The Scheduling and Energy Complexity of Strong Connectivity in Ultra-Wideband Networks

Transcription

1 The Schedulig ad Eegy Coplexity of Stog Coectivity i Ulta-Widebad Netwos Qiag-Sheg Hua Depatet of Copute Sciece The Uivesity of Hog Kog Pofula Road, Hog Kog, PR Chia qshua@cshuh Facis CM Lau Depatet of Copute Sciece The Uivesity of Hog Kog Pofula Road, Hog Kog, PR Chia fclau@cshuh ABSTRACT Recetly Mosciboda ad Wattehofe cae up with the otio of schedulig coplexity to captue the iiu aout of tie to successfully schedule all the tasissio equests ude the physical SINR odel Thei algoith featuig a oliea powe assiget ca schedule stogly coected 4 tasissios i aowbad etwos with O(log tieslots I this pape, we fist geealize this esult to ulta-widebad etwos We show the stog coectivity schedulig coplexity i UWB etwos to be O(log( log, whee is the pocessig gai Secodly, we show that both of these polylogaithic schedulig coplexity esults ae gaied at the expese of expoetial eegy coplexity with lowe boud ω( We also pove the uppe boud of the eegy coplexity i aowbad etwos to be O (, ad fo UWB etwos, this uppe boud ca be educed by a pocessig gai facto O the othe had, we show that ipovig the schedulig coplexity though abitay powe cotol has its liitatios, ad that diffeet powe assiget stategies have diffeet ipacts o the potocol itefeece odels, which was ofte eglected i the desig of wieless schedulig algoiths Copaed with aowbad etwos, although the effect of aggegate itefeeces i UWB etwos is geatly educed, we deostate that the costat ad liea powe assigets i UWB etwos ae still iefficiet i the wost case with espect to the schedulig coplexity ( Ω (, which suggests thee is a eed fo a bette abitay powe assiget Ou aalyses shed ew light o the desig of the powe assiget schee ad the pefoace aalysis of the wieless schedulig algoiths I eegy-costaied wieless etwos, a tadeoff betwee the schedulig coplexity ad eegy coplexity is a pactical cosideatio Ou esults i this pape ca be diectly applied to othe spead-spectu etwos icludig DS-CDMA ad FH-CDMA Categoies ad Subject Desciptos C [Copute-Couicatio Netwos]: Netwo Achitectue ad Desig wieless couicatio, etwo topology; Peissio to ae digital o had copies of all o pat of this wo fo pesoal o classoo use is gated without fee povided that copies ae ot ade o distibuted fo pofit o coecial advatage ad that copies bea this otice ad the full citatio o the fist page To copy othewise, o epublish, to post o seves o to edistibute to lists, equies pio specific peissio ad/o a fee MSWiM 06, Octobe 6, 006, Toeolios, Malaga, Spai Copyight 006 ACM /06/000 $500 F [Aalysis of Algoiths ad Poble Coplexity]: Noueical Algoiths ad Pobles - geoetical pobles ad coputatios, sequecig ad schedulig; Geeal Tes Algoiths, Theoy, Pefoace Keywods Ad hoc ad seso etwos, schedulig coplexity, eegy coplexity, itefeece odels, itefeece cotol, ultawidebad INTRODUCTION The schedulig coplexity poble i wieless etwos is to ty to use the iiu aout of tie to successfully schedule all the tasissio equests to eet soe topological popety equieet Recetly Mosciboda ad Wattehofe poposed a solutio of which the schedulig coplexity fo aowbad etwos with stog coectivity popety is oly 4 O(log tieslots [0], which is expoetially faste tha O ( if idividual tasissios ae scheduled oe by oe I this pape we geealize this esult to cove also ulta-widebad (UWB etwos which ae dawig iceasig attetio due to thei ay poisig featues [7] Sice a UWB etwo is iheetly a spead-spectu etwo [], the aggegate itefeeces caused by othe siultaeous tasissios at the iteded eceive ca be educed by a facto of the pocessig gai, aig it vey copetitive i wieless couicatios Ulie the aowbad etwos whee the itefeece age is lage tha the tasissio age, as show i Sectio, the itefeece age of UWB etwos aoud the eceive is salle tha the tasissio age, allowig oe siultaeous tasissios at the eceive Ad oe of the ecet fidigs i UWB etwo eseach [] is that the desig of the optial MAC ca be idepedet of the choice of outig Thus the use of ultawidebad ca e-itoduce the otio of laye sepaatio betwee the MAC ad the outig layes just lie i taditioal wie lie etwos This will ae the esultat etwo oe scalable ad a good choice fo geeic seso etwos Futheoe, UWB is ulti-path fadig esistat, ad as the followig SINR odel shows, it is oe flexible i tes of adaptig its paaetes to eet diffeet opeatioal equieets (eg, chage i pocessig gai Fo ou aalyses, we adopt the physical sigal-to-itefeeceplus-oise atio (SINR odel, with which oly whe the eceived powe is above the SINR atio theshold ca a essage be successfully eceived The SINR odel fo UWB etwos was fist give i [4], which is diffeet fo that fo aowbad 8

