Establishing Wireless Conference Calls Under Delay Constraints

Transcription

1 Establishing Wirlss Confrnc Calls Undr Dlay Constraints Aotz Bar-Noy Grzgorz Malwicz Novbr 17, 2003 Abstract A prvailing fatur of obil tlphony systs is that th cll whr a obil usr is locatd ay b unknown. Thrfor, whn th syst is to stablish a call btwn usrs, it ay nd to sarch, or pag, all th clls that it suspcts th usrs ar locatd in, to find th clls whr th usrs currntly rsid. Th sarch consus xpnsiv wirlss links and so it is dsirabl to dvlop sarch tchniqus that pag as fw clls as possibl. W considr cllular systs with c clls and obil usrs roaing aong th clls. Th location of th usrs is uncrtain as givn by probability distribution vctors. Whnvr th syst nds to find spcific usrs, it conducts a sarch opration lasting so nubr of rounds th dlay constraint. In ach round, th syst ay chck an arbitrary subst of clls to s which usrs ar locatd thr. In this stting th probl of finding on usr with iniu xpctd nubr of clls sarchd is known to b solvd optially in polynoial ti. In this papr w addrss th probl of finding svral usrs with th sa optiization goal. This task is otivatd by th probl of stablishing a confrnc call btwn obil usrs. W first show that th probl is NP-hard. Thn w prov that a natural huristic is an / 1-approxiation solution. Ky words: Location anagnt, confrnc call, NP-hardnss, approxiation algoriths, convx optiization. 1 Introduction In th last dcad, w hav witnssd two trnds: incrasing availability of popl and incrasing availability of inforation. Mobil tlphony systs ak it possibl to talk with popl vn if thy ar not rsiding in prdtrind locations as is th cas with convntional phon systs. Intrnt sarch ngins allow usrs to accuratly and fficintly accss inforation stord on wbsits that hav fixd location. Whn inforation is stord on obil dvics and nds to b rtrivd nw challngs occur. An intrinsic fatur of currnt obil tlphony systs is that location of th dvics is uncrtain. A challng hr is to dsign sarch algoriths that fficintly rtriv inforation givn liitd knowldg about its location. Prliinary vrsion of this work appard in th Procdings of th 21st ACM Syposiu on Principls of Distributd Coputing PODC 02. This is a pr-print vrsion of th articl that will appar in Journal of Algoriths. Coputr and Inforation Scinc Dpartnt, CUNY-Brooklyn Collg, 2900 Bdford Avnu, Brooklyn, NY Th work of th first author was don during his visit to th AT&T Shannon Lab, Florha Park, NJ. Dpartnt of Coputr Scinc, Univrsity of Alabaa, 116 Housr Hall, Tuscaloosa, AL Th work of th scond author was don during his sur intrnship at th AT&T Shannon Lab, Florha Park, NJ. 1

2 In this papr w focus on sarch tchniqus that ar otivatd by th probl of stablishing a confrnc call in a wirlss phon syst. Our goal is to find a givn collction of obil dvics insid a wirlss syst so as to iniiz th usag of wirlss links and control th aount of ti spnt on th sarch. 1.1 Background and otivation Our probl of stablishing wirlss confrnc calls is otivatd by th stat-of-th-art of wirlss tchnology. Currntly dployd wirlss prsonal counication systs ar coposd of a st of bas stations connctd by a wird backbon ntwork. Th rang of radio transission of a bas station dtrins an ara calld a cll. Eachobil dvic that roas insid th syst counicats with othr dvics obil or stationary through bas stations using radio signals. Whn a obil dvic is within th rang of a bas station, th dvic is said to b locatd in th cll dtrind by th bas station. On of th ain coponnts of a wirlss syst is a location anagnt srvic [2, 20]. Its goal is to track th locations of dvics that ar ndd in ordr to stablish calls. In so cass th syst knows th currnt location of a obil dvic and thn th syst dos not nd to sarch for th dvic to stablish a call. This occurs, for xapl, whn th dvic is participating in an ongoing call and thrfor is rpatdly counicating with bas stations. Howvr, in gnral, th location of a dvic ay b unknown for xapl bcaus th dvic has bn switchd off by th usr in ordr to prsrv battry powr. This givs ris to th nd of locating obil dvics which is th ai of a location anagnt srvic. A natural asur of fficincy of a location anagnt srvic is its usag of wirlss links. This is otivatd by th facts that wirlss transission uss radio frquncis that ar scarc and so th links ay bco congstd, that wirlss transission is svral ordrs of agnitud slowr and or pron to rrors copard to wird transission, and that wirlss transission involvs th usag of liitd battry powr of obil dvics. Th location tracking probl xhibits an inhrnt tradoff btwn th usag of wirlss links bcasu of dvics rporting thir locations and th usag bcasu of th syst sarching for dvics s for xapl [4]. To illustrat, lt us assu that ach dvic, or trinal, rports its location by snding a ssag ovr a wirlss link to th bas station vry ti it ntrs a nw cll. This ans that th syst has up-to-dat inforation about th locations of trinals and whn call is to b stablishd th syst dos not nd to sarch for any dvic. Assu on th othr hand that trinals nvr rport thir locations. Thn, whn th syst is to stablish a call with a trinal, th syst ust find th cll whr th trinal is currntly locatd. Sinc th trinal is obil, th syst ay nd to sarch all clls in th syst, which is don by broadcasting ovr wirlss links, or paging, fro bas stations. Aftr th trinal has rcivd a sarch ssag, it rsponds to th bas station. Svral factors affct th usag of wirlss links du to rporting and paging. Th volu of updat ssags ay b high spcially whn trinals ar highly obil and frquntly cross th boundaris btwn clls. Paging also ay induc high usag of wirlss links spcifically whn larg nubr of clls nds to b pagd to find th dvic and th incoing call frquncy is larg. An fficint location anagnt syst nds to achiv a balanc btwn rporting and paging dpnding on ths factors. Prsntly ajor wirlss systs us a sipl tchniqu to balanc btwn rporting and paging traffic. In GSM MAP [9] usd in Europ and IS-41 [8] usd in North Arica standards th st of all clls is partitiond into substs calld location aras ach containing any clls but only a fraction of all clls in th syst. Any cll broadcasts th idntifir of its location ara 2

