a probability of 1=K2 it holds,

1� 2M

K

W S�ðtÞð Þ � 4

ð1� "Þ2W [X ijðtÞ� �

: (11)

So by Proposition 1 we get that the approximation ratiofor the optimal is 4

ð1�2�MKÞð1�"Þ2. Ì

3.4 Time and Communication ComplexityWe define the time complexity as the time units required by therunning of Algorithm 1 inside each local area in the worst case[23]. It includes time units for local information collection andlocal computation time at the center node. At every time slot,the coordinators shall collect queue information and lastscheduling status of all links inside the super-subSquares.After computation, the coordinators broadcast the schedulingresults throughout the super-subSquares.

It may require multihop propagation to collect andbroadcast the needed information inside super-subSquares.To avoid collision, the coordinator can first compute a treethat determines the sequence of transmissions at each nodein the super-subSquare based on topology informationalready known. In the worst case each node has to transmitone by one at different mini slot, then it causes timecomplexity Oðn2

ijÞ where nij is the number of nodes insidesuper-subSquareði; jÞ in the worst case. Herein we just givea basic scheme, the time complexity may be further reducedwith better broadcast scheduling.

After all required information gathered, the localcomputation complexity at the center node is Oð2jLijjÞ.Since the time units for computation process is muchsmaller than the time unit for message propagation, thelocal computation complexity can be ignored comparing tothe time for information collection. Thus the total timecomplexity is OðnijÞ. The communication complexity is thetotal number of basic messages transmitted during eachscheduling in the worst case [23]. The communication

Fig. 2. Partitions at time slot t� � 1 and time slot t�.


complexity is OðjV jÞ, and the number of messages trans-mitted in each local area is OðnijÞ.

4 THE ALGORITHM IN THE UNIFORMPOWER SETTING

We now extend the framework to the uniform power set-ting. Instead of enumeration, we compute a MWISL of thecandidate links for each sub-square by adopting the meth-od proposed in [19], as the cardinality of the optimalMWISL in each sub-square is no longer bounded by a con-stant in the uniform power setting. Except for the dif-ference, the general structure of the algorithm is the samewith that of the oblivious power setting.

We first describe the main idea of Algorithm 1 in [19]for computation of MWISL inside each sub-square. Givena set of links and weights associated with the links, thealgorithm works as follows:

Phase IRemove every link whose associated weight is at most wmax

n

where wmax denotes the maximum weight among all linksand n is the number of all given links. Let wmin denote theminimum weight from the remaining links.

Phase IIPartition the remaining links into log wmax

wmingroups such that

the links of the i-th group Gi have weights within½2iwmin; 2

iþ1wmin�. For each group Gi of links, it finds anindependent set of links among it by adopting the methodin [27]. Totally it will get log wmax

wminISLs, one for each group.

Then it chooses the one with the maximum weight amongthe log wmax

wminISLs as the final solution.

As the resulted link scheduling set is an ISL by themethod in [27], we can prove that its size has a constantupper bound through the following lemma.

Lemma 5 ([27]). Consider a link l ¼ ðu; vÞ and a set N of nodesother than u whose distance from u is at most �kuvk. If link lsucceeds in the presence of the interference from N , then

jNj � ð�þ1Þ�� ½1� ðkuvkR Þ

��.

The corollary below asserts an upper bound of thenumber of successfully transmitting links in a local link setwith size JR � JR when utilizing Algorithm 1 in [19].

Corollary 1. The cardinality of the resulted link scheduling setby Algorithm 1 in [19] is upper bounded, i.e., jX ijðtÞj �ðffiffi2p

JRrþ1Þ�� ½1� ð rRÞ

�� within an area of JR � JR.

Proof. In the method presented in [27], it adds firstly theshortest link among all the candidates to the schedul-ing set. And the distance between any pair of nodes willbe no greater than

ffiffiffi2p

JR. Therefore, let r be the shortestlink length and then � ¼

ffiffi2p

JRr , we derive the upper

bound. Ì

We let jX ijðtÞjub denote an upperbound of the size of thelocal computed ISL for Zi by Algorithm 1 in [19]. Then wepresent the lemma below.

Lemma 6. In the uniform power setting, if the Euclidean dis-tance between any two disjoint local link sets is at leastM �R, then activated links in each local set suffer negligiblecumulative interference from all other activated link sets, i.e.,for each activated link l in local link set Zi,

Ilout � " � Imax; 0 G " G 1;

where M is a constant, satisfying M � ½2��P �jX ijðtÞjubð��2Þ"ImaxR� �1�.