2 etwos [6] Specifically, the achieved sigal-to-itefeeceplus-oise atio at the eceive of li i ca be epeseted as: Pi d( xi, xj SINRi = N0 Ri[ η + Tfσ P d( x, xj ] =, i whee Pi deotes the aveage tasissio powe of li i s tasitte, x i ; R i deotes li i s data ate, ad Ri = /( NNT s h c ; Ns deotes the ube of pulses pe sybol, N h the ube of tieslots pe Pulse Repetitio Iteval (PRI, ad T c the pulse duatio; Tf is the PRI, ad T f = N h T c ; σ is a paaete depedig o the shape of the oocycle; η is the bacgoud oise plus itefeece fo othe o-uwb systes; d( xi, xj deotes the Euclidea distace betwee tasitte xi ad x j ; is the path loss expoet ad is the SINR theshold; N0 deotes the ube of siultaeous tasissios with tasitte x i If we set N = η ( T f σ ad = ( RiTfσ, the above is tasfoed to a fo siila to the spead-spectu SINR odel i [5]: Pi d( xi, xj SINR = ( N0 N + P d( x, xj =, i Hee we use fo the pocessig gai If =, this becoes the taditioal aowbad SINR odel, as used i [0] The ogaizatio of the pape is as follows I Sectio, we eview itefeece cotol techiques fo both aowbad ad widebad etwos, ad fo both gaph coloig based ad physical SINR odels I Sectio, we exploe diffeet powe assigets ad thei ipacts o pai-wise itefeece odels which play a vey ipotat ole i the desig of wieless potocols ad wieless etwo capacity aalyses Futheoe, we discuss the liitatios of spatial euse though powe cotol, which ae the tas to apply pope powe cotol techiques to ipovig the schedulig coplexity o-tivial I Sectio 4, we give a foal defiitio of the schedulig coplexity poble ad show that both the costat ad liea powe assigets ae iefficiet with espect to the schedulig coplexity i UWB etwos I Sectio 5, we geealize the o-liea powe assiget algoith i [0], with the guaatee that all the siultaeous tasissios ca be successfully scheduled based o the SINR odel, ad show that the schedulig coplexity of stog coectivity i UWB etwos is O(log( log This esult epesets a ipoveet ove that fo the aowbad etwos I Sectio 6, we defie the eegy coplexity of the wieless schedulig pobles, ad show that the polylogaithic schedulig coplexity was achieved at the expese of the expoetial eegy Ad i UWB etwos, the uppe boud of the eegy coplexity ca be educed by a pocessig gai facto Sectio 7 cocludes the pape ad discusses soe futue tass that could ae ou algoith pactical To the best of ou owledge, this is the fist pape to study both the schedulig coplexity ad eegy coplexity of UWB (ad widebad etwos, as well as the fist to give eegy coplexity aalyses of the stog coectivity schedulig poble i aowbad etwos Futheoe, this is also the fist pape to aalyze the ipacts of powe cotol o the potocol itefeece odels RELATED WORK Itefeeces caused by cocuet tasissios i the shaed couicatio ediu could be ightaish i wieless etwos especially i ulti-hop ad-hoc o seso etwos If thee ae too ay siultaeous tasissios, eve if the itefeece fo a sigle ode is sall, the aggegate itefeece could be disastous O the othe had, if thee ae oly a few cocuet tasissios, o eve to schedule the oe by oe would waste valuable badwidth ad the thoughput capacity would suffe Thus fidig a pope itefeece cotol ethod is exteely ipotat i wieless etwos Ad sice itefeece cotol is based o the odel used, theefoe fidig a appopiate itefeece odel is equally ipotat Fo a ecet suvey o the algoithic odels of wieless ad-hoc ad seso etwos, please efe to [4] The ealy itefeece cotol techique used i pacet adio etwos is to avoid the so-called piay itefeece ad secoday itefeece pobles [5] By piay itefeece, a sigle ode caot pefo two opeatios at the sae tie, such as eceivig fo two sedes, tasittig to two eceives, o eceivig ad sedig at the sae tie By secoday itefeece, ode A is coveed i sede B s tasittig age but the iteded eceive is ode C which is diffeet fo ode A Obviously avoidig secoday itefeece pevets the captue effect, which should be a good thig i the physical eality The stadad itefeece cotol techique to deal with these itefeeces is the gaph-based schedulig algoith [] It was claied that the potocol itefeece odel i [6] is its geeic fo But as we will show i Sectio, the latte potocol odel is ideed diffeet fo the gaph-based oe Based o the piay/secoday itefeece odel, distace- atchig (o stog edge coloig fo li schedulig ad distace- coloig fo boadcast schedulig wee poposed [] We focus o li schedulig i this pape I distace- atchig, oly lis that ae at least of distace two apat ca be assiged the sae tieslot (o colo This atchig is actually i lie with the 80 DCF MAC potocol, whee RTS/CTS vitual caie sesig would bloc the lis withi a distace of two edges But ecet eseach idicates that both the physical caie sesig ad vitual caie sesig ethods i 80-ad-hoc ae ot thoughput capacity-efficiet [] So i this pape we will ot coside this id of distace- atchig odel Istead, we focus o the piay itefeece poble that is, all pais of lis shaig a coo edpoit will ot be scheduled at the sae tie, ad the eceptio ust satisfy the SINR iequality ( but o othe costaits Fo the gaph-based odels, it was show i [] that the schedulig of the axiu cadiality of the idepedet sets of the coflict gaph [8] caot guaatee the best thoughput capacity due to the aggegate itefeece effect as just etioed Theefoe ost of the ecet eseach has shifted to aalyzig 8