3 on a spcial radio channl. Whn a obil trinal finds out that it has ovd fro a cll to a cll blonging to a diffrnt location ara, it snds a wirlss signal rports to th bas station of th latr cll. This inforation is prsistd in a databas connctd to th backbon ntwork, that stors th ost rcntly visitd location ara for ach trinal. Whn a call to a obil trinal is to b stablishd, th syst broadcasts pags radio signals in paralll fro all bas stations in th location ara asking th trinal to rspond. This tchniqu rducs th nubr of clls pagd whn a call is to b stablishd, bcaus only th clls blonging to a location ara ar pagd. Howvr this cos at th cost that dvics ust rport whnvr thy cross boundaris btwn location aras. Th choic of location aras affcts th rporting traffic.g., [1, 5]. A tchniqu has bn dvlopd [11, 16, 17] to rduc th nubr of clls pagd insid a location ara during th sarch for a singl obil dvic, at th cost of incrasd aount of ti ndd for paging. Th tchniqu considrs a odl of a location ara with c clls whr th probability distribution of th dvic across th clls is givn. Thr ar svral thods for approxiating th distribution; s [15, 16] for xapls. Hr an arbitrary subst of clls of th location ara can b pagd in paralll in unit ti to s if th dvic is locatd thr. Also paging can b prford for d c units of ti. Paging is carrid out according to a d-round stratgy, whr in ach round i a subst S i [c] ={1,...,c} of clls is pagd until th dvic is found. It is assud that th dvic dos not ov during th sarch. Authors show how to fficintly find a stratgy that has at ost d rounds and that iniizs xpctd nubr of clls pagd. For xapl, if th dvic is uniforly distributd across all c clls, c vn, and w hav at ost 2 units of ti for paging, thn th bst paging stratgy is to pag half of th clls in th first round and whn th dvic is not found in ths clls thn pag th othr half in th scond round. This givs 3c/4 xpctd nubr of clls pagd a c/4 iprovnt ovr th tchniqu spcifid by GSM MAP and IS-41 standards. Whn th wirlss syst has to stablish a confrnc call btwn obil dvics, thn th location anagnt syst ay nd to find or than just on obil dvic. On can attpt to gnraliz th tchniqu of [11, 16, 17], so as to iniiz th xpctd usag of wirlss links during th sarch and at th sa ti control th aount of ti spnd on th sarch. Gnralization of th tchniqu is th subjct studid in this papr. 1.2 Modl and probl statnt Assu a odl of a wirlss syst with c clls and obil dvics. W assign all th dvics to clls by slcting a cll j for a dvic i with probability p i,j indpndntly fro othr dvics. W assu that ach probability is positiv and that p i, p i,c = 1, for all i. In this stting w can prob or pag an arbitrary group of clls in a unit of ti and dtct th location of dvics insid ths clls. W also assu that w can afford to prfor probing for d units of ti. Th dvics do not ov during th sarch. Th goal is to dvlop fficint algoriths that can find all th dvics within at ost d units of ti and that pag th last xpctd nubr of clls. In gnral on can considr two typs of algoriths: oblivious and adaptiv. Oblivious algoriths pag, in ach unit of ti, a prdtrind subst of clls. Adaptiv algoriths dcid which clls to pag during a unit of ti basd on th dvics that hav bn found so far. Each typ has its advantags. Adaptiv algoriths ay achiv lowr xpctd nubr of clls pagd to find all trinals. Oblivious algoriths hav vry low coputational cost during th sarch procss. In this work w focus on oblivious algoriths. Spcifically, a stratgy is a squnc S 1,...,S t of nonpty sts that partition [c] ={1,...,c}, whr t is calld th lngth of th stratgy. W call ach st a group. For a givn stratgy clls ar pagd in rounds such that in round r, 1 r t, all clls in th group S r ar pagd. Clls 3

4 in S r+1,...,s t ar not pagd whn all obil dvics hav bn found i.., if and only if r is th sallst round nubr for which all obil dvics ar locatd in clls S 1... S r. For ach stratgy w can find th xpctd nubr of clls pagd until all obil dvics hav bn found. W call this nubr th xpctd paging of th stratgy. W sk a stratgy that iniizs this xpctation. W call th probl of finding such stratgy th Confrnc Call probl. Confrnc Call Instanc: Nubr of obil dvics 1, nubr of clls c 1, positiv rational probabilitis p i,j of finding obil dvic i in cll j such that for all i, p i, p i,c = 1, and a axiu nubr of rounds d, 1 d c. Objctiv: Find a stratgy that iniizs xpctd paging for all stratgis of lngth at ost d. For xapl whn d = 1, th probl is trivial sinc all th clls ust b pagd in th first and only round. Th probl bcos intrsting whn d = 2, bcaus w ar looking for a subst of th clls to b pagd in th first round to iniiz th xpctd nubr of clls pagd until all dvics hav bn found if thy ar not locatd in th clls pagd in th first round, thn w nd to pag all th raining clls. Whn d = c, w ar looking for a prutation of th clls that dictats th squnc in which clls ar pagd that iniizs th xpctd nubr of clls pagd until all dvics hav bn found. 1.3 Contributions Th probl of sarching for on dvic = 1 is known [11, 16, 17] to b solvabl optially for any d in polynoial ti using a dynaic prograing algorith. In this papr w addrss th gnral probl of sarching for 1 dvics. Our contributions ar as follows: Coplxity of th probl. W show that th Confrnc Call probl is NP-hard bcaus it is NP-hard whn rstrictd to =2andd = 2. Our nontrivial rduction uss a spcial typ of th Partition probl. Th rsult stablishs a thrshold: th probl is asy whn =1orwhnd = 1, but bcos difficult whn =2andd = 2. W also show that th spcial cas of th Confrnc Call probl for any constant nubr of dvics 2and any constant nubr of rounds d 2isNP-hard. Constant factor approxiation. W prsnt a natural huristic. In this huristic w squnc th clls in a non-incrasing ordr of th xpctd nubr of dvics locatd in ach cll, and us dynaic prograing to find th partition of this squnc into d subsquncs. W show that th xpctd nubr of clls pagd by this huristic is at ost / 1 tis th inial nubr of clls pagd by any stratgy. W also show a lowr bound of 320/317 on th prforanc ratio of our huristic. 1.4 Papr organization Th rst of th papr is structurd as follows. In Sction 2, w prsnt so prliinaris. In Sction 3.1, w prov that th probl is NP-hard by showing that th probl rstrictd to two obil dvics and two rounds of paging is NP-hard. Thn, in Sction 3.2, w show that rstrictions to any fixd nubr of obil dvics and rounds ar also NP-hard. In Sction 4.1, w study 4/3 approxiation for th probl rstrictd to two dvics and two rounds, to donstrat ky idas of our analysis on a siplr probl. Thn, in Sction 4.2, w prsnt th ain rsult of th papr: an / 1-approxiation for th Confrnc Call probl. Finally, in Sction 5, w discuss futur work and rlatd work. 4

5 2 Prliinaris W show that th Confrnc Call probl is a cobinatorial optiization probl and rstat so xisting rsults fro convx optiization that w us throughout. In th la blow w show how to find th xpctd paging for a stratgy. La 2.1. Lt S 1,...,S t b a stratgy. Thn th xpctd nubr of clls pagd until all obil dvics ar found is t 1 EP = c S r+1 p i,j, j L r whr L r = S 1... S r. Proof. Th sarch lasts xactly r rounds whnvr not all obil dvics ar found in rounds 1,...,r 1, but all ar found on or bfor round r. LtF r b th vnt that all obil dvics ar found on or bfor round r. Obsrv that by indpndnc Pr [F r ]= j L r p i,j. Hnc Pr [paging lasts xactly r rounds] = Pr [F r ] Pr [F r 1 ]. Sinc if paging stops in round r w pag S S r clls, th xpctd nubr of clls pagd until all obil dvics ar found is EP = t S S r Pr [F r ] Pr [F r 1 ] = c Pr [F t ] which coplts th proof. t S r Pr [F r 1 ], It can b sn that for any stratgy of lngth t 1 <cthr xists a stratgy of lngth t with strictly lowr xpctd paging. Thus, aong all stratgis of lngth at ost d, a stratgy that iniizs xpctd paging ust hav lngth d. Now w rstat so basic notions and proprtis fro th thory of ultidinsional convx optiization [18] and linar algbra [14] that w us throughout. A subst D of spac R k is convx if for any two points x and y in D th sgnt xy is containd in D. Thatisλx +1 λ y D, for all x, y D, λ [0, 1]. For a convx st D, a function f : D R is strictly convx if for any x and y in D and λ 0, 1 w hav f xλ +1 λ y <fx λ + f y1 λ. A atrix H of siz k by k ovr R is positiv dfinit if for any x R k, x T Hx > 0, unlss x = 0. W rport a standard fact that charactrizs strict convxity of a function in trs of its Hssian th atrix of scond drivativs s [18] for a proof and or discussion. Thor 2.2 [18]. Lt D b an opn convx st in R k. Suppos that a diffrntiabl function f : D R has continuous scond partial drivativs in th st D. Lt th Hssian Hx b positiv dfinit for vry point x D. Thnf is strictly convx. Using th abov thor it is strightforward to show that th axiu of a strictly convx function ar attaind at th boundary of th doain as prsntd in th nxt la. La 2.3. Lt f : D R b a strictly convx and continuous function dfind ovr an opn, convx and boundd subst D R k,andlth baclosdsubstofd. Thn th axiu of f on H is achivd at a point that blongs to th boundary of H. 5

