College of William & Mary Department of Computer Science

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1 1 Techncal Repor WM-CS College of Wllam & Mary Deparmen of Compuer Scence WM-CS Onlne Vecor Schedulng and Generalzed Load Balancng Xaojun Zhu, Qun L, Wezhen Mao, Guha Chen March 5, 2012

2 Onlne Vecor Schedulng and Generalzed Load Balancng Xaojun Zhu, Qun L, Wezhen Mao and Guha Chen Sae Key Laboraory for Novel Sofware Technology, Nanjng Unversy, Nanjng, P. R. Chna Deparmen of Compuer Scence, he College of Wllam and Mary, Wllamsburg, VA, USA Emal: Absrac We gve an approxmaon-preservng reducon from vecor schedulng problem (VS) o generalzed load balancng problem (GLB). The reducon brdges exsng resuls of he wo problems. Specfcally, he hardness resul for VS holds also for GLB and any algorhm for GLB can be used o solve VS. Based on hs, we ge wo new resuls. Frs, GLB does no have consan approxmaon algorhms ha run n polynomal me unless P = NP. Second, here s an onlne algorhm (vecors comng n an onlne fashon) ha solves VS wh approxmaon bound e log(md), where e s he narual number, m s he number of parons and d s he dmenson of vecors. The algorhm s borrowed from GLB leraure and s very smple n ha each vecor only needs o mnmze he L ln(md) norm of he resulng load. However, s unclear wheher hs algorhm runs n polynomal me, due o he rraonal and non-neger naure of ln(md). We address hs ssue by roundng ln(md) o he nex neger. We prove ha he resulng algorhm runs n polynomal e log(e) ln(md)+1 me wh approxmaon bound e log(md) + whch s n O(ln(md)). Ths mproves he O(ln 2 d) bound of he exsng polynomal me algorhm for VS. I. INTRODUCTION Schedulng wh coss s a very well suded problem n combnaoral opmzaon. The radonal paradgm assumes sngle-cos scenaro: each job ncurs a sngle cos o he machne ha s assgned o. The load of a machne s he oal cos ncurred by he jobs serves. The objecve s o mnmze he makespan, he maxmum machne load. Vecor schedulng and generalzed load balancng exend he scenaro n dfferen drecons. Vecor schedulng assumes ha each job ncurs a vecor cos o he machne ha s assgned o. The load of a machne s defned as he maxmum cos among all dmensons. The objecve s o mnmze he makespan. Vecor schedulng s a mul-dmensonal generalzaon of he radonal paradgm. I fnds applcaon n mul-dmensonal resource schedulng n parallel query opmzaon [1], where s no suable o represen dfferen requess (such as CPU, memory, nework resource, ec.) as a sngle scalar. To solve vecor schedulng, here are hree approxmaon soluons [1]. Two of hem are deermnsc algorhms based on derandomzaon of a randomzed algorhm, wh one provdng O(ln 2 d) approxmaon 1, where d s he dmenson The work was done when he frs auhor was vsng he College of Wllam and Mary. 1 In hs paper, e denoes he naural number, ln( ) denoes he naural logarhm, and log( ) denoes he logarhm base 2. of vecors, and he oher provdng O(ln d) approxmaon wh runnng me polynomal n n d, where n s he number of vecors. The hrd algorhm s a randomzed algorhm, whch assgns each vecor o a unformly and randomly chosen paron. I gves O(ln dm/ ln ln dm) approxmaon wh hgh probably, where m s he number of parons (servers). For fxed d, here exss a polynomal me approxmaon scheme (PTAS) [1]. A PTAS has also been proposed for a wde class of cos funcons (raher han max) [2]. Generalzed load balancng s recenly nroduced o model he effec of wreless nerference [3][4]. Each job ncurs coss o all machnes, no maer whch machne s assgned o. The