3 physical SINR odels [][0] Fo a latest suvey of usig physical SINR odels i cellula etwos, please efe to [9] Regadig the use of SINR odels i ulti-hop etwos, ost of the itefeece cotol techiques focus o aowbad etwos, ad oly a few of the would tae o the spead-spectu SINR odel [5][8] As we have etioed, the tasfoed UWB SINR odel is siila to the spead-spectu SINR odel, so all the esults i this pape ca be diectly applied to othe speadspectu etwos, such as DS-CDMA ad FH-CDMA POWER ASSIGNMENT IMPACT ON PROTOCOL INTERFERENCE MODELS AND LIMITATIONS I this sectio, we fist discuss the ipact of diffeet powe assigets o the pai-wise itefeece odels Ad the we discuss the liitatios of ipovig the spatial euse though powe cotol Copaig with aowbad etwos, thee is oe oo fo UWB etwos to tae advatage of powe cotol to ipove the schedulig coplexity Potocol Itefeece Models Ude Diffeet Powe Assigets We focus o the ipact of the powe assigets o the paiwise itefeece odels, which was ofte eglected i wieless schedulig algoith desig Basically, we ca divide the cuet powe assiget stategies ito thee categoies: Costat powe assiget; liea powe assiget ; ad abitay powe assiget (eg, o-liea powe assiget We fist coside aowbad etwos Accodig to iequality ( ad because = i aowbad etwos, i ode to esue a successful tasissio, x, the followig iequality ust hold Px d, x d( y s, x P y > ( ( N + P d( y, x d( x, x P y s s x a Potocol itefeece odel with costat powe assiget i aowbad etwos: With the costat powe assiget, P x = P y, so by iequality (, we have / d( ys, x > d, x ( If we set (+ Δ = /, this becoes the sae as the potocol odel give i [6] Sice i aowbad etwos, usually the / theshold > ad cosequetly the age d, x is geate tha the sede s tasissio age d, x Thus to esue a successful tasissio, a disc of adius at / least d, x aoud each successful eceive x ust ot / cotai othe tasittes So we deote d, x as the itefeece age (o exclusio egio aoud each eceive x Fo exaple, i Fig (a, assuig costat powe assiget, sice d, y < d( ys, y, tasissio ( ys, y is ot successful; wheeas, sice d( ys, x > d, x, tasissio, x is successful With this we ca distiguish the gaph-based itefeece odel fo the potocol itefeece odel which was cosideed the sae i [] Notice that the potocol itefeece odel oigiates fo the physical SINR odel, ad so it ca eflect the physical eality icludig the captue effect, while all the gaph-based itefeece odels caot eflect the sae Fo exaple, sice ode x is i the tasissio age of y s, it suffes fo the secoday itefeece poble [5], so tasissio ( x, x is ot successful s y s y s y s y s x s y x s (a d(x s,x =, d(y s,y =4, d(x s,y =,d(y s,x = x (b d(x s,x =, d(y s,y =4, d(x s,y =,d(y s,x = (c d(x s,x =, d(y s,y =, d(x s,y =,d(y s,x = (d d(x s,x =, d(y s,y =4, d(x s,y =,d(y s,x = x s Figue Pai-wise tasissios exaples b Potocol itefeece odel with liea powe assiget i aowbad etwos: With the liea powe assiget, P x = ρ d, x ad P y = ρ d( ys, y, ad so accodig to iequality (, we have d( y, s x d( ys, y > d( ys, x > d( ys, y (4 d, x d, x This potocol odel was used i [4] But copaed with (, it has attacted uch less attetio ostly because ay capacity aalysis papes assue the costat powe assiget Note that the itefeece age of eceive x has bee chaged fo / d, x to / d( ys, y Fo exaple, i Fig (a, assuig liea powe assiget, sice d( ys, x < d( ys, y, tasissio, x is ot successful Ad sice d, y > d, x, tasissio ( ys, y is successful Now we tu to UWB etwos Accodig to iequality (, i ode to esue a successful tasissio, x, the followig iequality ust hold Px d, x d( y s, x P y > ( ( (5 N + Py d( ys, x d, x Px c Potocol itefeece odel with costat powe assiget i UWB etwos: With the costat powe assiget, siila to pat a, by iequality (5 we have / d( ys, x > ( d, x (6 / The itefeece age d, x aoud the eceive x / is eplaced with ( d, x Hece the itefeece age becoes salle tha the tasissio age Fo exaple, i Fig (a, if =4, =, =00, sice / d, y => ( d( ys, y 5, the peviously usuccessful tasissio ( ys, y with costat powe assiget i aowbad etwos becoes successful i UWB etwos As a esult, the two tasissios ca be scheduled i paallel d Potocol itefeece odel with liea powe assiget i UWB etwos: With the liea powe assiget, siila to pat b, by iequality (5 we have d( y, s x d( ys, y > ( d( ys, x > ( d( ys, y (7 d, x d, x The itefeece age aoud eceive x is chaged / / fo ( d, x to ( d( ys, y Fo exaple, i Fig (a, if =4, =, =00, sice / d( ys, x => ( d( ys, y 5, the peviously usuccessful tasissio ( x, x with liea powe assiget s x x x x s y y y 84

4 i aowbad etwos becoes successful i UWB etwos So the two tasissios ca be siultaeously scheduled Fo the above aalyses, o oe had, due to the lage pocessig gai whe usig the costat o liea powe assiget, ay usuccessful siultaeous tasissios i aowbad etwos becoe successful i UWB etwos, thus leadig to iceased spatial euse i UWB etwos O the othe had, as the exaples i [6] have show, eve i aowbad etwos, the usuccessful siultaeous tasissios ca also becoe successful with a pope abitay powe assiget Fo exaple, fo Fig (a, if =4, =, N=, ad P x =80, P y =50, the two tasissios ca be successfully scheduled i paallel Ad fo Fig (c, if =, =4, N=, ad P x =4, P =64, the two tasissios ca also tae place siultaeously Nevetheless, i Sectio 4, we will show that eve i UWB etwos, both the costat ad liea powe assigets ae still iefficiet i tes of schedulig coplexity The followig two theoes show that the powe cotol techique has its liitatios i educig the schedulig coplexity Liitatios of Powe Cotol i Ipovig Spatial Reuse THEOREM I aowbad etwos, fo ay two tasissios ( x s, x ad ( y s, y, if d( x s, y d( y s, x / d( x s, x d( y s, y, the thee exists o feasible powe assiget fo siultaeous tasissios; othewise, thee always exists a feasible powe assiget to have a siultaeous schedule PROOF If the two tasissios ca be successfully scheduled, based o iequality ( with pocessig gai equal to, the followig two iequalities ust follow: Px d, x Py d( ys, y N + Py d( ys, x N + Px d, y Fo these iequalities, we have d, x d, y Py < Px < Py d( ys, x d( ys, y d, x d, y Theefoe, if, thee