6 3 Coplxity of th probl This sction is ddicatd to showing that th Confrnc Call probl is NP-hard. W donstrat that it contains a subprobl that is NP-hard. This subprobl is th Confrnc Call probl rstrictd to =2andd = 2. Our rsult stablishs a thrshold bcaus th Confrnc Call probl is in P for = 1 [11, 16, 17] or for d = 1. In our proof w rduc a crtain variation of th Partition probl to th rstrictd Confrnc Call probl. Nxt w gnraliz th tchniqus usd in th proof and show that any rstriction of th Confrnc Call probl to constant 2 and d 2isalsoNP-hard. 3.1 Th cas whn d =2and =2 W show that th Confrnc Call probl is NP-hard by a transforation fro th following Partition probl which is NP-coplt as prsntd by Gary and Johnson [10] on pag 223. Partition Instanc: Anubrg divisibl by 2 and positiv intgr sizs s 1,...,s g. Objctiv: Dcid if thr is a subst P of [g] such that P = g/2 and k P s k = 1 2 sk. Th Partition probl can b rducd to th following variation of th Partition probl calld Quasipartition1. W oit th proof bcaus Quasipartition1 is on of probls fro th faily Quasipartition2 dfind in th following sction, and thr w show that any of ths probls is NP-coplt. Quasipartition1 Instanc: A list of non-ngativ rational sizs s 1,...,s c,whrc is divisibl by 3. Objctiv: Dcid if thr is a subst I of [c] such that I = 2 3 c and k I s k = 1 2 sk. W us th Quasipartition1 probl to show NP-hardnss of th Confrnc Call probl. Th following tchnical la that analyzs xtra of a function is usd in th proof. La 3.1. For any c 1 th function f :[0, 1] [0,c] dfind as fx, y =c y 1 3 y + x y x 2c achivs th global axiu at a singl point of th doain x =1/2 and y = 2 3 c. Proof. Lt us xtnd th doain of f to th st D = {x, y :0 x y +1and0 y c}. Th function f is continuous ovr closd and boundd doain D so it achivs axia ithr in th intrior or at th boundary of D. Lt us focus on th intrior first. Obsrv that f 3y x =0ifandonlyifx = 4c,soanxtru in th intrior can b achivd only whn this condition holds. Th first drivativ of th function f 3y 4c,y is zro only whn y = 2 3c,ory = 0. Thus if thr is xtru in th intrior it is for x =1/2, and y = 2 3 c, whr th function achivs th valu f1 2, 2 3 c= 4 27 c3 2 9 c2 + 1 Now lt us considr th valus of th function at th boundary. If y = c or y = 0 thn fx, y 0forx 0. If x =0thn f0,y y is zro only whn y =0ory = 2 3c. Sinc f0, 0 = 0, f0, 2 3 c= 4 27 c3 2 9 c2,andf0,c = 0, for th part of boundary whr x = 0 th function achivs valus strictly sallr than f 1 2, 2 3 c. Finally if x = y +1, 2 fy+1,y =4 3 y 2 c > 0, and so th axiu is achivd whn y =0orwhny = c, in which cass th function has non-positiv valus c and 0 rspctivly. 12 c. 6

7 Thus th function f dfind for a sallr doain 0 x 1, 0 y c achivs th global axiu at a singl point x =1/2, and y = 2 3 c. La 3.2. Th Confrnc Call probl rstrictd to =2and d =2is NP-hard. Proof. W show how to transfor th Quasipartition1 probl to th rstrictd Confrnc Call probl. Lt s 1,...,s c b any list of non-ngativ rational sizs such that c is divisibl by 3, and lt S = s s c.ifthrxistsi such that s i = S, thn thr is no partition. For th raindr of th proof assu that s i <S, for all i. W dfin an instanc of th Confrnc Call probl rstrictd to d = 2and = 2. Sinc thr ar only two obil dvics in this instanc for clarity w dnot th probabilitis that obil dvics ar locatd in cll i by p i and q i rspctivly. Lt th probabilitis p i and q i b dfind as p i = 1 c 1/ c + s i S, qi = 1 c 1 1 s i S,for1 i c. Not that th probabilitis ar positiv, that p i =1and q i = 1, and that th siz of th instanc of th Confrnc Call probl is polynoial in th siz of th instanc of th Quasipartition1 probl. W show that th Quasipartition1 probl for th squnc s 1,...,s c has an answr if and only if w can answr whthr th inial xpctd paging for th probabilitis achivs a crtain valu dfind blow. Bfor w procd with th proof, w not a lowr bound on th valu of xpctd paging for th instanc =2,d =2,p 1,...,p c, q 1,...,q c. Tak any stratgy and lt I [c], I = y, bth st of clls pagd in th first round. By La 2.1 th xpctd paging for this stratgy is EP = c c I j I = c s j p j c y c 1/2c 1 q j j I 1 3 2c y + j I s j S y j I Lt us dnot j I S by x, and obsrv that 0 x 1. Consquntly by La 3.1 th xpctd nubr of clls pagd until all obil dvics ar found is boundd fro blow by 1 1 EP LB = c c 1/2c 1 f 2, 2 3 c Suppos that P is a partition of cardinality 2 3 c of th squnc s 1,...,s c. Thn s j j P S =1/2, and so a stratgy that pags clls P in th first round, and clls [c] \ P in th scond round has xpctd paging qual to LB. Morovr, by La 3.1, this is th sallst valu possibl, so th answr to th instanc of th Confrnc Call is a stratgy with xpctd paging qual to LB. Suppos that th answr to th instanc of th Confrnc Call probl is a stratgy with xpctd paging qual to LB. LtP b th st of clls pagd in th first round. Obsrv that th cardinality of P ust b 2 3c bcaus othrwis by La 3.1 th xpctd paging of th stratgy would b strictly gratr than LB. For th sa rasons s j j P S ust b 1/2. Thus P is a partition of cardinality 2 3 c of th squnc s 1,...,s c. Corollary 3.3. Th Confrnc Call probl is NP-hard. s j S. 7

8 3.2 Th cas whn d 2 and 2 ar fixd It is intrsting to considr rstrictions of th Confrnc Call probl to constant d and bcaus in spcific applications on ay wish to hav an algorith that constructs paging stratgis for so fixd nubr of obil dvics and so fixd axiu dlay but for any location ara so that c and th probabilitis ar th input to th algorith. It turns out that such rstrictions ar also NP-hard and w show it in this sction by gnralizing tchniqus fro th prvious sction. Lt us bgin with a tchnical la that gnralizs La 3.1. Th la blow will b usd to show that th xpctd paging is iniizd whn th sts pagd in ach round hav spcific cardinalitis. La 3.4. Lt 2 d c, andi 1,...,i d satisfy constraints i j 0, i i d = c, andx 1,...,x d satisfy constraints x i 0, x x d =1. Lt 2, andα 1 = +1, α k = +1 α,for k 1 2 k d 1, b d = c, b k 1 = α k 1 b k,for2 k d, b 0 =0. Thn th function fi 1,...,i d,x 1,...,x d 1 d 1 = c i d r r r i k + x k i k cc 1 c k=1 k=1 k=1 r k=1 r k=1 x i 2 k k c is strictly gratr than 2c 12 d 1 c 4c 1 c +1 b r+1 b r b r, with quality only whn i j = b j b j 1,for1 j d, andx j = b j 2c,for1 j d 1 and x d =1 d 1 k=1 x k. Proof. Lt b r = r k=1 i k. Thn by th thor of arithtic and gotric ans w hav 1 1 r c k=1 i k + r k=1 x k r k=1 i k r k=1 x c b r k 2, and so f>c 1 d 1 cc 1 b r+1 b r b r 2c = c 2c 12 d 1 c 2 4c 1c +1 b r+1 b r b r, with quality only whn r k=1 x k = br 2c, for all 1 r d 1. Our goal now is to find an uppr bound on th su d 1 b r+1 b r b r undr constraints 0 b 1... b d = c. Sin th doain for b 1,...,b d is boundd and closd, and d 1 b r+1 b r b r is a continuous function of b i s, it achivs th axial valu at a point of th doain. W first show that th axiu ust b achivd in th intrior of th doain. Lt us tak a point p fro th boundary and us two actions to find a point in th intrior with strictly largr su. Action 1 Lt k b th largst indx for which b k =0sothatk<d. W can ak th su strictly largr by taking b k = b k+1 2. Aftr applying this action at ost d tis w hav that b i > 0 for all i. Action 2 Lt k b th sallst indx for which b k = b k+1.ifk =1 thn b 1 = b 1 + b 1 1 b 2 b 1 < 0 and so w can ak th su largr by slightly rducing b 1.If1<k<dthn b k = b k 1 b k + b 1 k b k+1 b k < 0, bcaus b k 1 <b k. Again w can incras th su by slightly rducing b k. Aftr applying this action at ost d tis w obtain a point fro th intrior that has th valu of th su strictly gratr than th valu of th su for th boundary point p. 8