Proof. The proof is available in Appendix C availableonline. Ì

In light of Lemma 6, the partition strategy to enabledistributedimplementation will remain effective in theuniform power setting. Similarly, we have:

Lemma 7. The weight of the newly computed link scheduling setinside each sub-squareði; jÞ, i.e., W ðX ijðtÞÞ, has an approx-

imation ratio of 4

ð1�"Þ2 to the weight of the intersection set by

the corresponding local link set Lij and the global optimalMWISL S�ðtÞ.

Proof. The newly computed local scheduling set for eachsub-square should be a ð1� "Þ-signal set, so its weight is

at least ð1�"Þ2

4 times the corresponding 1-signal set. Since

the solution for MWISL with uniform power assignmentproposed in [19] finds an ISL with affectness no greaterthan 1, it is obvious that

Fig. 3. Total backlog vs. average arrival rate vs. different values of KM at time slot 1000. in the oblivious power setting. (a) " ¼ 0:2. (b) " ¼ 0:4. (c) " ¼ 0:8.


W S�ijðtÞ� �

�W OPTijðtÞ� �

� 4

ð1� "Þ2W X ijðtÞ� �

; (12)

where ¼ log jV j is the approximation bound of thealgorithm in [19]. Ì

Then, we will state an exact bound on the throughputperformance of our algorithm in Theorem 2.

Theorem 2. The union of the local computed scheduling linksets, i.e., SðtÞ ¼

SSijðtÞ, is feasible under the uniform power

setting. The weight of SðtÞ achieves a fraction of Oðlog jV jÞtimes the optimal solution.

Proof. We first show that the resulted scheduling link setby our algorithm is an independent link set with uni-form power at every time slot. Using the same tech-nique in the proof of Theorem 1, we can inductivelyconclude that each global scheduling set SðtÞ iscomposed by disjoint link sets which are kept at leastM cells away from each other. Therefore, by Lemma 6,we can get that for each link l 2 SðtÞ, the total inter-ference it receives satisfies that,

IlSðtÞ � ð1� "Þ � Ilmax þ " � Imax � Ilmax; (13)

indicating SðtÞ is an independent link set. Using Lemma6 and the same techniques in proof of Theorem 1, we get

P W SðtÞð Þ � ð1� "Þ2ðK � 2MÞ4K

S�ðtÞ !

� 1

K2; (14)

where ¼ log jV j. Ì

The time complexity and communication complexity isthe same as that of the oblivious power assignment.

5 PERFORMANCE EVALUATION

We do simulation experiments to evaluate the throughputperformance of our proposed algorithms. The generalsetting of our experiments is as follows. We consider anetwork with 500 nodes, half of which as senders randomlylocated on a plane with size 200 � 200 units, the other halfas receivers positioned uniformly at random inside disks ofradius R around each of the senders. Packets arrive at each

link independently in a Poisson process1 with the sameaverage arrival rate �. Initially, we assign each link kpackets where k is randomly chosen from [100, 300]. Thepath loss exponent is set to be 3.

We conduct two series of experiments to focus on theevaluation of average throughput performance in terms oftotal backlog (the total number of unscheduled packets).In the first series, we study how some related variablesaffect the performance of the algorithms. In the secondseries, we further study the performance efficiency of theproposed algorithms by comparisons with two distri-buted algorithms:

1. Distributed Greedy Maximal Schedule (DGMS) [21],it needs to pre-computation network wide todetermine a neighborhood of each link. If a linkhas maximum length in its neighborhood, it will getscheduled.

2. Distributed Randomized Algorithms (DRA), whereeach link determines to be active with a probability.To ensure that the global scheduling set is feasibleor almost feasible, the probability has to be set quitesmall.

5.1 Under Oblivious Power SettingWe set the first series of experiments to study theperformance of Algorithm 1 alone under different valuesof variables that may impact the average throughputperformance actually. The variables K

M and " dominate thetheoretical bound of Algorithm 1. When " is fixed, a largerKM implies a bigger fraction of the optimal capacity region,but with a smaller probability of 1

K2 to achieve it. " denotes aweighting factor between the interference a activated linksuffers inside and outside the sub-square. A bigger value of" indicates a smaller value of M theoretically. Therefore,though under the fixed value of K

M a smaller " leads to abigger fractional capacity region, the probability to achievethis region gets smaller because of the resulted bigger M.Since the running time of the simulation is much shorterthan the time for the algorithm to achieve the theoreticalvalue, the probability will impact the actual average

Fig. 4. Total backlog vs. average arrival rate vs. different values of " at time slot 1000 in the oblivious power setting. (a) KM ¼ 3. (b) K

M ¼ 4.