is o feasible d( ys, x d( ys, y powe assiget fo siultaeous schedulig; othewise, thee always exists a feasible powe assiget to schedule these two tasissios i paallel Actually this is a special case of the Peo-Fobeius theoy used i the powe cotol of cellula etwos [9], but it has bee paid little attetio i ulti-hop etwos Fo exaple, i Fig (d, if =4, =, ad N=, thee will be o feasible powe assiget to siultaeously schedule tasissio ( x s, x ad ( y s, y The sae is tue of Fig (b THEOREM I UWB (o ay spead-spectu etwos, fo ay two tasissios ( x s, x ad ( y s, y, if / d( x s, y d( y s, x > ( d( x s, x d( y s, y, thee always exists a powe assiget to schedule these tasissios i paallel; o feasible powe assigets fo siultaeous schedule, othewise PROOF Siila to the poof of Theoe, if the two tasissios ca be successfully scheduled, the followig two iequalities ust follow: Px d, x Py d( ys, y N + Py d( ys, x N + Px d, y Fo these iequalities, we have y d, x d, y Py < Px < Py d( ys, x d( ys, y d, x d, y Theefoe, if <, thee always exists d( ys, x d( ys, y a powe assiget to siultaeously schedule these two tasissios; othewise, thee is o valid powe assiget to give a paallel schedule Fo exaple, i Fig (d, if =4, =, N=, ad =0, P x = P y =000, the two tasissios ca be siultaeously scheduled Theefoe, give ay two tasissios i aowbad etwos whee powe cotol caot guaatee a siultaeous schedule, they ca be scheduled i paallel i UWB etwos as / log as d( x s, y d( y s, x > ( d( x s, x d( y s, y Give this esult, we will discuss the schedulig coplexity poble i UWB etwos i Sectio 5 4 SCHEDULING COMPLEXITY AND INEFFICIENCY OF CONSTANT AND LINEAR POWER ASSIGNMENTS IN UWB NETWORKS 4 Schedulig Coplexity We coside a abitaily distibuted etwo with odes X={ x 0, x,, x } i the Euclidea plae Fo ay lis f ij =( x i, x j, ( f ij =d( x i, x j deotes the distace betwee ode x i ad ode x j DEFINITION 4 A powe assiget φt is a fuctio + φ t : X which aps evey ode i the etwo to a cetai powe level φ t( xi = P i deotes the powe level of ode x i i tieslot t A schedule S= ( φ, φ,, φ T ( S is a sequece of T(S powe assigets, whee φ i deotes the powe assiget i tieslot i DEFINITION 4 Give a tieslot t ad a powe assiget φ t, we say that the diected li, x is successfully scheduled i tieslot t if x eceives a essage fo x s accodig to the SINR iequality ( Let Et be the set of all successful lis i tieslot t, we have DEFINITION 4 The schedulig poble fo a etwo popety Ψ is to fid a schedule S of iial legth T(S, such T( S that the uio of the set E t ( Et satisfies the popety Ψ t = DEFINITION 44 The schedulig coplexity of a etwo popety Ψ is the iial ube of tieslots T, such that thee always exists a valid schedule S fo Ψ of legth T =T(S 4 The Iefficiecy of Costat ad Liea Powe Assigets i UWB Netwos We fist defie the followig popetyψ i which is the sae as that i [0]: Fo evey ode x i X, it ca successfully sed at least oe essage to ay othe ode This is the siplest popety to chec the algoith s schedulig coplexity We also coside the expoetial ode chai [7], whee all the odes ae placed o a staight lie with expoetially iceasig distaces betwee the Fig below is a exaple i i+ i+ i+ i+ 4 i+ 5 i+ Figue : Expoetial ode chai, whee i+ is the distace betwee odes x ad x + 85

5 THEOREM 4 Fo both costat ad liea powe assigets, the schedulig coplexity fo poble ψ i is at least /( + Ω (, eve i the absece of abiet oise, whee is the ube of the odes, ad is the pocessig gai PROOF a Costat powe assiget: I this case, fo all odes, tasissio powe P i = P =P Now coside the exaple i Fig ; we assue thee ae at ost L siultaeous tasissios i a schedulig tieslot Suppose ode x s is the ight-ost tasitte i this tieslot, ad ode x is its eceive The othe (L- siultaeous tasissios will cause aggegate itefeeces to ode x Accodig to the popety of the expoetial ode chai, if ode x is o the left side of ode x s, the distace fo evey othe siultaeous tasitte to the eceive x is d( x i, x d( x s, x ; ad if ode x is o the ight side of ode x s, the distace fo evey othe siultaeous tasitte to the eceive x is d( x i, x d( x s, x Theefoe the aggegate itefeece caused by these (L- siultaeous tasittes is at least ( L P ( d, x Accodig to the SINR iequality (, we have: P d, x N + ( L P ( d, x Fo this, it follows that the axiu ube of siultaeous tasissios L i each tieslot is ( + / Theefoe, the costat powe assiget ethod equies at least /( + tieslots to schedule all odes at least oce b Liea powe assiget: With liea powe assiget, the sede xs will sed to its eceive x with powe P s = ρ d, x, whee ρ deotes the iiu eceived powe to decode the essage Siila to the costat powe assiget aalysis, we assue thee ae at ost L siultaeous tasissios i a schedulig tieslot Accodig to the popety of the expoetial ode chai, fo all odes x i, it will cause at least the itefeece ρ to its left side odes [6] Now suppose x is the left-ost eceive, ad x s is soe tasitte i the L siultaeous tasissios The othe (L- siultaeous tasissios will cause at least the aggegate itefeece (L- ρ to this left-ost eceive x Accodig to the SINR iequality (, we have ρ d, x / d, x N + ( L ρ Fo this, it follows that the axiu ube of siultaeous tasissios L i each tieslot is ( + / Ad theefoe the liea powe assiget ethod equies also at least /( + tieslots to schedule all odes at least oce 5 THE SCHEDULING COMPLEXITY OF STRONG