9 To coplt th proof it is nough to show that thr is xactly on point in th intrior for which all partials vanish. Lt us invstigat whn b 1 =... = b k =0. Ifk =1thn b 1 = b1 1 b 2 b 1 b 1 vanishs if and only if b 1 = α 1 b 2,forα 1 = +1. Using this quation for 2 k d 1whavthat b k = b k +b 1 k b k+1 b k +b k 1 = b k +b 1 k b k+1 b k +α k 1 b k = b 1 k bk+1 b k +α k 1 b k b k vanishs if and only if bk = α k b k+1,forα k = +1 α. This k 1 stablishs a rcursiv forula for b i s, and rcall that by assuption b d = c. Obsrv that α i s ar onotonically incrasing, +1 = α 1 <α 2 <... < α d 1 < 1, and consquntly b i s ar also incrasing 0 <b 1 <...<b d 1 <b d = c. Thusb 1,...,b d dfin a uniqu point in th intrior of th doain whr all partials vanish. This coplts th proof. In th raindr of th sction w assu that 2andd 2arconstants. Lti j = r j c. Obsrv that thn th nubrs r 1,...,r d and x 1,...,x d dfind in th statnt of th La 3.4 ar so rational nubrs that dpnd only on d and. LtM b th last coon ultipl of th dnoinators of all r j s. For any c that is a ultipl of M th corrsponding nubrs i 1,...,i d ar natural nubrs. W dfin a Multipartition probl paratrizd with M, th fractions r 1,...,r d,andx 1,...,x d. Multipartition Instanc: Anubrc = M k, forsonaturalk, and non-ngativ rational sizs s 1,...,s c. Objctiv: Find a partition P 1,...,P d of [c] such that P j = r j c and k P j s k = x j s k. Using ssntially th sa argunt as in La 3.2 togthr with La 3.4 instad of La 3.1 w can rduc th Multipartition probl to th Confrnc Call probl rstrictd to th and d. La 3.5. Th Confrnc Call probl rstrictd to th 2 and d 2 is hardr than th Multipartition probl. Our goal now is to show that th Multipartition probl is hardr than an NP-coplt probl. W show this using two rductions. W first dfin a variation of th Partition probl and thn show that it can b rducd to th Multipartition probl. Lt π b a prutation on [d] which sorts th squnc of x i s in a non-incrasing ordr x π1... x πd. Lt u b th indx of th sallr of r πd 1 and r πd or πd if thy ar qual, and lt v b th othr indx. Quasipartition2 Instanc: Anubrn = M r u + r v h, for so natural h, and non-ngativ rational sizs s 1,...,s n. Objctiv: Dcid if thr a subst P of [n] such that P = M r v h and k P s k = xv x sk u+x v. La 3.6. Th Multipartition probl is hardr than th Quasipartition2 probl. Proof. Without loss of gnrality lt us assu that π is th idntity prutation bcaus w can always sort th squncs x j, r j,andi j at th bginning of th proof. Lt ŝ 1,...,ŝ n b an instanc of th Quasipartition2 probl. W dfin an instanc of th Multipartition probl for c = n r u+r v in a fw stps. First w lt s j = ŝj ŝk x d 1 + x d, for 1 j i d 1 + i d. Lt s b th axiu nubr which is sallr or qual to any of s 1,...,s n and also sallr or qual to any diffrnc x j x j+1 for j for which this diffrnc is non-zro. Nxt for ach x j,1 j d 2, w dfin i j additional sizs such that th first big is qual to x j si j 1 2c, and th raining i j 1 sall sizs ar qual to s 2c. Obsrv that this instanc can b constructd in ti polynoial to th siz of th input squnc. 9

10 k P If th instanc of th Quasipartition2 probl has a positiv answr thn x v = ŝk ŝk x u + x v for so subst of [n] of cardinality Mr v h. Thus k P s k = x v and k [n]\p s k = x u, and sinc sk = x x d = 1 a ultipartition can b found. Suppos that a ultipartition has bn found for th instanc s 1,...,s c.obsrvthatforany 1 j d 2 th first siz constructd for x j is so larg that it can only fit into a partition P k of siz x j. Th raining spac in th partition is so sall that it can only accoodat xactly i j 1 sall sizs s 2c. Consquntly th cardinality of P k ust b i j. Aftr possibly swapping partitions P j with P k w can assu that P j = i j. W can carry out this procss for j =1,j =2, andsoonuntilj = d 2. Consquntly xactly two partitions on of siz x d and th othr of siz x d 1 contain rspctivly i d 1 and i d sizs and ths ust b all th n sizs s 1,...,s n. As a rsult thr xists a subst P of [n] of cardinality ax{i d,i d 1 } such that k P s k = x v,and k [n] aftr ultiplying both sids by ŝk x d 1 +x d w s that th instanc ŝ 1,...,ŝ n of th Quasipartitio2 probl has a positiv answr. To coplt th squnc of rductions and show NP-hardnss of th rstriction of th Confrnc Call probl it is nough to show that th Quasipartition2 probl is NP-coplt. W will show a rduction fro th Partition probl dfind arlir which asks if hr is a subst of half of th sizs that sus up to half of th total su of th sizs. La 3.7. Th Quasipartition2 probl is NP-coplt. Proof. Lt us first assu that x v >x u and lt Ŝ =ŝ 1,...,ŝ g b an instanc of th Partition probl. W dfin an instanc of th Quasipartition2 probl with n = M r u + r v h sizs, g for h = 2 2Mr u. This n is larg nough so that M r v h 1 M r u h 1 g 2. Lt ū = Mr u h 1 g/2 0, and v = Mr v h 1 g/2 0, and U = {g +1,...,g +ū}, and V = {g +ū +1,...,g+ū + v}. Ltp = log 2 ŝ ŝ g +1. W dfin a list S = s 1,...,s n as: s k =ŝ k +2 p,for1 k g, ands k =0,fork U W. Thn w rscal ths sizs so that thy and two additional sizs s n 1 = xv 1 3 xu x u+x v and s n = 2 3 x u x u+x v su to 1 that is, w tak S = g k=1 s k andthnlts k bco s k S 1 s n 1 s n, for 1 k g. W show that th instanc of th Partition probl has a positiv answr if and only if th instanc of th Quasipartition2 probl has an answr. Th forward iplication is a sipl consqunc of th abov dfinitions. Lt us invstigat th opposit iplication. Suppos that thr xists a subst P of cardinality M r v h such that k P s k = xv x u+x v. Obsrv that s n 1 is strictly largr than th siz of th partition [n] \ P and so s n 1 ust blong to P. Morovr th raining spac in th partition P is too sall to accoodat s n and so s n ust b a br of th partition [n] \ P. But thn th raining spac in ach partition is qual to xactly 1 3 x u x v+x u, and so thr xists a subst of M r v h 1 sizs othr than th s n 1 and th s n that su up to half of th total su of ths othr sizs. Sinc th sizs s 1,...,s g hav a suand with factor 2 p, thr ust xist a subst I of [g] of cardinality g/2 that sus up to half of g k=1 ŝk, and so th instanc of th Partition probl has a positiv answr. If x u >x v thn th proof is obtaind utatis utandis. If x u = x v thn th raining spac in ach partition is 1/6. Thus th rsult follows. Obsrv that whn M =3,r u =1/3, r v =2/3, x u = x v =1/2 th Quasipartition2 probl bcos th Quasipartition1 probl, and so th lattr is NP-coplt by th abov la. Th squnc of rductions shown abov lads to th following thor. Thor 3.8. Th rstriction of th Confrnc Call probl to fixd 2 and d 2 is NP-hard. 10