1. The results hold under any arrival process satisfying the stronglaw of large numbers, using the fluid model approach [28], [10].


throughput in our experiments. Typically, a small proba-bility of 1

K2 may cause poorer throughput performance. Anexperiment study of the two variables are illustrated inFigs. 3 and 4. In Fig. 3 we compare the total backlog byincreasing the feasible values of K

M where " serves as aconstant. Figs. 3a, 3b, 3c respectively shows the compar-isons of total backlog at time slot 1000 with increasingarrival rate when " ¼ 0:2, " ¼ 0:4, " ¼ 0:8. Note that thevalue of K

M must be greater than 2, or the size of the sub-squares will be 0. It shall also be noticed that the feasiblevalues of K

M are different when " varies, since " affects thevalue of M. It then explains why we set different values forKM for varying ". All the three figures show that thethroughput performance generally improves as K

M in-creases, which coincides with the theoretical results wederive. Though the relevant probability shall becomesmaller as K

M increases, there is no obvious sign shown inFigs. 3a and 3b. We can see an obvious impact in Fig. 3cwhere K

M can be set bigger values. It shows the averagethroughput becomes a little worse at K

M ¼ 7 than KM ¼ 6.

We briefly illustrate the impact of " at fixed KM in Fig. 4.

Though the theoretically achievable capacity region shallhave been greater with a smaller ", it seems to showcontradicted results in Fig. 4a. The apparent contradictionlies behind the probability of 1

K2 which becomes quite small

because of a much bigger M caused by a smaller ". Fig. 4bshows a similar consequence caused by the crucial impactof " both on the achievable capacity region and the relevantprobability.

We then focus on comparisons with DGMS and DRA inFigs. 5 and 6. We set " ¼ 0:8, KM ¼ 6 to conduct the followingsimulations. These simulation results indicate that ourdistributed scheduling algorithm (denoted by DS infigures) achieves much better performance than DGMSand DRA.

In Fig. 5 we compare the total backlog changes of thethree algorithms as average arrival rate increases. It showsthat our algorithm may approximately support a maximumaverage rate no greater than 0.2, and the other two supportno greater than 0.1. We zoom in a subgraph of the Fig. 5a inthe Fig. 5b, where the average arrival rate is in [0.16, 0.18]. Itshows that our algorithm has a total queue length muchsmaller than the two. Fig. 6 then illustrates detailedcomparisons of achievable capacity region for the threealgorithms. It shows that our algorithm can support alarger traffic arrival rate vector. From the three subfigureswe can see that our proposed algorithm can keep the totalbacklog stable at an arrival rate no greater than 0.18, whilethe counterpart of the distributed greedy algorithm andrandom algorithm is 0.07 and 0.05.

Fig. 5. Total backlog vs. average arrival rate vs. different algorithms at time slot 1000 in the oblivious power setting. (a) Total backlog vs. averagearrival rate. (b) Zoom in of (a).

Fig. 6. Achievable capacity region of different algorithms in the oblivious power setting. (a) DS. (b) DGMS. (c) DRA.


5.2 Under Uniform Power SettingIn the first series of experiment, we show the effect of K

M atdifferent values of " where " ¼ 0:2, 0.4, 0.8 respectively inFig. 7. Figs. 7a, 7b, and 7c plot the total backlog changes asincreasing average arrival rate under different values of KM.The trends are a little different from those in the obliviouspower setting. From the three figures we can see that theaverage throughput performance gets better as increasingKM under a fixed ". The decreasing probability of 1

K2 showsno obvious influence on the results. It may be partlycaused by the algorithm for computing new schedulingsinside sub-squares since it selects candidate links basedon their distance with previous selected links. Thus alarger area implies more links get scheduled. This im-provement remits the influence of the decreasing proba-bility of 1

K2.The similar trend occurs when " increases at different

values of KM. We give a brief illustration in Fig. 8. Both the

figures show that a larger " generates better performance atfixed K

M. Since the affectness of local ISLs computed by thealgorithm of [19] inside each sub-square may be muchsmaller than 1� ", it explains why " has little influence on thetheoretical bound actually. But " has much more influence onthe value of M and the corresponding probability. Therefore,in the uniform power setting, the algorithm achieves betterperformance with a larger ".