CONNECTIVITY IN UWB NETWORKS 5 The NPAW Schedulig Algoith It is ipotat to distiguish betwee li legth class ad li legth class set which ae used i ou algoith A li legth class is a set of tasissio lis such that the legths of these lis diffe by at ost a facto of (lie 6 of the ai algoith A ube of li legth classes fo a li legth class set The thee ids of li legth class set L, S ad I used i ou algoith, ad thei elatioships, ae descibed i Fig I : I 0 I I I I q S : L : Figue : Thee ids of li legth class set ad thei elatioships I Fig, L, S ad I deote the espective legth classes i each set L is eaed to S because the epty legth classes (cotaiig o tasissio lis i L wee deleted (lie 7 of the ai algoith Fo exaple, the legth classes L ad L wee deleted S is eaed to I because i each oud, the schedulig algoith oly selects the legth classes i S with a cetai legth class sepaatio The sepaatio value is log(4 i [0] but we use log(/ i ou algoith (lie 9 of the ai algoith The solid aows fo S to I ea we select the legth classes S 0 S j S i the fist oud, while the dashed aows ea we select the legth classes S S j+ S + i the secod oud (the details ae i Tables ad Note that oly lis i L have the + popety ( <, but ot those i S o I (because f ij + uppe boud would ot hold fo the Ou schedulig algoith also uses a o-liea powe assiget Fo coveiece, we efe to the schedulig algoith i [0] as NPAN (o-liea powe assiget fo aowbad etwos, ad ou algoith NPAW (o-liea powe assiget fo (ulta-widebad etwos The ai algoith of NPAW poceeds i phases I each phase, the algoith costucts a diected eaest eighbo foest Fp o the cuet active odes (lies, 4 ad 5 Iitially all the odes i X ae active, but afte evey diected li i Fp has bee scheduled, all the tasittig odes of these lis becoe passive i the ext phase (lie Whe thee is oly oe active ode, the while loop (lie teiates ad the ode set X becoes a diected spaig tee with the active ode as the si (A si ode eas a ode havig o outgoig li The the si ode tasits with axiu powe to cove all the odes i X (lie, so that the esultat topology would becoe stogly coected The challegig pat of the algoith is how to schedule F p both successfully ad efficietly Just as Fig has deostated, we fist patitio all the lis i F p ito legth classes of L which is the eaed to S (lies 6 ad 7 The we use the suboutie Schedule( to schedule the lis i legth classes Sh log( S 0 S S S j S j+ S S + L 0 L L L L 4 L L - + i the th oud (lies 8, 9 ad Table The tic of this algoith lies i two aspects: oe is the o-liea powe assiget schee (lie 9 of the suboutie This powe assiget uses a powe scalig facto τ which depeds o the positio of the schedulig lis i li legth class set I (lies ad of the suboutie ad Fig Because shot lis have a high τ value ad log lis have a low τ value, this powe assiget ca icease the powe of the shot lis elative to the log oes so that it aes siultaeous tasissios of vey diffeet legths possible Futheoe, because this powe assiget taes the paaete (total ube of the odes ito accout, it ca boud the aggegate itefeeces though the popely desiged potocol itefeece odel (lies ad of 86

6 the suboutie But as discussed i Sectios ad, taditioal pai-wise potocol itefeece odels caot guaatee the successful tasissio due to the aggegate itefeece effect The secod pat of the tic is the selectio of the siultaeous tasittig lis i legth class set I (Fig With the pope legth class sepaatio, the algoith ca boud the total ube of deletig lis at O(log i each tieslot of evey schedulig oud (lie 9 of the ai algoith ad lies ad of the suboutie, thus guaateeig that at least Ω ( /log of the cuet cadidate lis ca be siultaeously tasitted i a sigle tieslot Theefoe the polylogaithic schedulig coplexity ca be aived at Thee ae thee ipotat diffeeces of ou NPAW algoith fo the NPAN algoith: a We desig a ew o-liea powe assiget fo UWB etwos (lie 9 of the suboutie b We desig two ew potocol itefeece odels which ca boud the aggegate itefeeces though deletig fewe o at ost equal tasittig lis tha the NPAN algoith The deleted lis cosist of two goups: the fist ae i the sae legth class li schedulig (lie of the ai algoith, ad lie of the suboutie ad the secod ae i diffeet legth class li scheduligs (lie of the suboutie The poofs will be give i Coollay 55 ad Lea 56 c As show i Fig ad Table, thee ae oe legth classes to be scheduled i the sae oud i li legth class set I (lie 9 of the ai algoith, ad theefoe the total iteatios of the fo loop is educed as copaed to the NPAN algoith (lie 8 of the ai algoith By this eductio, we educe the 4 schedulig coplexity fo O(log to O(log( log Mai Algoith: Stog Coectivity Schedulig Algoith fo UWB Netwos (NPAW Iput: A abitaily distibuted set of odes X (cf sectio 4 Output: A schedule S satisfyig stog-coectivity : Defie a costatυ :=4N ad a vaiable μ which is a fuctio of the pocessig gai such that ( μ :=+ ε +4 (7 ; >; t:=; ( {N is the bacgoud oise fo iequality ( ad ε is a sall positive paaete} : while X > do : F p := ; 4: Fo each x i X fid its closest eighbo x j such that F p := F p f ij ; { f ij is a diected edge fo x i to x j } 5: If Fp cotais bi-diectioal edges the eove oe edge of the; { To ae F p a diected eaest eighbo foest } 6: Patitio all the tasissio lis i F p ito legth class set L = { L0, L,, L Δ }, such that L cotais all lis f ij of + legth ( f ij < ; { Δ= log( lax, ad lax eas the