11 4 Constant factor approxiation In th prvious sction w showd that finding an optial stratgy for th Confrnc Call is an NP-hard probl. Thrfor, this sction focuss on finding a polynoial ti stratgy that approxiats th xpctd paging of an optial stratgy. Intuitivly, a paging stratgy that finds all dvics and pags fw clls on avrag, first pags clls that hav larg chancs of finding all obil dvics in th clls. To raliz this intuition, w squnc th clls in a non-incrasing ordr of th su of probabilitis of finding all th obil dvics in th clls. In particular, th first cll in th squnc is a cll j that axiizs p 1,j p,j and th last cll in th squnc is a cll j that iniizs p 1,j p,j. Thn w us dynaic prograing to find th bst stratgy that pags c clls in d rounds according to th squnc. W can show that if an optial paging stratgy has high xpctd paging, thn any stratgy is a good approxiation, whil if an optial stratgy has low xpctd paging, thn th stratgy that our dynaic prograing algorith finds also has low xpctd paging. Th proof of th prforanc of th approxiation algorith has two stps. W considr th faily of all stratgis of lngth d that pag clls according to th squnc. First w show that thr xists a stratgy T in this faily whos xpctd paging is gratr than th optial xpctd paging by at ost / 1 factor. Howvr, stratgy T nds to prob, in ach round, th sa nubr of clls as an optial stratgy S. Sinc w do not know th sizs of groups for stratgy S, w cannot straightforwardly and fficintly find T. In th scond stp, w show that with th hlp of dynaic prograing, w can fficintly find a stratgy G in th faily that iniizs xpctd paging across all stratgis in th faily. Thus stratgy G is no wors than stratgy T, which coplts th proof. Th xposition of th prforanc analysis argunts is split into two parts for clarity of prsntation. W first discuss a spcial cas of th Confrnc Call probl with two obil dvics and th axiu dlay of two, as th proofs ar siplr to follow but at th sa ti thy xplify th ky idas. Thn w show how to gnraliz ths proofs to obtain an approxiation rsult. 4.1 Approxiation of th spcial cas =2and d =2 Fro La 3.2 w know that this spcial cas is NP-hard. Hr w show that thr is a 4/3 approxiation solution that can b found in Oc tiando1 spac. Lt us prsnt a roadap for th argunt bfor w prsnt foral proofs. Considr an optial paging stratgy S 1,...,S d and lt us focus on what has happnd up until a round inclusiv. If th xpctd nubr of dvics found by th optial stratgy until th round is low th assuption x<1 in La 4.3, thn th optial stratgy will unlikly find all dvics until this round this is anifstd by th bound that contains x/2 2 in La 4.3, and so th sarch will likly continu and th xpctd paging will b high. Thus paging any subst of th sa nubr of clls up until this round should b good nough to achiv good approxiation. A probl ay occur whn th optial stratgy pags fw clls arly in th sarch and has high chancs of finishing th sarch until thn. This is bcaus th xpctd paging of th optial stratgy will b low, and so w should ak sur that our stratgy will also hav low xpctd paging. Spcifically, w will nd to nsur that our stratgy also initially pags fw clls and has high chancs of finishing th sarch thn if our stratgy pags rlativly any clls or has rlativly low chancs of finding all dvics, thn its xpctd paging ay b too high copard to th low xpctd paging of th optial stratgy. W can show that this probl dos not actually occur, if our stratgy pags clls in th ordr fro th on that has th ost dvics on avrag to th on that has th last dvics on 11

12 avrag. Indd, if th xpctd nubr of dvics found by th optial stratgy until th round is high th assuption x 1 in La 4.3, thn it is possibl that th optial stratgy will hav high chancs of finding all dvics until th round. Th chancs cannot b arbitrary high, howvr, as thy ust b boundd fro abov by a function of th xpctd nubr dvics found until th round again th bound that contains x/2 2 in La 4.3. If our stratgy has pagd, up until th round, clls that axiiz th xpctd nubr of dvics found, w can show that our stratgy will also hav high chancs of finding all dvics until this round, and ths chancs will b boundd fro blow by th xpctd nubr of dvics that th optial stratgy has found Proposition 4.1. Th last stp in th argunt is to show that th ratio of th xpctd paging of our stratgy and th optial stratgy is boundd wll no attr how any clls ar pagd in ach round and no attr what th actual valus of th xpctd nubr of dvics found by th optial stratgy up until ach round ar. Controlling th ratio is particularly iportant in th cas whn th optial stratgy alost crtainly finds all dvics aftr fw clls hav bn pagd. In so sns, th xpctd paging of a good approxiation stratgy should quickly or convxly approach a lowr bound on th xpctd paging of an optial stratgy, whn th chancs of finding all dvics approach 1 fro blow, and th nubr of clls that nd to b pagd to attain ths chancs approachs 1 fro abov Proposition 4.2. Proposition 4.1. Lt 1 x 2, andvariablsa i and b i satisfy constraints a i,b i 0, a i + b i 1, a 1 + a 2 x b 1 + b 2,thn a 1 + b 1 a 2 + b 2 x 1. Proof. Notic that by dcrasing so a i th valu of th product a 1 + b 1 a 2 + b 2 can only b rducd. Thus it is nough to bound th product fro blow with th additional constraint a 1 + a 2 = x b 1 + b 2. But thn a 1 + b 1 a 2 + b 2 =x a 2 + b 2 a 2 + b 2, and w can rwrit this xprssion as x v v for v = a 2 + b 2. By th assuption that a 2 + b 2 1whavthat v 1, and by th assuptions that a 1 + a 2 = x b 1 + b 2 andthata 1 + b 1 1whavthat v = a 2 + b 2 = x a 1 + b 1 x 1. Not that for th doain x 1 v 1, th xprssion x v v tratd as a function of v is iniizd at th boundary of th doain whn v = x 1, or whn v = 1. This yilds a lowr bound of x 1. = 2 3s>0, th function is a strictly convx func- Proposition 4.2. Lt 0 <s c and 1 x 2, thnc s x c s x 2. Proof. Sinc c 2 s x 1 4 x 2 3 c s x 2 2 tion of x, and so it is boundd fro abov by th gratr of th valus achivd for x =1orforx =2. For x = 2 its valu is c s 4 3 c s < 0, and for x = 1 its valu is c 4 3 Thus th function is boundd fro abov by 0, and th rsult follows. 2 c 1 4 s c c =0. Suppos that S 1,S 2 is a stratgy that iniizs xpctd paging aong all stratgis of lngth 2. W do not know how to find this stratgy, but dspit this lt us assu that w know S 1 = s 1. Lt us pick s 1 clls T 1 which axiiz th xpctd nubr of dvics locatd in th clls i.., for any cll j in T 1 and any cll j not in T 1 w hav p 1,j +p 2,j p 1,j +p 2,j. Thn th following la shows that th stratgy T 1,T 2 has dsird approxiation, whr T 2 =[c] \ T 1. La 4.3. EP T /EP S 4/3. Proof. Rcall that by th slction of th stratgy T 1,T 2 th st T 1 contains clls j that axiiz th su p 1,j + p 2,j. W divid th clls in S 1 T 1 into thr sts B = S 1 T 1, A = T 1 \ B, and 12