exac cos ncurred by a job o a specfc machne s dependen on whch machne he job s assgned o. The load of a machne s he oal cos ncurred by all he jobs, nsead of jus he jobs serves. Ths model s well sued for wreless ransmsson, snce, n wreless nework, a user may nfluence all APs n s ransmsson range due o he broadcas naure of wreless sgnal. To solve he generalzed load balancng problem, he curren soluon s an onlne algorhm, adaped from he recen progress n onlne schedulng on radonal model [5]. The soluon, hough provdes good approxmaon, s raher smple: each job selecs he machne o mnmze he L norm of he resulng loads a all machnes where s a parameer. To avod confuson, we keep he wo erms job and machne unchanged for generalzed load balancng, whle refer o job and machne n he vecor schedulng model as vecor and paron respecvely. We make wo conrbuons. Frs, we presen an approach o encode any vecor schedulng nsance by an nsance of generalzed load balancng problem. Ths encodng mehod brngs he recen progress n generalzed load balancng no he vecor schedulng doman, and akes he hardness resul of vecor schedulng o generalzed load balancng problem. Second, he exsng soluon o GLB needs o compue he L ln l norm (l s he number of machnes), bu s unclear wheher hs norm can be compued n polynomal me. We elmnae hs uncerany by roundng ln l o he nex neger, guaraneeng polynomal runnng me. In addon, we prove ha he approxmaon loss due o roundng s small. We conclude hs secon by he followng wo defnons.

3 A. Vecor Schedulng We are gven posve negers n, d, m. There are a se V of n raonal and d-dmensonal vecors p 1, p 2,..., p n from [0, ) d. Denoe vecor p = (p 1,..., p d ). We need o paron he vecors n V no m ses A 1,..., A m. The problem s o fnd a paron o mnmze max 1 m A where A = j A p j s he sum of he vecors n A, and A s he nfny norm defned as he maxmum elemen n he vecor A. For he case m n, here s a rval opmal soluon ha assgns vecors o dsnc parons. Therefore, we only consder he case m < n. For ease of presenaon, we gve an equvalen neger program formulaon. Le x j be he ndcaor varable such ha x j = 1 f and only f vecor p s assgned o paron A j. Then A j x j p k 1 k d The vecor schedulng problem can be rewren as mn max x j,k x j p k subjec o x j = 1, j x j {0, 1}, B. Generalzed Load Balancng, j (VS) Ths formulaon frs appears n [4]. We reformulae wh slghly dfferen noaons. There are a se M of ndependen machnes, and a se J of jobs. If job s assgned o machne j, here s non-negave cos c jk o machne k. The load of a machne s defned as s oal cos. The problem s o fnd an assgnmen (or schedule) o mnmze he makespan, he maxmum load of all he machnes. Ths problem can be formally defned as follows. mn max x k x j c jk,j subjec o x j = 1, j x j {0, 1}, J J, j M (GLB) J M where x {0, 1} s he assgnmen marx wh elemens x j = 1 f and only f job s assgned o machne j. The wo consrans requre each job o be assgned o one machne. II. ENCODING VECTOR SCHEDULING BY GENERALIZED LOAD BALANCING We frs creae a GLB nsance for any VS nsance, hen prove her equvalence. A las, we dscuss he hardness of GLB and exend he VS model. A. Creang GLB Insances Comparng VS o GLB, we can fnd ha hey manly dffer n he subscrps of max and. Our consrucon s nspred by hs observaon. Gven as npu o VS he vecor se V and m parons, we consruc he GLB nsance as follows. We se he jobs J = V. For each paron A j and s k-h dmenson, we consruc a machne, denoed by he par (j, k). Thus, he consruced machne se M s {(j, k) j = 1, 2,..., m and k = 1, 2,..., d}. We refer o a machne as a par of ndces so ha we can map he machne back o s correspondng paron and dmenson easly. For a machne = (j, k) where M, we refer o he paron j as 1, and he dmenson k as 2,.e., = ( 1, 2 ). We can see ha here are oally d machnes ( ncluded) correspondng o he same paron as he machne. We denoe [] as he se of hese machnes,.e., [] = {(j, 1), (j, 2),..., (j, d)}, where j = 1. Among hese d machnes, we selec he frs one (j, 1) as he anchor machne, denoed by, such ha a vecor chooses paron A j n VS f and only f he correspondng job chooses n he new GLB problem. The ncurred cos c s of job o machne f chooses machne s s defned as p 2 f s = (1) c s = f s [] s (2) 0 f s / [] (3) where (1)(2) are for he suaon where s and correspond o he same paron. They force a job o selec only he anchor machnes. (3) s for he suaon where s and correspond o dfferen parons. In hs case, here s no load ncrease. The resulng GLB nsance s defned as VS-GLB: mn max x,s x sc s subjec o x s = 1, s x s {0, 1}, J J, s M (VS-GLB) To avod he confuson wh he general GLB problem, we nenonally use dfferen noaons x, s and. The noaon s kep snce s n 1-1 correspondence wh he vecors n VS. As an example, consder he case when d = 1. All vecors n VS have only one elemen, and here s only one machne n VS-GLB represenng a paron n VS. The objecve of VS becomes max j x jp 1. On he oher hand, he objecve of VS-GLB s max x c x p 1. Snce any machne corresponds o a dsnc paron A j, smply changng subscrps shows ha he wo problems are equvalen. For he case when d > 1, he proof s much nvolved, whch we delay o Secon II-B. Theorem 1. The consrucon of VS-GLB can be done n polynomal me.

4 Proof: An nsance of VS needs Ω(nd) bs. The consruced VS-GLB nsance has n jobs, md machnes and n(md) 2 coss. Snce m < n, all hree erms are polynomals n n and d. The heorem follows mmedaely. The followng heorem shows ha he consruced VS-GLB problem s equvalen o s correspondng VS problem. Le T be a posve consan. Theorem 2. There s a feasble soluon x o VS wh objecve value T f and only f here s a feasble soluon x o VS-GLB wh he same objecve value T. Ths heorem shows ha VS and s correspondng VS- GLB have he same opmal value. In addon, any c- approxmaon soluon o VS-GLB, afer ransformaon, s also a c-approxmaon soluon o VS, vce versa. We prove hs heorem n Secon II-B. I s worh menonng ha VS-GLB s a specal nsance of GLB. Snce VS-GLB s convered from VS, VS s a specal nsance of GLB, whch mples ha VS should have approxmaon algorhms a leas as good as GLB. Unforunaely, on he conrary, he leraure shows beer approxmaon algorhm for GLB han ha for VS. Hence, s worh applyng algorhms of GLB o VS. B. Proof of Equvalence We frs sudy he properes of feasble soluons o VS- GLB n Lemma 1 and Lemma 2, and hen prove Theorem 2. Lemma 1. Gven a feasble soluon x o VS-GLB yeldng objecve value T, for any J, we have 1) s s, x s = 0; 2) j such ha for s = (j, 1), x s = 1. Proof: For 1), suppose x s = 1 for some s wh s s. Then x s c ss = > T, conradcng wh max,s x sc s = T. For 2), snce s x s = 1, here exss some s such ha = 1. Due o 1), we mus have s = s. x s Lemma 2. Gven a machne, a job, and a feasble soluon x o VS-GLB yeldng objecve value T, we have s x s c s = x p 2. Proof: Recall ha [] = {( 1, 1), ( 1, 2),..., ( 1, d)}. We have x sc s = x sc s + x sc s s s [] s/ [] = s [] x sc s (4) = x c (5) = x p 2 (6) where (4) s due o (3), (5) s due o Lemma 1, and (6) s due o (1). Wh he wo