axiu li legth i F p } 7: Delete all epty legth classes L i Fp ad eae L to S = { S0, S,, S,} such that S is the th sallest o-epty legth-class i S; 8: fo =0 to log( do 9: Schedule all the lis / log( ij S h= 0 h log( + f usig suboutie Schedule(; 0: ed fo : Delete all the odes fo ode set X except the si ode i each tee of the diected eaest eighbo foest F ; : ed while : φ t( xi :=N/ l ax fo xi X ; { Hee l ax eas the lagest Euclidea distace betwee two odes i ode set X} 4: S:={ φ,, φt }; Suboutie Schedule(: : Let F be the set of lis to be scheduled, eae these li legth classes i S to I = { I0, I,, I q } with at ost q+ legth classes whee q= / log( I is the th sallest legth-class i I; {lie 9 of the ai algoith} : fo each f uv I do τ ( xu : = q + ; {Lis withi the sallest legth class I 0 have the highest τ value / log(, ad lis withi the lagest legth class I q have the lowest τ value } : while F do 4: fo each xi X do φ t( xi: = 0; ed fo {Set the powes of the tasittes i the peviously scheduled lis to 0} 5: F : = F; E : = ; t 6: while F t do t 7: Choose the li f * ij Ft of iial legth; 8: * * E : = E { f }; F : = F \{ f }; t t ij t t ij 9: τ ( * ( : ( / x i φt xi υ ( fij = ; {Schedule li 0: fo each f l F t do : δ : = τ( x τ( x ; i i : if δ i =0 ad d( x i, x l μ ( f ij the F : = F \ { f }; t t l i : else if d( x i, x l ( ( f ij the F : = F \ { f }; 4: ed if 5: ed fo 6: ed while 7: F : = F \ E t ; t:=t+; 8: ed while t t l * p ( δ + / * * f ij } 5 Coectess Aalysis I Sectio 5, we have show that i each tieslot of evey schedulig oud, the deletig lis i the NPAW algoith ust be fewe tha o at ost equal to the deletig lis i the NPAN algoith This eas that thee ae oe tasissio lis that should be scheduled i the sae tieslot, thus povig that the successful siultaeous tasissios of all available lis is of fudaetal ipotace LEMMA 5 Coside a scheduled li fx with iteded sede xs ad eceive x Let I( y i be the itefeece caused at x by siultaeously tasittig odes yi fo which τ ( yi < τ It holds that I( yi υ( τ PROOF I ou ai algoith, because evey ode y i tasits essages to its eaest eighbo, we have d( y i, x ( f y Hece the itefeece at x caused by y i is at ost I ( y = i 87

7 τ ( y i P υ( ( fy i d( y, x ( f i y τ( yi τ υ = υ( ( LEMMA 5 Coside a scheduled li fx with iteded sede xs ad eceive x Let I( y i be the itefeece caused at x by siultaeously tasittig odes yi fo which τ ( yi > τ It holds that I( yi υ( τ PROOF Assue fo cotadictio that thee exists a ode y i with τ ( yi > τ ad I( yi υ( τ > The τ ( y ( / i P υ ( f i y τ I( y i = > υ( d( yi, x d( yi, x ( δis + / Fo this, we have d( yi, x < ( ( fy Howeve, this cotadicts the defiitio of ou algoith I lie of the suboutie, if ode y i has bee scheduled (because it has shot li legth, lie 7 of the suboutie, fo the above iequality, ode x s should have bee deleted, which establishes the cotadictio Theefoe, I( yi υ( τ holds LEMMA 5 Coside a scheduled li fx with iteded 0 sede xs ad eceive x Let I be the total itefeeces caused at x by siultaeously tasittig odes yi fo which τ ( yi = τ The followig holds: 0 τ τ I ( υ/ ( ( PROOF The poof of this lea is siila to that of Lea 44 i [0] The ai idea is that because the legths of the lis i the sae legth class diffe by at ost a facto of, accodig to a siple geoetic aea aguet, the deletig lis ust be bouded by a cetai ube The diffeece is that we chage the ig width fo ( μ ( f x to ( μ ε ( f x Ad oe ipotatly, the μ value is geatly educed due to the itoductio of the pocessig gai i the deoiato Thus the deletig lis i the sae legth class ae geatly educed Pluggig i the value of μ i lie of the ai algoith, the esults follow We oit the detailed poof because of the lac of space THEOREM 54 Fo a abitay tieslot t, all scheduled tasissios Et i t ae eceived successfully by the iteded eceives, ad thus the coputed schedule is coect PROOF Coside a scheduled li fx with iteded sede xs ad eceive x The aggegate itefeeces at this eceive x ca be calculated though Leas 5, 5 ad 5 By Leas 5 ad 5, we ow that fo all yi with τ ( yi > τ ad τ ( yi < τ, the itefeece I( yi is bouded by υ( τ Hece, because thee ae at ost odes i these sets, it holds that τ υ τ τ I( yi υ( = ( ( yi: τ τ( yi Theefoe the aggegate itefeece at x is τ τ τ τ I = ( υ/ ( ( + ( υ/ ( ( τ τ = ( υ/ ( ( Ad SINR at x is τ SINR υ ( ( fx / ( fx = N ( / ( / τ ( τ + υ Sice υ :=4N (lie of the ai algoith τ υ ( SINR = τ τ N + ( υ/ ( ( Fo this, we coclude that the coputed schedule is coect 5 Efficiecy Aalysis COROLLARY 55 I each tieslot, the deleted lis i the sae legth class i the NPAW algoith ae stictly fewe tha the deleted lis i the NPAN algoith PROOF This coclusio is fo the poof of Lea 5 LEMMA 56 I each tieslot, the deleted lis i diffeet legth classes i the NPAW algoith ae fewe tha o at ost equal to the deleted lis i the NPAN algoith PROOF Fo lie of the suboutie, o oe had, if the diffeece of the powe scalig factos betwee diffeet legth classes is the sae, because we have itoduced the pocessig gai as the deoiato i the base, the deleted lis ust be fewe tha its coutepat i NPAN O the othe had, sice ( / ( δi + ( /log( / / ( / + / =, ad sice ( δ / (4 i + ( /log(4 / (4 + / =, the deleted lis ust be at ost equal to its coutepat i NPAN THEOREM 57 The schedulig coplexity fo stog coectivity i UWB etwos is O(log( log PROOF Fist of all, accodig to the costuctio of the