13 C = S 1 \ B. For ach obil dvic i w dfin a i to b th probability that th dvic is in th clls A i.., a i = j A p i,j. Siilarly w dfin b i = j B p i,j and c i = j C p i,j. By th thor of th arithtic and gotric ans [12] w hav 2 j S 1 p i,j x 2 2, whr x =b 1 + c 1 +b 2 + c 2, 0 x 2. Rcall that for ach dvic i w hav p i, p i,c =1, consquntly a i + b i 1. By th choic of T 1 w hav that a 1 + a 2 c 1 + c 2 = x b 1 + b 2. W divid th analysis of th approxiation ratio into two cass. If x<1 thn w bound th probability that all dvics ar found by th stratgy T 1,T 2 in th first round by 0. Rcall that s 1 is th nubr of clls pagd in th first round. Thus EP T EP S c c c s 1 2 j S 1 p i,j c c c s c c c 1 4 = 4 3. For th scond cas assu that 1 x 2. By Proposition 4.1 w can bound th ratio as EP T c c s 1x 1 EP S c c s 1 x 2. 2 It rains to show that th nurator is boundd fro abov by 4 3 this follows fro Proposition 4.2, which coplts th proof. tis th dnoinator. But Not that w do not nd to know th siz of S 1 bcaus w can afford to valuat xpctd paging of a stratgy T 1,T 2 constructd for s 1 = 1, thn valuat it for s 1 = 2, and so on until s 1 = c 1, and pick th stratgy that iniizs xpctd paging. Thus w can find a stratgy that is a 4 3 approxiation without th knowldg of th siz of th st of clls pagd in th first round by a two-round stratgy that iniizs xpctd paging. 4.2 Gnral cas This sction is dvotd to gnralizing th idas givn in th prvious sction. For this purpos w ploy tchniqus fro ultidinsional convx optiization and dynaic prograing. Spcifically w show an / 1-approxiation algorith for th Confrnc Call probl. In th raindr of th sction w assu that all clls ar ordrd so that for all 1 j<j c p 1,j p,j p 1,j p,j. W considr a faily F of stratgis such that a stratgy S 1,...,S d fro th faily has th proprty that any group S j, j 2, contains clls gratr than all prcding groups in th stratgy i.., for all 1 i<j d, for all i S i and j S j,whav i <j Existnc of approxiat solution In this sction w donstrat that th faily F contains a stratgy T that has th xpctd nubr of clls pagd until all obil dvics ar found largr by at ost th factor of 1 than th xpctd paging of a stratgy of lngth d that iniizs th xpctation. Th ky ida bhind th proof is to obsrv that if an optial stratgy yilds sall chancs of finding all obil dvics until a round thn th stratgy T is trivially good nough, whil whn ths chancs ar high thn th stratgy T ust also hav high chanc of finding all obil dvics. For th analysis of prforanc of our huristic w nd two tchnical inqualitis, which w show nxt. Ths inqualitis gnraliz Proposition 4.1 and Proposition 4.2 givn in th prvious sction. W prov th or gnral facts using tchniqus fro ultidinsional convx optiization. 13

14 La 4.4. Lt 2, 1 x, andvariablsa i and b i satisfy constraints a i,b i 0, a i + b i 1, a i x b i thn a i + b i x +1. Proof. Notic that th valu of th product a i + b i can only b rducd by dcrasing so a i. Thus it is nough to bound th product fro blow with an additional constraint a i = x b i. Th proof is by induction on. Th bas cas for = 2 was shown in Proposition 4.1. For th inductiv stp tak +1 a i + b i =x, such that a i s and b i s ar non-ngativ, that a i +b i 1and that x + 1. Lt us ak two obsrvations that will allow us to ak us of th inductiv assuption. Sinc a i +b i 1, th su a i + b i isatost, andsox a +1 + b +1. By assuption, x, so 1 x 1 x a +1 + b +1. Suarizing ths obsrvations, w hav 1 x a +1 + b +1, and so w can us th inductiv assuption, with x a +1 + b +1 instad of x, to bound th product by +1 a i + b i =a +1 + b +1 a i + b i a +1 + b +1 x a +1 + b Rcall that a +1 +b +1 rangs btwn x and 1, and so w can look at th rightost xprssion in th inquality abov as a function of a +1 + b +1 inthsawaywdidwhnwwrproving Proposition 4.1: th iniu is achivd th boundary, ithr whn a +1 +b +1 is qual to x or to 1. In both cass th xprssion is qual to x, as dsird, and so th rsult follows. La 4.5. Lt x 1,...,x k b variabls satisfying 1 x i, forso 2. Lts 2,...,s d, k d 1, b strictly positiv and satisfy s s d c, forsoc 0 thn k c s r+1 x r +1 k xr s k s d c s r+1. 1 Proof. W considr a function k fx 1,...,x k =c s r+1 x r +1+ c + 1 k xr s k s d s r+1 + dfind ovr doain H =[ 1,] k i.., k-dinsional cub, and show that its axiu is at ost 0, which will coplt th proof. For this purpos lt us xtnd th doain of f to D = 1 ɛ, + ɛ k,forso0<ɛ<1. Obsrv that D is an opn and convx subst of R k,andthatf has continuous scond partial drivativs in D. Also for vry point x D, th Hssian atrix Hx off is diagonal, and that th ntris on th diagonal ar strictly positiv 2 f x x 2 r 2 > 0. Thus Hx is positiv r dfinit, and so by Thor 2.2 f is strictly convx. Sinc H is a closd subst of D, La 2.3 nsurs that th valus of f on H ar boundd fro abov by th valus achivd at th boundary of H. Aftr ths prliinary obsrvations lt us prov th statnt of th la by induction on k. For th bas cas assu that k = 1. Thr ar two boundary cass. Whn x 1 = thn fx 1 =c s = s r c + s 2 + s s d 14 = s s d c 1 0.

15 Whn x 1 = 1thn fx 1 =c + 1 c + s s s d s s d c 1 0. Thus f is boundd fro abov by 0. For th inductiv stp w considr th valus of f at th boundary of H. Du to sytricity of th function and constraint conditions w can focus on two cass x k = and x k = 1. Whn x k = thn k 1 f = c s r+1 x r +1+ c + 1 k 1 = c s r+1 x r +1+ c + 1 k 1 k 1 xr s r+1 + sk s r+1 xr + s k s d + s k s d and w can us th inductiv hypothsis to bound f fro abov. For th scond cas considr x k = 1. Thn k 1 f c s r+1 x r +1+ k 1 xr s k s d c + s r+1 +, 1 and w can us th inductiv hypothsis again to bound f fro abov. Suppos that w hav gussd th sizs of groups pagd in ach round by a stratgy that iniizs xpctd nubr of clls pagd until all obil dvics ar found, and lt s 1,...,s d b ths sizs i.., th stratgy pags s r clls in round r, s s d = c. In th nxt la w donstrat that whn w us our huristic with th sa sizs of groups pagd in corrsponding rounds thn th xpctd paging ay b gratr by at ost factor La 4.6. Lt s 1,...,s d b positiv intgrs such that s s d = c. Lt S 1,...,S d b any stratgy for which S r = s r,andlt T 1,...,T d b th stratgy in F that has groups of th sa cardinality T r = s r. Thn th ratio of xpctd nubr of clls pagd until all obil dvics ar found is EP T EP S 1. Proof. Whn = 1 thn th ratio EP T EP S 1, which was studid by othr rsarchrs [11, 16, 17]. Our proof focuss on cas whn 2. Rcall that by th construction th st Z r = T 1... T r contains clls j that axiiz th su p 1,j p,j. Lt us dfin U r = S 1... S r. For any r w divid th clls in U r Z r into thr sts B r = Z r U r, A r = Z r \ B r,andc r = U r \ B r. For ach obil dvic i w dfin a i,r to b th probability that th dvic i is in th clls A r i.., a i,r = j A r p i,j. Siilarly w dfin b i,r = j B r p i,j and c i,r = j C r p i,j. By th thor of th arithtic and gotric ans [12] w hav j U r p i,j x r, whr x r = b i,r + c i,r, 0 x r. Rcall that for ach obil dvic i w hav p i, p i,c =1,consquntlya i,r + b i,r 1. By th choic of Z r w hav that a i,r c i,r = x r b i,r. If x r < 1 thn w can bound th product T r+1 j Z r p i,j fro blow by 0, and th product S r+1 j U r p i,j fro abov by S r+1 1 Sr ,