lemmas, we can now prove he equvalence. Proof of Theorem 2: = Gven a feasble soluon x o VS, consruc a feasble soluon x o VS-GLB as follows. Se x s = x s 1 and all ohers o be 0. We frs show ha x s a feasble soluon o VS-GLB. Obvously, x s an neger assgnmen. We wll check ha s x s = 1. Observe ha x s = 0 f s s. We only need o consder m machnes (1, 1), (2, 1),..., (m, 1). Snce x s a feasble soluon o VS, hen for any V, here exss one and only one paron A j such ha x,j = 1. Our ransformaon ses x s = 1 where s = (j, 1). So s x s = 1. Second, we prove ha he objecve values of he wo feasble soluons are equal. max x sc s x sc s (7),s j,k s [] x c (8) x 1 p 2 x j p k, where (7) s due o ha c s = 0 f s / [], and (8) s due o our assgnmen of x ha x s = 0 f s s. = Gven x for VS-GLB, consruc x for VS as follows. Se x j = x s where s 1 = j. We show ha x s a feasble soluon o VS. Due o Lemma 1, for any, here exss one s such ha x s = 1 and s = s. Therefore, here exss one j such ha x j = 1. On he oher hand, here canno be wo js boh wh x j = 1, oherwse x s no a feasble soluon o VS-GLB. For he objecve value, we have max x j p k x 1 p 2 j,k =(j,k) x p 2 x sc s, (9) where (9) s due o Lemma 2. Ths complees our proof. C. Inapproxmably for GLB I has been proved ha no polynomal me algorhm can gve c-approxmaon soluon o VS for any c > 1 unless NP = ZP P [1]. Combnng hs resul wh Theorem 2, we have he followng heorem. Theorem 3. For any consan c > 1, here does no exs a polynomal me c-approxmaon algorhm for GLB, unless NP = ZP P. Proof: Snce VS-GLB s a specal nsance of GLB, any c-approxmaon algorhm for GLB can be used o oban c- approxmaon soluon o VS-GLB. By Theorem 2, any c- approxmaon soluon o VS-GLB s also a c-approxmaon soluon o he correspondng VS. Therefore, he approxmaon algorhm s also a c-approxmaon algorhm for VS, a conradcon.,s

5 We can oban a sronger resul by relaxng he assumpon N P ZP P o P N P. (I s a relaxaon because P ZP P N P.) Ths can be done by examnng he napproxmably proof for VS [1]. The napproxmably proof reles on he resul ha no polynomal me algorhm can approxmae chromac number o whn n 1 ɛ for any ɛ > 0 unless NP = ZP P. Recenly, has been proved ha no polynomal me algorhm can approxmae chromac number o whn n 1 ɛ for any ɛ > 0 unless P = NP [6]. Thus, we can change he assumpon NP ZP P o P NP safely. Theorem 4. For any consan c > 1, here does no exs a polynomal me c-approxmaon algorhm for GLB, unless P = NP. D. Exendng o generalzed VS Our consrucon of VS-GLB and proof can be easly exended o a general verson of vecor schedulng. In he curren VS defnon, all machnes (parons) are dencal so ha any job (vecor) ncurs he same vecor cos o all machnes. The machnes can be generalzed o be heerogeneous so ha each job ncurs a dfferen vecor cos o dfferen machnes. Formally, job ncurs vecor cos p (j) o machne j f s assgned o machne j. The formulaon and ransformaons can be slghly changed as follows. In he neger program formulaon of VS, change he objecve o max j,k x jp (j) k. Change p 2 n equaon (1) o be p (1) 2. For Lemma 2, change x p 2 o x p(1) 2. I can be verfed ha he proof of Theorem 2 s sll vald wh mnor modfcaons. The onlne algorhm adoped laer s also vald for hs general verson of vecor schedulng. For smplcy, we manly focus on he orgnal VS model. III. ONLINE ALGORITHM FOR VS Based on Theorem 2, we can solve VS by s correspondng VS-GLB. We revew he approxmaon algorhm [3] for GLB, and hen modfy o solve VS. Gven a GLB nsance and a posve