We conduct the second set of experiments to comparewith DGMS and DRA on average throughput where we set" ¼ 0:9, KM ¼ 9. The results in Figs. 9 and 10 convey the samekind of information as those in Figs. 5 and 6. These graphsjointly show that Algorithm 1 still outperforms the DGMSand DRA in terms of total backlog. For example, themaximum supportable average arrival rate of our algo-rithm is around 0.12, comparing with 0.07 by DGMS and0.04 by DRA.

6 RELATED WORK

Numerous literatures consider different optimizationmeasures and assume different interference models. Herewe focus on related works on physical interference model.A complete review is available in our technique report [29].

Goussevskaia et al. [12] firstly present the NP-completenessproofs for the scheduling problem under the physicalinterference model. It firstly proposes algorithms withlogarithmic approximation ratio OðgðjEjÞÞ under the uni-form power setting without noise, where OðgðjEjÞÞ presentsthe link diversity among all links. Goussevskaia et al. fur-ther [13] attempt to develop a constant approximation-ratio algorithm for the problem of maximum independentset of links (MISL), a special case of MWISL with uniformweight. However, it is valid only without noise as pointed

Fig. 7. Total backlog vs. average arrival rate vs. different values of KM at time slot 1000 in the uniform power setting. (a) " ¼ 0:2. (b) " ¼ 0:4. (c) " ¼ 0:8.

Fig. 8. Total backlog vs. average arrival rate vs. different values of " at time slot 1000 in the uniform power setting. (a) KM ¼ 3. (b) K

M ¼ 4.


in [30]. Wan et al. finally succeed in developing a constantapproximation-ratio algorithm for MISL with the existenceof noise in [27].

Under the oblivious power setting, by utilizing partitionand shifting strategies, Xu et al. get a constant approxima-tion algorithm for MWISL subject to physical interferences[15]. Very recently Xu et al. [19] propose another constantapproximation algorithm for the same problem based onthe solution for the maximum weighted independent setof disks problem in [31]. They also develop a logarithmicapproximation algorithm for the uniform power setting.Chafekar et al. [17] provide algorithms for maximizingthroughput with logarithmic approximation ratio in thetwo power settings as well. However, the attained bound isnot relative to the original optimal throughput capacity, butto the optimal value by using slightly smaller power levels.

Despite the main concern of this paper is on linkscheduling for throughput maximization, we also make areview on the closely related problem of minimum lengthscheduling, which seeks a link schedule of minimumlength that satisfies all link demands. A very recent work in[11] gives an overall analysis on link scheduling problemsunder the physical interference model from an algorithmview. It reveals that the algorithmic reduction from theminimum length scheduling to the throughput maximiza-tion scheduling is approximation-preserving.

The NP-completeness proofs of the minimum lengthscheduling problem, and a logarithmic approximationalgorithm without noise in the uniform power setting, isavailable in [12]. With noise taken into consideration,Moscibroda et al. present a scheduling algorithm for theproblem with power control, but without provable guar-antee in [32]. They subsequently get a linear approximationbound in [33]. An attempt on a constant approximationbound with uniform power setting fails in [16], pointed outby Wan. et al. [18]. They [18] then propose a logarithmicapproximation algorithm for the minimum length sched-uling problem with power control.

Blough et al. [20] claim that the so-called black links,with length exactly at the maximum transmission range ofthe sender, hinder a tighter approximation bounds for theminimum length scheduling problem. They try to get aconstant approximation ratio by limiting scheduling ofsuch kind of links. Thus they revise the algorithm GOWproposed in [12] by partitioning links according to theSNR diversity, instead of the length diversity. However,their revised algorithm GOW* can only guarantee aconstant approximation ratio when the number of blacklinks is bounded by a constant. They have also recognizedthe necessity of distributed implementation for the sched-uling algorithm. Some discussions on the suitability of thealgorithm for distributed execution are then presented.

Fig. 9. Total backlog vs. average arrival rate vs. different algorithms at time slot 1000 in the uniform power setting. (a) Total backlog vs. averagearrival rate. (b) Zoom in of (a).