16 If 1 x r thn by La 4.4 w can bound th product T r+1 blow by T r+1 x r + 1, and th product S r+1 Rcall that by La 2.1 EP T = c T 2 j Z 1 p i,j... T d j Z d 1 p i,j EP S c S 2 j U 1 p i,j... S d. j U d 1 p i,j j Z r p i,j fro j U r p i,j fro abov by S r+1 x r. Not that 0 x 1 x 2... x d 1. So w can partition th x i s into ths strictly sallr than 1 and ths btwn 1and: sayx 1,...,x k 1 < 1, and ach of x k,...,x d 1 is btwn 1and. Thus w can bound th ratio fro abov by EP T EP S c s k+1 x k s d x d 1 +1 c s xk xd 1 k+1... sd s s k. 1 It rains to show that th nurator is boundd fro abov by 1 tis th dnoinator. If non of th x i s is btwn 1and i.., k = d, thn th bound is trivial, so assu that k d 1. But thn, whn w rna variabls and apply La 4.5, w conclud that th nurator is nvr gratr than 1 tis th dnoinator. Thus th xpctd paging of th stratgy T 1,...,T d is at ost 1 tis th xpctd paging of th stratgy S 1,...,S d. This la idiatly tlls us that for any fixd nubr of rounds d w can find an approxiat paging stratgy in polynoial ti bcaus thr ar Oc d 1 stratgisthatsatisfy s s d = c bcaus any slction of valus for so d 1 variabls dtrins th valu of th raining on variabl. Howvr this thod is not satisfactory whn d and c grow, and so w sk a or scalabl solution. W achiv this by using dynaic prograing Finding approxiat solution using dynaic prograing Now our goal is to find a stratgy in F that iniizs xpctd paging aong all stratgis in F. For this purpos w dvlop a dynaic prograing algorith that gnralizs an approach givn by [11]. Tak any 1 l k c and considr a class of stratgis of lngth l that ay pag th last k clls and only ths clls during th l rounds. Lt S 1,...,S l b a stratgy fro this class so that S 1... S l = {c k +1,...,c 1,c}. Lt P b a rando variabl qual to th nubr of clls pagd by this stratgy givn that at last on obil dvic is locatd aong th last k clls thn P is at last 1. In th class w can find a stratgy that iniizs th xpctd valu of P and dnot this valu by El, k. Obsrv that th valu of Ed, c is xactly th inial xpctd paging across all stratgis in F. W nd to show that w can fficintly find a stratgy that achivs Ed, c, and as th first stp w prov a rcursiv forula for finding th valu of El, k. La 4.7. Th valu of El, k, 1 l k c, isqualto: E 1,k=k, whn 1 k c, and whn 2 l k c. in 1 x k l+1 { x E l, k = c k+x j=1 p i,j c k j=1 p i,j E l 1,k x }, 16

17 01 approxiation in: c,, d, p i,j, 02 1 i c, 1 j ; 03 out: g r,1 r d 04 array 05 X[1,...,d;1,...,c], F [1,...,c] 06 E[1,...,d;1,...,c], S[1,...,] 07 for i =1to 08 S[i] =0 09 for j =1toc 10 for i =1to 11 S[i] =S[i]+p i,j 12 F [j] =1 13 for i =1to 14 F [j] =F [j] S[i] 15 for k =1toc 16 E[1,k]=k 17 X[1,k]=k 18 for l =2tod 19 for k = l to c 20 E[l, k] = 21 for x =1tok l v = x + 23 if v<e[l, k] thn 24 E[l, k] =v 25 X[l, k] =x 26 w = c 27 for l = d downto 1 28 g d l+1 = X[l, w] 29 w = w X[l, w] 1 F [c k+x] 1 F [c k] E[l 1,k x] Figur 1: Algorith for finding th sizs g 1,...,g d of groups for a stratgy that achivs approxiatation factor. 1 Proof. W nd to show that in ach of th two quations abov th xprssion on th right sid of th qual sign is qual to th xprssion on th lft sid of th sign. Th first quation is trivial bcaus any stratgy of lngth 1 that pags xactly k clls during 1 round givn that thr is at last on obil dvic aong ths k clls has xpctd nubr of cll pagd qual to k. In ordr to show th scond quation w prov two inqualitis. First w show that th xprssion on th right sid of th qual sign is nvr gratr than El, k. Lt S 1,...,S l b a stratgy that ay pag th last k clls and only ths clls during l rounds and that iniizs th xpctd valu of th nubr of clls P pagd during th l rounds givn that at last on obil dvic is locatd in on of th last k clls. Obsrv that 1 S 1 k l + 1 bcaus non of th groups can b pty. Lt A b th vnt that at last on obil dvic is locatd aong th last k clls, and lt B b th vnt that at last on obil dvic is locatd aong th last k S 1 clls. Obsrv that th xpctd valu of P is qual to But El, k =Exp [P ]= S 1 + Pr [ B A ] El 1,k S 1. Pr [ B A ]= 1 c k+ S 1 j=1 p i,j c k / 1 j=1 and so th xprssion on th right sid of th scond quation is at ost El, k. To show that th xprssion on th right sid of th scond quation is nvr sallr than El, k obsrvthatforanyx, 1 x k l + 1, th valu of th xprssion is qual to th xpctd nubr of clls pagd by a stratgy of lngth l that ay pag th last k clls and only ths clls, x of which in th first round, givn that at last on obil dvic is aong th last k clls. Hnc th valu of th xprssion can nvr b sallr than th valu El, k thatisth iniu of th xpctation. This coplts th proof of th la. During th calculation of Ed, c w can find th sizs of groups for a stratgy G 1,...,G d that has xpctd paging qual to Ed, c s Figur 1 for a psudocod of an algorith. Lins 07 p i,j, 17

18 through 14 calculat th probabilitis that all dvics ar found by round r, for r = 1,...,c.Lins 15 through 25 valuat th rcursiv forula givn in La 4.7. Lins 26 through 29 find th sizs of groups for th approxiation stratgy that wr calculatd during th valuation of th rcursiv forula. This lads to th ain thor of th papr. Thor 4.8. For any instanc of th Confrnc Call probl th stratgy G 1,...,G d has xpctd paging EP G 1 EP MIN, and it can b found in O c + dc ti and O + dc spac. Proof. Tak any instanc of th Confrnc Call probl. By La 4.6 th faily F contains a stratgy T 1,...,T d that has xpctd paging at ost 1 tis th inial xpctd paging of any stratgy of lngth d. By La 4.7 th dynaic prograing finds a stratgy G 1,...,G d that iniizs xpctd paging across all stratgis in th faily. Thus th xpctd paging of th stratgy G 1,...,G d is at ost 1 tis th xpctd paging of any stratgy of lngth at ost d. W rark that th dynaic prograing approach dvlopd in this sction allows us to find a stratgy that iniizs xpctd paging across a faily of stratgis that pag clls in any prdfind squnc. 4.3 Lowr bound on prforanc Finally w giv a lowr bound on th prforanc ratio of our huristic. Considr an instanc of th Confrnc Call probl with =2,c =8,andd =2. Ltp 1,1 = 2 7, p 2,1 = p 1,7 = p 1,8 =0, and th raining probabilitis ar st to 1 7. By sipl cas analysis w can show that th bst stratgy pags clls 2 through 6 in th first round and achivs xpctd paging, whil th huristic chooss to pag clls 1 through 5 in th first round and achivs xpctd paging. This stablishs a lowr bound of on th prforanc ratio of our huristic. It is possibl to show ssntially th sa lowr bound without rlying on bad ti braking: w can prturb th probabilitis by a tiny ɛ, thus forcing th huristic to choos th clls 1 through 5, and only slightly affcting th lowr bound. 5 Futur work and discussion In this sction w list so opn probls and rport so of our rsults on our work in progrss and thn ntion so rlatd work. Our approxiation solution has a sall 1.58 approxiation factor and cannot b bttr than W bliv that our algorith actually achivs a lowr approxiation factor. Is thr a bttr approxiation algorith or vn an approxiation sch? So far w know an approxiation sch for a subclass of th probl. Hr w assu that th st of probabilitis {p i,j :1 i, 1 j c} can b covrd by a constant nubr of ral intrvals of constant lngth. This allows us to sarch th spac of solutions xhaustivly in polynoial ti. Thr is an altrnativ way to show th NP-hardnss rsult prsntd in this papr. W can solv th probl with paratrs c, 2,d using a solution to c +1,,d+ 1. Th ida is to add on cll, st th probability of 2 dvics to 1 for this xtra cll, and scal th probabilitis of ach of th raining two dvics by th factor 1 a such that a 1 1/c 2. This nsurs that 18