number, he algorhm [3] consders jobs one by one (n an arbrary order) and assgns he curren job o a machne o mnmze he L norm 2 of he resulng load of all machnes. Specfcally, suppose jobs are numbered as 1, 2,..., n, he same as he consdered order. Suppose he load of machne k afer jobs 1, 2,..., 1 are assgned s l 1 k. Then job s assgned o he machne arg mn j ( k (l 1 k + c jk ) ) 1/. The above opmzaon problem can be solved by ryng each possble machne. Durng he opmzaon, he compuaon of he las sep of L norm, ( ) 1/, can be omed. In addon, because he algorhm does no requre he order of jobs and each job s assgned once, can be mplemened n an onlne fashon. Ths algorhm was orgnally proposed 2 L norm of a vecor x = (x 1, x 2,..., x ) s defned as ( x )1/. for he radonal load balancng problem [5], and recenly exended o he GLB problem [3]. The parameer conrols he approxmaon rao of he algorhm, as shown n he followng lemma. Lemma 3 ([5], [3]). Mnmzng L norm gves ln(2) l1/ approxmaon rao o solve GLB where l s he number of machnes. Seng = ln l yelds he bes approxmaon rao e log l. However, s unclear wheher he compuaon of L ln l can be done n polynomal me. We consder hs ssue laer. A. Adapng o VS To apply he above algorhm o VS, we can frs solve VS-GLB and ransform he soluon o VS. Ths process can be smplfed by omng he ransformaon beween VS and VS-GLB. Recall ha he algorhm s o assgn vecors one by one. Consder a vecor p n VS. To solve VS-GLB, hs vecor should choose a machne o mnmze he L norm of he resulng load. Due o he consrucon of VS-GLB, hs vecor can only choose from he anchor machnes, oherwse, he resulng L norm would be nfne (defnely no he opmal choce). Thus, hs s equvalen o pckng from he correspondng parons n VS. Afer he assgnmen of any number of vecors ha leads o parons A 1, A 2,..., A m, he L norm of he load of machnes n VS-GLB s, n fac, equal o f () (A 1,..., A m ) = ( A A m ) 1/ where A j = k A j p k Suppose he assgnmen of vecors p 1, p 2,..., p 1 leads o parons A 1, A 2,..., A m. Le f (),j be L norm of he resulng load f vecor p chooses paron A j,.e., f (),j = f () (A 1,..., A j {p },..., A m ). Then, accordng o he algorhm, vecor p should be assgned o he paron arg mn f (),j. j The procedure s descrbed n Algorhm 1. For each ncomng vecor, only needs o execue Lnes 5-9. Algorhm 1 wh = ln(md) s an e log(md) approxmaon algorhm o solve he correspondng VS-GLB. Thus, we have he followng resul due o Theorem 2. Lemma 4. Algorhm 1 wh = ln(md) s an e log(md) approxmaon algorhm o solve VS. However, s unclear wheher Algorhm 1 wh = ln(md) can ermnae whn polynomal me. The algorhm requres he compuaon of x ln(md) for some x. Frs, he number ln(md) s rraonal, hus canno be represened by.

6 Algorhm 1: Vecor Schedulng Inpu: m, he number of parons; d, he dmenson of each vecor; p 1, p 2,..., p n, he n vecors o be scheduled;, he norm Oupu: A 1,..., A m, he m parons 1 begn 2 for j from 1 o m do 3 A j ; 4 for from 1 o n do 5 f A j, A j = hen 6 A j A j {p }; 7 else 8 fnd j o mnmze f (),j ; 9 A j A j {p }; polynomal bs o acheve arbrary resoluon. Second, even hough we can approxmae by a raonal number wh accepable resoluon, he number x may sll be rraonal, where s he raonal approxmaon o. For example, when = 1.5, here are los of values of x such ha x 1.5 are rraonal. Though we can sll approxmae by a raonal number, s complcaed o heorecally analyze wheher he approxmaon rao sll holds and how he runnng me ncreases wh respec o raonal number approxmaon accuracy. Ths