Fig. 10. Achievable capacity region of different algorithms in the uniform power setting. (a) DS. (b) DGMS. (c) DRA.


7 CONCLUSION

We tackle the problem of throughput-optimum localizedlink scheduling subject to physical interference constraints.Our work provides theoretical guarantee for this practicallink scheduling problem for multihop wireless networks,keeping them away from arbitrarily bad throughput perfor-mance. We believe that our work can find applications in sometime-slotted wireless networks, e.g., time-slotted wirelesssensor networks or wireless mesh networks. Understandingthese factors that determine the throughput of a network alsohelps to better deploy a multihop wireless network andenhance the overall throughput performance.

ACKNOWLEDGMENTS

The research of authors is partially supported by NSFECCS-1247944, China 973 Program under Grant2011CB302702, NSF CNS-0832120, NSF CNS-1035894,National Natural Science Foundation of China under Grants61170216, 61228202, 61132001, 61120106008, 61070187,61272474, 61202410, 61272475, and 61003225, China 973Program under Grant 2011CB302705. A previous conferencevision, ‘‘Distributed Link Scheduling for Throughput Maxi-mization under Physical Interference Model’’ [1], has beenpublished in Infocom 2012. It mainly focuses on theoreticalanalysis in oblivious power setting.

REFERENCES

[1] Y.Q. Zhou, X.-Y. Li, M. Liu, Z.C. Li, S.J. Tang, X.F. Mao, andQ.Y. Huang, ‘‘Distributed Link Scheduling for ThroughputMaximization under Physical Interference Model,’’ in Proc. IEEEINFOCOM, 2012, pp. 2691-2695.

[2] M. Li and Y. Liu, ‘‘Underground Coal Mine Monitoring withWireless Sensor Networks,’’ ACM Trans. Sens. Netw., vol. 5, no. 2,p. 10, Mar. 2009.

[3] Y. Liu, K. Liu, and M. Li, ‘‘Passive Diagnosis for Wireless SensorNetworks,’’ IEEE/ACM Trans. Netw., vol. 18, no. 4, pp. 1132-1144,Aug. 2010.

[4] X.-H. Xu, X.-Y. Li, P.-J. Wan, and S.-J. Tang, ‘‘Efficient Schedulingfor Periodic Aggregation Queries in Multihop Sensor Net-works,’’ IEEE/ACM Trans. Netw., vol. 20, no. 3, pp. 690-698,Aug. 2011.

[5] C. Joo, X. Lin, and N.B. Shroff, ‘‘Understanding the CapacityRegion of the Greedy Maximal Scheduling Algorithm in Multi-Hop Wireless Networks,’’ in Proc. IEEE INFOCOM, 2008,pp. 1103-1111.

[6] L. Tassiulas and A. Ephremides, ‘‘Linear Complexity Algorithmsfor Maximum Throughput in Radio Networks and Input QueuedSwitches,’’ in Proc. IEEE INFOCOM, 1998, pp. 533-539.

[7] X. Lin and S.B. Rasool, ‘‘Constant-Time Distributed SchedulingPolicies for Ad Hoc Wireless Networks,’’ in Proc. IEEE CDC,2006, pp. 1258-1263.

[8] S. Sanghavi, L. Bui, and R. Srikant, ‘‘Distributed Link Schedulingwith Constant Overhead,’’ in Proc. ACM SIGMETRICS, 2007,pp. 313-324.

[9] S.-J. Tang, X.-Y. Li, X. Wu, Y. Wu, X. Mao, P. Xu, and G. Chen,‘‘Low Complexity Stable Link Scheduling for MaximizingThroughput in Wireless Networks,’’ in Proc. IEEE SECON,2009, pp. 1-9.

[10] E. Modiano, D. Shah, and G. Zussman, ‘‘Maximizing Through-put in Wireless Networks via Gossiping,’’ in Proc. ACMSIGMETRICS, 2006, pp. 27-38.

[11] P.-J. Wan, O. Frieder, X.-H. Jia, F. Yao, X.-H. Xu, and S.-J. Tang,‘‘Wireless Link Scheduling Under Physical Interference Model,’’in Proc. IEEE INFOCOM, 2011, pp. 838-845.

[12] O. Goussevskaia, Y. Oswald, and R. Wattenhofer, ‘‘Complexityin Geometric SINR,’’ in Proc. ACM MobiHoc, 2007, pp. 100-109.