diected eaest eighbo foest F p i each phase of the ai algoith, the ube of the odes i the foest F p + (/ F p Hece at ost log diected eaest eighbo foests Fp exist util thee eais oly a sigle active ode (lie of the ai algoith Ad accodig to Coollay 55 ad Lea 56, the total ube of the deletig lis i each tieslot of the sae schedulig oud ust ot exceed O(log which is the esult of the NPAN algoith Hece, at least a factio of Ω (/log of the tasissio lis that eai to be scheduled i the th schedulig oud ca be siultaeously scheduled i a sigle tieslot The afte at ost x tieslots, the ube of eaiig odes that still eed to be scheduled /log is y ( / log x y e x, whee y is the iitial ube of odes that eed to be scheduled i F p ( y So whe x = l y log, the ube of eaiig odes i this schedulig oud is Based o this obsevatio, each schedulig oud (the suboutie Schedule( oly equies at ost O(log tieslots Ad accodig to lie 8 of ou ai algoith, thee ae at ost log( schedulig ouds; theefoe the total schedulig coplexity of this algoith is: T( S O(log log log( O(log( log 6 THE ENERGY COMPLEXITY OF WIRELESS SCHEDULING PROBLEMS We defie the eegy coplexity of the wieless schedulig poble to be the total eegy cost fo successfully schedulig all the tasissio equests to eet soe topological popety equieet THEOREM 6 Fo the stog coectivity schedulig algoith i aowbad etwos, the lowe boud of the eegy coplexity is ω( ; ad the uppe boud of the eegy coplexity is O (, whee is the ube of the odes PROOF I the NPAN algoith, oly lis i li legth class Sh log(4 + ca be scheduled siultaeously i the th schedulig oud ( is fo 0 to log(4, epeseted by the colus of Table Ad h is fo 0 to / log(4 (epeseted by the ows of Table I paticula, let s coside the li legth classes S ad S log(4 +, which ae the shotest legth class ad the logest legth class i the th schedulig oud, espectively 88

8 Accodig to Fig, suppose the legth class S is apped fo L u,we have u ; Ad suppose the legth class S log(4 + is apped fo L log(4 + v,we have v u Accodig to the powe scalig factoτ of thei algoith, legth class S has the highest τ value / log(4 ; ad legth class S log(4 + has the lowestτ value, of So accodig to the o-liea powe assiget schee i the algoith, the powe PS ( assiged to the lis i S has the popety /log(4 /log(4 ( (4 u ( (4 P S u + υ < υ u ( u+ υ PS ( < υ The powe PS ( log(4 + assiged to lis i S log(4 + has the log(4 + v popety PS ( log(4 + υ (4 ( ad log(4 + v+ log(4 + < υ v ( v + P S log(4 + PS ( (4 ( υ /(4 ( < υ /(4 Because 0 u v log(4, we have 0 u ( u+ υ υ PS ( < υ ( v /(4 ( log(4 P S v + υ + < υ /(4 (4 υ Fo this, ad because the si ode of the fial diected spaig tee tasits with the powe N l ax, which could be N, we get the lowe boud of the eegy coplexity fo the stog coectivity schedulig poble i aowbad etwos, which is ω(, ad the uppe boud of the eegy coplexity, which is O ( THEOREM 6 Fo the stog coectivity schedulig algoith i UWB (o ay spead-spectu etwos, the lowe boud of the eegy coplexity is still ω( ; but the uppe boud of the eegy coplexity is educed to O(, whee is the ube of the odes ad is the pocessig gai PROOF With ou ai algoith, oly lis i li legth class Sh log( + ca be scheduled siultaeously i the th schedulig oud ( is fo 0 to log(, epeseted by the colus of Table ; ad h is fo 0 to / log( (epeseted by the ows of Table I paticula, let s coside the li legth classes S ad S log( +, which ae the shotest legth class ad the logest legth class i the th schedulig oud, espectively Accodig to Fig, suppose the legth class S is apped fo L u, we have u ; ad suppose the legth class S log( + is apped fo L log( + v,we have v u Fo lie of the suboutie Schedule(, the legth class S has the highest τ value / log(,ad the legth class S log( + has the lowestτ value, of So accodig to the o-liea powe assiget schee i ou algoith, the powe PS ( assiged to the lis i S has the popety /log( u PS ( υ( ad /log( ( u+ PS ( < υ( u ( u+ υ PS ( < υ The powe PS ( log( + assiged to lis i S has the popety log( / log( + v PS ( log( + υ ( ( ad + log( + v+ log( + < υ ( + v ( + v+ P S log( + PS ( ( ( υ /( / ( < υ /( / Because 0 u v log(, we have 0 u ( u+ υ υ PS ( < υ v v υ /( P( S < υ /( ( + ( + + log( + υ ( Fo this, ad because the fial si ode tasits with the powe N/ l ax (lie of the ai algoith, which could be N/, we get the lowe boud of the eegy coplexity fo the stog coectivity schedulig poble i UWB etwos is ω(, ad the uppe boud of the eegy coplexity is O( Table Li legth classes schedulig (i ode i aowbad etwos (fo left to ight, fo top to botto S 0 Slog(4 S log(4 S log(4 S Slog(4 + S log(4 + S log(4 + S Slog(4 + log(4 S + S log(4 + Slog(4 Slog(4 S log(4 S Table Li legth classes schedulig i UWB etwos (fo left to ight, fo top to botto S 0 Slog( S log( S log( S Slog( + S log( + S log( + S Slog( + log( S + S log( / + Slog( Slog( S log( Fo these two theoes, we ca see that the polylogaithic schedulig coplexity coes at the expese of expoetial eegy coplexity The esults ca also be applied to the situatio i [] because they also adopt the NPAN algoith Copaed with aowbad etwos, by Theoe 6, we ca see that the uppe boud of the eegy coplexity ca be educed by a pocessig gai facto i UWB etwos 7 CONCLUSIONS I this pape, we geealize the schedulig coplexity i aowbad etwos to UWB etwos, ad educe it 4 fo O(log to O(log( log By cosideig the ipact of the abitay