problem has no been addressed n leraure. Our soluon s o round ln(md) o he nex neger ln(md) and compue he L ln(md) norm. Ths guaranees polynomal runnng me, bu causes he loss of approxmaon rao. We show n he followng ha he loss s very small. B. Guaraneeng Polynomal Runnng Tme To deal wh he rraonal number ssue, we round ln(md) o he nex neger ln(md). In he followng, we analyze he resulng approxmaon rao. Theorem 5. Le l be he number of machnes. Mnmzng L ln l norm gves e log(l) + e log(e) ln l+1 approxmaon rao o solve GLB. Proof: Ths resul s obaned from Lemma 3 by performng calculus analyss. Le g(x) = x ln(2) l1/x. Consder he dervave of g, ( g (x) = l1/x 1 ln l ). ln 2 x For x ln l, holds ha g (x) 0 so ha he funcon g(x) s monooncally ncreasng. Snce ln(l) ln(l) ) ln(l) + 1, we have g( ln(l) ) g(ln l) g(ln l + 1) g(ln l). In addon, consder he wo pons (ln l, g(ln l)) and (ln l + 1, g(ln l + 1)). Due o Langrange s mean value heorem n calculus, here exss ξ [ln l, ln l + 1] such ha g(ln l + 1) g(ln l) = g (ξ). Snce ξ ln l, we have l 1/ξ e. Addonally, ( ξ ln l ) + 1, so 1 ln l ξ 1 ln l+1. Therefore, g (ξ) e ln(2) 1 ln l ξ. We have e log(e) ln l+1 g( ln l ) g(ln l) g(ln l + 1) g(ln l) e log(e) ln l + 1 Noe ha g(ln l) = e log(l). Ths complees our proof. Ths heorem holds for general GLB problem, such as he one consdered n [3] [5] and [4]. Of course, holds for VS- GLB as well. To have an nuon on he loss, we plo he wo approxmaon raos wh respec o he number of machnes n Fgure 1. We can see ha he loss s small. approxmaon rao before roundng afer roundng number of machnes Fg. 1. Approxmaon rao loss due o roundng. Before roundng, he approxmaon rao s y = e log(x) and becomes y = e log(x) + e log(e) ln(x)+1 afer roundng. We have he followng corollary due o Theorem 5. Corollary 1. Wh = ln(md), Algorhm 1 s an e log(md) + e log(e) ln(md)+1 approxmaon algorhm o VS, and runs n polynomal me. The polynomal runnng me can be shown by he followng analyss. We assume = ln(md) f no specfed. The man me consumng sep s o mnmze f (),j over j for gven. We can om he compuaon of he ouer 1/ power snce funcon x y s monoonc for x 0 and y > 0. In compung L norm, here s a basc operaon, he neger power of a number, a, where a s an elemen n any vecor A j. The nave approach, whch mulples a eravely, nvolves 1 mulplcaons. Ths can be mproved by ulzng paral mulplcaon resuls. For example, compung a 8 as ((a 2 ) 2 ) 2 only needs 3 mulplcaons. Generally, compung a requres log() +ν() 1 mulplcaons, where ν() s he number of 1s n he bnary represenaon of (Chaper n [7]). In he followng, we pu an upper bound 2 log o he number of mulplcaons needed o compue a. To compue f (),j for gven and j, needs d + m 1 addons (addng p o A j, suppose A j s mananed n each eraon) and 2md log() mulplcaons (md numbers, each needs o compue s power). To fnd he opmal j for gven, we need o compue f (),j for all j, and selec he opmal one by comparson. Ths procedure needs m(d + m 1)

7 addons, 2d log()m 2 mulplcaons, and m 1 comparsons. In summary, akes O(d log()m 2 ) me for one vecor. For he overall algorhm, akes O(d log()nm 2 ) me. The compuaons can be sped up by explong he problem srucure. The complexy can be reduced o O(d log()mn), droppng one m facor, as shown n he followng. C. Compuaon Speedup Towards VS-GLB, we have he followng lemma. Noe ha hs lemma does no hold for he general GLB problem. Lemma 5. For any j 1, j 2, holds ha f (),j 1 only f A j1 {p } Aj1 > Aj2 {p } > f (),j 2 Aj2 f and