[13] O. Goussevskaia, R. Wattenhofer, M.M. Halldorsson, and E. Welzl,‘‘Capacity of Arbitrary Wireless Networks,’’ in Proc. IEEE INFOCOM,2009, pp. 1872-1880.

[14] X.-Y. Li, ‘‘Multicast Capacity of Wireless Ad Hoc Networks,’’IEEE/ACM Trans. Netw., vol. 17, no. 3, pp. 950-961, June 2009.

[15] X.-H. Xu, S.-J. Tang, and P.-J. Wan, ‘‘Maximum WeightedIndependent Set of Links under Physical Interference Model,’’in Proc. Wireless Algorithms, Syst., Appl., 2010, vol. LNCS 6221,pp. 68-74.

[16] M. Halldorsson and R. Wattenhofer, ‘‘Wireless Communicationis in APX,’’ in Proc. 36th Int’l Colloquium Automata, Lang. Program.,2009, pp. 525-536.

[17] D. Chafekar, V. Kumar, M. Marathe, S. Parthasarathy, andA. Srinivasan, ‘‘Approximation Algorithms for ComputingCapacity of Wireless Networks with SINR Constraints,’’ in Proc.IEEE INFOCOM, 2008, pp. 1166-1174.

[18] P.-J. Wan, X.-H. Xu, and O. Frieder, ‘‘Shortest Link Schedulingwith Power Control under Physical Interference Model,’’ in Proc.IEEE MSN, 2010, pp. 74-78.

[19] X.H. Xu, S.J. Tang, and X.-Y. Li, Stable Wireless Link Scheduling Subjectto Physical Interferences with Power Control, 2011, manuscript.

[20] D.M. Blough, G. Resta, and P. Santi, ‘‘Approximation Algorithmsfor Wireless Link Scheduling with SINR-Based Inteference,’’IEEE/ACM Trans. Netw., vol. 18, no. 6, pp. 1701-1712, Dec. 2010.

[21] L.-B. Le, E. Modiano, C. Joo, and N.B. Shroff, ‘‘Longest-Queue-First Scheduling Under SINR Interference Model,’’ in Proc. ACMMobiHoc, 2010, pp. 41-50.

[22] G. Brar, D. Blough, and P. Santi, ‘‘The SCREAM Approach forEfficient Distributed Scheduling with Physical Interference inWireless Mesh Networks,’’ in Proc. IEEE ICDCS, 2008, pp. 214-224.

[23] D. Peleg, ‘‘Distributed Computing: A Locality-Sensitive Ap-proach,’’ in SIAM Monographs on Discrete Mathematics andApplications, vol. 5. Philadelphia, PA, USA: SIAM, 1987.

[24] L. Tassiulas and A. Ephremides, ‘‘Stability Properties ofConstrained Queueing Systems and Scheduling Policies forMaximum Throughput in Multihop Radio Networks,’’ IEEE/ACM Trans. Autom. Control, vol. 37, no. 12, pp. 1936-1948, Dec. 1992.

[25] G. Sharma, R. Mazumdar, and N. Shroff, ‘‘On the Complexity ofScheduling in Wireless Networks,’’ in Proc. ACM MobiCom, 2006,pp. 227-238.

[26] X. Lin and N.B. Shroff, ‘‘The Impact of Imperfect Scheduling onCross-Layer Rate Control in Multihop Wireless Networks,’’ inProc. IEEE INFOCOM, 2005, pp. 1804-1814.

[27] P.-J. Wan, X.-H. Jia, and F. Yao, ‘‘Maximum Independent Set ofLinks under Physical Interference Model,’’ in Proc. WASA, 2009,pp. 169-178.

[28] S. Shakkottai and R. Srikant, ‘‘Network Optimization andControl,’’ Found. Trends Netw., vol. 2, no. 3, pp. 271-379, Jan. 2007

[29] Y.Q. Zhou, X.-Y. Li, M. Liu, X.F. Mao, S.J. Tang, and Z.C. Li,Throughput Optimizing Localized Link Scheduling for MultihopWireless Networks Under Physical Interference Model, Jan. 2013, ArXive-prints. [Online]. Available: http://arxiv.org/abs/1301.4738

[30] X.-H. Xu and S.-J. Tang, ‘‘A Constant Approximation Algorithmfor Link Scheduling in Arbitrary Networks Under PhysicalInterference Model,’’ in Proc. ACM FOWANC, 2009, pp. 13-20.