powe assiget o pai-wise tasissios schedulig, we explicitly show that whe soe ode distace fuctio is satisfied, thee does ot exist ay powe assiget fo siultaeous li schedulig, ad thus the schedulig coplexity caot be futhe ipoved via the eas of powe assiget Theefoe, the schedulig algoith ust tae full advatage of the powe assiget schees so that it ca siultaeously schedule as ay lis as possible without violatig the physical SINR odel Copaed to aowbad etwos, we show that thee is oe oo fo UWB etwos to tae full advatage of powe cotol to ipove the schedulig coplexity Besides the powe cotol liitatios, we also give a detailed aalysis o the powe cotol ipact o the pai-wise potocol itefeece odels Moe ipotatly, we explicitly pove that the polylogaithic schedulig coplexity is gaied at the expese of expoetial eegy coplexity i both aowbad etwos ad UWB etwos I ode to tu ou algoith ito a pactical etwo potocol, soe pobles eed to be solved fist, icludig the followig Although i UWB etwos, the uppe boud of the eegy coplexity ca be educed by a pocessig gai facto, the S 89

9 expoetial eegy coplexity lowe boud would ot chage Thus educig the eegy coplexity without sacificig the schedulig coplexity is a vey iteestig ad challegig tas To tae up this challege, a ew abitay powe assiget ay eed to be desiged Ou pape assues that the taffic dead of all lis is, but i soe ealistic wieless etwos, diffeet lis expeiece diffeet taffic deads Fo exaple, the odes ea the wieless oute i wieless esh etwos expeiece oe taffic deads tha the odes o the bode of the etwo [] Theefoe, devisig soe efficiet li schedulig to eet both topological equieets ad abitay li taffic deads is aothe vey challegig ad iteestig poble Costaitpogaig techiques ay be useful With the o-liea powe assiget, evey tasittig ode ust ow its ow powe scalig facto τ, which is based o soe global pictue, thus aig it difficult to ipleet the algoith i a distibuted ae To tae up this challege, ipleetig a clusteig algoith is a possible ethod 4 Ou algoith assues oe chael is used, but actually i MIMO etwos (eg, 80, a ode ca be equipped with ultiple adios ad opeate o ultiple chaels Thus extedig ou algoith to ulti-adio ulti-chael sceaios is a atual idea 8 ACKNOWLEDGMENTS We would lie to tha Pof Roge Wattehofe ad D Thoas Mosciboda fo thei valuable suggestios We also tha the aoyous eviewes fo thei coets to ipove the pesetatio of this pape 9 REFERENCES [] H Balaisha, C Baett, V S Ail Kua, M Maathe, ad S Thite The distace -atchig poble ad its elatioship to the MAC laye capacity of ad-hoc wieless etwos IEEE J Selected Aeas i Couicatios, (6: , August 004 [] A Behzad ad I Rubi O the pefoace of gaph-based schedulig algoiths fo pacet adio etwos I Poc of IEEE Global Telecouicatios Cofeece (GLOBECOM, 6:4-46, Sa Facisco, Decebe 00 [] G Ba, D Blough, ad P Sati Coputatioally efficiet schedulig with the physical itefeece odel fo thoughput ipoveet i wieless esh etwos I Poc Twelfth ACM Aual Iteatioal Cofeece o Mobile Coputig ad Netwoig (MOBICOM, Los Ageles, CA, US, Sept 006 [4] F Cuoo, C Matello, A Baiocchi ad F Capiotti Radio esouce shaig fo ad-hoc etwoig with UWB IEEE Joual o Selected Aeas i Couicatios, 0(9:7-7, Decebe 00 [5] M Gossglause ad D Tse Mobility iceases the capacity of ad-hoc wieless etwos IEEE/ACM Tasactios o Netwoig, 0(4: , August, 00 [6] Piyush Gupta ad P R Kua The capacity of wieless etwos IEEE Tasactios o Ifoatio Theoy, 46(:88-404, 000 [7] F Meye auf de Heide, C Schidelhaue, K Volbet, ad M Gueewald Eegy, cogestio ad dilatio i adio etwos I Poc of the 4th Aual ACM Syp o Paallel Algoiths ad Achitectues (SPAA, Wiipeg, Caada, August 00 [8] K Jai, J Padhye, V N Padaabha, L Qiu Ipact of itefeece o ulti-hop wieless etwo pefoace Wieless Netwos, (4: , 005 [9] S Kosie ad Z Gajic Sigal-to-itefeece-based powe cotol fo wieless etwos: a suvey, Dyaics of Cotiuous, Discete ad Ipulsive Systes B: Applicatios ad Algoiths, (:87-0, 006 [0] T Mosciboda ad R Wattehofe The coplexity of coectivity i wieless etwos I Poc 5th Aual Joit Cofeece of the IEEE Copute ad Couicatios Societies (INFOCOM, Baceloa, Spai, Apil 006 [] T Mosciboda, R Wattehofe, ad A Zollige Topology cotol eets SINR: the schedulig coplexity of abitay topologies I Poc 7th ACM Iteatioal Syposiu o Mobile Ad Hoc Netwoig ad Coputig (MOBIHOC, Floece, Italy, May 006 [] RC Qiu, H Liu, ad X She Ulta-widebad fo ultipleaccess couicatios IEEE Couicatios Magazie, 4(:80-87, Feb 005 [] B Raduovic ad J-Y Le Boudec Optial powe cotol, schedulig ad outig i UWB Netwos IEEE Joual o Selected Aeas i Couicatios, (7:5-70, Sept 004 [4] S Schid ad R Wattehofe Algoithic odels fo seso etwos I Poc 4th Iteatioal Woshop o Paallel ad Distibuted Real-Tie Systes (WPDRTS, Islad of Rhodes, Geece, Apil 006 [5] A Se ad M L Huso A ew odel fo schedulig pacet adio etwos Wieless Netwos, :7-8, 997 [6] R Wattehofe MACbeth: The thee witches of edia access theoy I st IEEE Iteatioal Woshop o Foudatio ad Algoiths fo Wieless Netwoig (FAWN, Pisa, Italy, Mach 006 [7] M Z Wi ad R A Scholtz Ulta-wide badwidth tiehoppig spead-spectu ipulse adio fo wieless ultiple-access couicatios IEEE Tasactios o Couicatios, 48(4:679-69,000 [8] X Yag ad G de Veciaa Iducig spatial clusteig i MAC cotetio fo spead spectu ad hoc etwos I Poc 6th ACM Iteatioal Syposiu o Mobile Ad Hoc Netwoig ad Coputig (MOBIHOC, Ubaa- Chapaig, IL, USA May