19 thr ar vry high chancs that all dvics will b locatd in th xtra cll. Consquntly, an optial stratgy will pag only th xtra cll in th first round, and thn follow an optial stratgy for two dvics, d rounds and c clls, for th raindr of th sarch. In this papr w considr only oblivious stratgis i.., whr th st of clls to b probd in ach round is fixd in advanc. It is intrsting to also considr adaptiv stratgis which dtrin, in ach round, th st of clls to pag dpnding on th dvics found in arlir rounds. Our NP-hardnss rsult applis to adaptiv stratgis as wll sinc for d = 2 any adaptiv stratgy is oblivious. W do not know if th probl of finding an optial adaptiv stratgy for fixd 2 and fixd d 2 is NP-hard. On can asily xtnd th huristic prsntd in Sction 4 to for an adaptiv stratgy whr, in ach round, w calculat conditional probabilitis and basd on thir valus w dtrin th group of clls to pag in th nxt round using th algorith prsntd in Figur 1. Th analysis of th prforanc ratio of th rsulting algorith stands as an opn probl. Thr ar othr intrsting typs of sarchs to considr. A dual probl to th Confrnc Call probl is th Yllow Pags probl in which w ar sarching for on out of possibl dvics. W showd an -approxiation algorith basd on a huristic that is diffrnt fro th on considrd in this papr. W also know that th huristic considrd in this papr dos not offr constant factor approxiation. A probl that gnralizs th two probls is th Signatur probl whr w ar looking for any k dvics out of th dvics. Th confrnc call probl is th cas whr k = and th yllow pags probl is th cas whr k = 1. Solutions to th signatur probl can b applid to th task of finding k anagrs out of anagrs to sign a docunt. Anothr intrsting dirction is to xtnd th odl. For xapl du to bandwidth liitations in ral systs it ay b rasonabl to assu that at ost a fixd nubr of b clls can b pagd at any unit of ti. Hr w obsrv that our approxiation rsult gnralizs to yild rsults in this odl: w can us La 4.6 to show th xistnc of an approxiat stratgy, and in La 4.7 w can liit th rang for x accordingly and considr only E l, k whnk bl to find an appropriat stratgy. Anothr possibl xtnsion of th odl is to assu that whn thr is a dvic at a cll and w pag th cll w do not always find out if th dvic is thr siilar assuptions hav bn considrd [19], and that th chancs of finding out dcras with th incras of th nubr of dvics in th cll. This odls collision of rspons signals to th paging signal ittd fro th bas station. 5.1 Rlatd work Burkard t al. [6] rviw a cobinatorial optiization probl rlatd to th Confrnc Call probl. Th probl is calld th Quadratic Assignnt Probl [13] and it is forulatd as follows. Givn two sytric atrics A =a i,j andb =b i,j ofsizc by c with non-ngativ ntris find a prutation π that axiizs i,j a i,jb πi,πj. It can b shown that on can us a solution to th Quadratic Assignnt Probl to solv th Confrnc Call probl for two obil dvics. If d is constant thn th rduction is polynoial ti. Thr ar svral rsults fro Sarch Thory [19] that can b usd in location anagnt. Sarch Thory dals with finding a singl objct locatd aong a st of clls as givn by a probability distribution. Sarching consists of a squnc of lookups of th clls. Thr is a cost associatd with looking up a cll. It is assud that with so probability a lookup ay not find an objct in a cll vn though th objct is locatd thr. Th goal is to dcid which clls and whn to look up to axiiz chancs of finding th objct undr constraints on th total cost of sarch. Awduch 19

20 t al. [3] considr a odl whr fixd groups of clls ar pagd and a probability is givn that paging a cll dos not dtct a dvic vn though it is locatd at th cll. Authors show how to apply rsults fro Sarch Thory to iniiz th xpctd nubr of clls pagd. Anothr rlatd sarch probl is th Cobinatorial Group Tsting probl [7]. Using th trinology of our papr, th odl of CGT assus that a paging of a subst of clls calld its in CGT rturns whthr a obil dvic calld dfctiv in CGT xists aong th clls pagd. Thus th sarch ay nd to continu rcursivly by rpaging sallr and sallr sts in ordr to locat th singl cll whr th obil dvic xists is locatd. In th odl that w study in this papr w assu that a paging of any subst of clls rturns all th dvics that ar locatd in on of th clls that hav bn pagd and for ach of th dvics that hav bn found th cll whr th dvic is locatd. Hnc no rcursiv rpaging is ndd. Acknowldgnts. Th work of th scond author would not b possibl without gnrous support and ncouragnt fro his anagr Michal Mrritt whil an intrn at th AT&T Shannon Lab, who h would lik to thank. Th scond author would also lik to thank his advisor Alx Shvartsan for vry valuabl discussions. Th authors would lik to thank David Johnson, Jff Lagarias, and S. Muthukrishnan for discussion, and th anonyous PODC 03 and Journal of Algoriths rviwrs for conts that iprovd th prsntation of th papr. W would lik to particularly acknowldg a Journal of Algoriths rviwr for th carful rading of th papr, sharing constructiv criticis with us, and ncouraging us to xplain th intuition bhind tchnical rsults. Rfrncs [1] Abutalb, A., Li, V.O.K.: Location updat optiization in prsonal counication systs. Wirlss Ntworks, Vol [2] Akyildiz, I.F., McNair, J., Ho, J.S.M., Uzunalioglu, H., Wang, W.: Mobility Managnt in Nxt Gnration Wirlss Systs. IEEE Procdings Journal, Vol [3] Awduch, D.O., Gaylord, A., Ganz, A.: On Rsourc Discovry in Distributd Systs with Mobil Hosts. MOBICOM [4] Bar-Noy, A., Ksslr, I., Sidi, M.: Mobil Usrs: To Updat or not to Updat? Wirlss Ntworks journal, Vol [5] Bar-Noy, A., Ksslr, I.: Tracking Mobil Usrs in Wirlss Ntworks. IEEE Transactions on Inforation Thory, Vol [6] Burkard, R.E., Cla, E., Pardalos, P.M., Pitsoulis L.S.: Th Quadratic Assignnt Probl. In Handbook of Cobinatorial Optiization D.-Z. Du and P.M. Pardalos, ds., Vol. 2, Kluwr Acadic Publishrs [7] Du, D.-Z., Hwang, F.K.: Cobinatorial Group Tsting and Its Applications 2nd Edition. World Scintific, Sris on Applid Mathatics Vol [8] EIA/TIA: Cllular radio-tlcounications intrsyst oprations. EIA/TIA Tchnical Rport IS-41 Rvision C