[31] X.-Y. Li and Y. Wang, ‘‘Simple Approximation Algorithms andPTASs for Various Problems in Wireless Ad Hoc Networks,’’ J.Parallel Distrib. Comput., vol. 66, no. 4, pp. 515-530, Apr. 2006.

[32] T. Moscibroda and R. Wattenhofer, ‘‘The Complexity ofConnectivity in Wireless Networks,’’ in Proc. IEEE INFOCOM,2006, pp. 1-13.

[33] T. Moscibroda and R. Wattenhofer, ‘‘Topology Control MeetsSINR: The Scheduling Complexity of Arbitrary Topologies,’’ inProc. IEEE MobiHoc, 2006, pp. 310-321.

Yaqin Zhou received the BS and MS degreesin computer science from China AgriculturalUniversity, China, in 2008 and 2010, respectively.She is currently pursuing the PhD degree at Instituteof Computing Technology, Chinese Academy ofSciences, China. Her research interests include linkscheduling and channel accessing in wirelessnetworks.


Xiang-Yang Li received the BS degrees incomputer science and business managementfrom Tsinghua University, China, both in 1995,and the MS and PhD degrees from Departmentof Computer Science, University of Illinois atUrbana-Champaign, in 2000 and 2001, respec-tively. He has been a Professor since 2012,Associate Professor from 2006 to 2012, and As-sistant Professor from 2000 to 2006 of ComputerScience at the Illinois Institute of Technology. Heis a recipient of China NSF Outstanding Over-

seas Young Researcher (B). He published a monograph ‘‘Wireless AdHoc and Sensor Networks: Theory and Applications’’. He also co-editedthe book ‘‘Encyclopedia of Algorithms’’. His research has been sup-ported by USA NSF, HongKong RGC, and China NSF. His researchinterests span the wireless networks, game theory, computationalgeometry, and cryptography and network security. Dr. Li is an editor ofseveral journals, including IEEE Transaction on Parallel and DistributedSystems (2009 to present). He is a Senior Member of the IEEE.

Min Liu received the BS and MS degrees incomputer science from Xian Jiaotong University,China, in 1999 and 2002, respectively, and thePhD degree in computer science from the Grad-uate University of the Chinese Academy of Sci-ences in 2008. She is currently a Professor at theNetworking Technology Research Center, Insti-tute of Computing Technology, Chinese Acad-emy of Sciences. Her current research interestsinclude mobility management, wireless resourcemanagement and delay/disruptive-tolerant net-

works. She is a member of the IEEE.

Xufei Mao received the BS degree in computerfrom Shenyang University of Technology in 1999,the MS degree in computer science from North-eastern University in 2003, and the PhD degree incomputer science from Illinois Institute of Technol-ogy, Chicago in 2010. He is with the School ofSoftware and TNLIST, Tsinghua University, Beij-ing,China.His research interests spanwireless ad-hoc networks, wireless sensor networks, pervasivecomputing, mobile cloud computing and gametheory. He is a member of the IEEE.

Shaojie Tang received the BS degree in radioengineering from Southeast University, China, in2006, and the PhD degree from Department ofComputer Science at Illinois Institute of Technologyin 2012. He is an Assistant Professor in the Depart-ment of Computer and Information Science (Re-search) at Temple University. His main researchinterests focus on wireless networks (including sen-sor networks and cognitive radio networks), socialnetworks, pervasive computing, mobile cloud com-puting and algorithm analysis and design. He has

recently served as guest editor of Journal of Tsinghua Science and Tech-nology. He also served as TPCmember of a number of conferences suchas IEEE ICPP, IEEE IPCCC, MSN. He is a member of the IEEE.

Zhongcheng Li received the BS degree incomputer science from Peking University in1983, and theMS and PhD degrees from Instituteof Computing Technology, Chinese Academy ofSciences, in 1986 and 1991, respectively. From1996 to 1997, he was a visiting professor at Uni-versity of California at Berkeley. He is currently aprofessor of Institute of Computing Technology,Chinese Academy of Sciences. His current re-search interests include next generation Internet,dependable systems and networks, and wireless

communication. He is a member of the IEEE.

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.


Documents

Throughput Optimizing Localized Link Scheduling for Multihop Wireless Networks under Physical Interference Model