Proof: Addng A A m o boh sdes proves he lemma. Algorhm 2 shows he fnal desgn. For each paron A j, he algorhm manans wo varables, he vecor A j (µ j n he algorhm) and s norm A j (δ j n he algorhm). If here s no empy paron, hen each ncomng vecor searches over all parons o fnd he j o mnmze A j {p } A j (Lnes 12-24). As Lemma 5 shows, hs s equvalen o mnmze f (),j. For he runnng me, consder a new vecor ha canno fnd an empy paron. There are md addons (Lnes 13,16), 2md log() mulplcaons (Lnes 14,17), 2(m 1) subracons and m 1 comparsons (Lne 18). The domnang facor s md log(). Ths s for one vecor. For all n vecors, he runnng me s O(mnd log()), compared o O(m 2 nd log ) before speedup. Subsung = ln(md) no he formula yelds O(nmd ln ln(md)) runnng me, polynomal n he npu lengh (noe m < n). Ths analyss, ogeher wh Corollary 1 and Lemma 5, gves he followng heorem. Theorem 6. Algorhm 2 s an e log(md) + e log(e) ln(md)+1 approxmaon algorhm o VS. I runs n O(nmd ln ln(md)) me. I should be noed ha we rea mulplcaons as basc operaons n he above runnng me analyss. The runnng me wll be dfferen f we furher consder he complexy of compung mulplcaons. Mulplyng wo n-b negers akes me O(n 1.59 ) for a recursve algorhm (Chaper 5.5 n [8]). Applyng such analyss o Algorhm 2, however, requres he consderaon of he lengh of he bnary represenaon of each numerc value n he vecors, whch may be complcaed. Neverheless, s clear ha mulplcaons run n polynomal me n he npu lengh. Thus Algorhm 2 ermnaes ceranly n polynomal me. IV. CONCLUSION In hs work, we connec he vecor schedulng problem wh he generalzed load balancng problem, and oban new resuls by applyng exsng resuls o each oher. On one hand, we show ha generalzed load balancng does no adm consan approxmaon algorhms unless P = N P. On he oher hand, Algorhm 2: Sped-up Vecor Schedulng wh = ln(md) Inpu: m, he number of parons; d, he dmenson of each vecor; p 1, p 2,..., p n, he n vecors o be scheduled Oupu: A 1,..., A m, he m parons 1 begn 2 for j from 1 o m do 3 A j ; 4 µ j 0; // vecor A j 5 δ j 0 ; // scalar A j 6 for from 1 o n do 7 8 f A j, A j = hen A j A j {p }; 9 µ j p ; 10 δ j p ; 11 else 12 j mn 1; // paron ndex 13 µ mn µ 1 + p 1 ; 14 δ mn µ mn ; 15 for j from 2 o m do 16 µ µ j + p ; // vecor addon 17 δ µ ; 18 f δ mn δ jmn > δ δ j hen 19 j mn j; 20 µ mn µ; 21 δ mn δ; 22 µ jmn µ mn ; δ jmn δ mn ; A jmn A jmn {p }; we desgn a polynomal me algorhm for vecor schedulng by usng an exsng algorhm for GLB and modfyng o guaranee polynomal runnng me. The resulng algorhm has wo advanages. I can be performed n an onlne fashon, and provdes beer approxmaon bound o solve VS han exsng offlne polynomal me algorhm. Noe ha we round ln(md) o an neger o guaranee polynomal runnng me, whch causes a small approxmaon rao loss. I s unclear wheher compung L ln(md) norm drecly, whou roundng, can be done n polynomal me. REFERENCES [1] C. Chekur and S. Khanna, On muldmensonal packng problems, SIAM J. Compu., vol. 33, pp , Aprl [2] L. Epsen and T. Tassa, Vecor assgnmen problems: a general framework, J. Algorhms, Sepember [3] F. Xu, C. C. Tan, Q. L, G. Yan, and J. Wu, Desgnng a praccal access pon assocaon proocol, n Proceedngs of INFOCOM 10. [4] F. Xu, X. Zhu, C. C. Tan, Q. L, G. Yan, and W. Je, Smarassoc: Decenralzed access pon selecon algorhm o mprove hroughpu, under submsson. [5] I. Caraganns, Beer bounds for onlne load balancng on unrelaed machnes, n Proceedngs of SODA 08.

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