Multi-organization scheduling approximation algorithms

Transcription

1 CONCURRENCY AND COMPUTATION: PRACTICE AND EXPERIENCE Concurrency Computat.: Pract. Exper. (2011) Publshed onlne n Wley Onlne Lbrary (wleyonlnelbrary.com) Mult-organzaton schedulng approxmaton algorthms Johanne Cohen 1, Danel Cordero 2, Dens Trystram 3,4,, and Frédérc Wagner 3 1 Laboratore d Informatque PRSM, Unversté de Versalles St-Quentn-en-Yvelnes, 45 Avenue des États-Uns, Versalles Cedex, France 2 LIG, Grenoble Unversty, 51 Avenue Jean Kuntzmann, Montbonnot Sant-Martn, France 3 Grenoble Techncal Unversty, 51 Avenue Jean Kuntzmann, Montbonnot Sant-Martn, France 4 Insttut Unverstare de France, Pars, France SUMMARY In ths paper we consder the problem of schedulng on computng platforms composed of several ndependent organzatons, known as the Mult-Organzaton Schedulng Problem (MOSP). Each organzaton provdes both resources and jobs and follows ts own objectves. We are nterested n the best way to mnmze the makespan on the entre platform when the organzatons behave n a selfsh way. We study the complexty of the MOSP problem wth two dfferent local objectves makespan and average completon tme and show that MOSP s strongly NP-Hard n both cases. We formally defne a selfshness noton, by means of restrctons on the schedules. We prove that selfsh behavor mposes a lower bound of 2 on the approxmaton rato for the global makespan. We present varous approxmaton algorthms of rato 2 whch valdate selfshness restrctons. These algorthms are expermentally evaluated through smulaton, exhbtng good average performances and presentng good farness to organzatons local objectves. Copyrght 2011 John Wley & Sons, Ltd. Receved 15 November 2010; Revsed 15 March 2011; Accepted 15 March 2011 KEY WORDS: schedulng; approxmaton algorthms; mult-organzaton schedulng 1. INTRODUCTION 1.1. Motvaton and presentaton of the problem The new generaton of many-core machnes and the now mature grd computng systems allow the creaton of unprecedented massvely dstrbuted systems. In order to fully explot such large numbers of processors and cores avalable and reach the best performances, we need sophstcated schedulng algorthms that encourage users to share ther resources and, at the same tme, respect each user s own nterests. Many of these new computng systems are composed of organzatons that own and manage clusters of computers. A user of such systems submts hs/her jobs to a schedulng system that can choose any avalable machne n any of these clusters. However, each organzaton that shares ts resources ams to take the maxmum advantage of ts own hardware. In order to mprove cooperaton between the organzatons, local jobs should be prortzed to ensure that local performance objectves wll be satsfed. To fnd an effcent schedule for the jobs usng the avalable machnes s a crucal problem. Although each user submts jobs locally n hs/her own organzaton and has ts own performance Correspondence to: Dens Trystram, Grenoble Techncal Unversty, 51 Avenue Jean Kuntzmann, Montbonnot Sant-Martn, France. E-mal: Dens.Trystram@mag.fr Copyrght 2011 John Wley & Sons, Ltd.

2 J. COHEN ET AL. goals, t s necessary to optmze the allocaton of the jobs for the whole platform n order to acheve good performance. Ths work extends the paper presented n the 2010 Euro-Par conference [1], by provdng a new algorthm to solve the problem and deeper results on the qualty of the performances obtaned. We also study the noton of farness on the results obtaned by our algorthms Related work From the classcal schedulng theory, the problem of schedulng parallel jobs s related to Strp packng [2]. It corresponds to packng a set of rectangles (wthout rotatons and overlaps) nto a strp of machnes n order to mnmze the heght used. Then, ths problem was extended to the case where the rectangles were packed nto a fnte number of strps [3, 4]. More recently, an asymptotc (1+ε)-approxmaton AFPTAS wth addtve constant O(1) and wth runnng-tme polynomal n n andn1/ε was presented n [5]. Schwegelshohn et al. [6] studed a very smlar problem, where the jobs can be scheduled n non-contguous processors. Ther algorthm s a 3-approxmaton for the maxmum completon tme (makespan) f all jobs are known n advance, and a 5-approxmaton for the makespan on the onlne, non-clarvoyant case. The Mult-Organzaton Schedulng problem (MOSP) was ntroduced by Pascual et al. [7, 8] and studes how to effcently schedule parallel jobs n new computng platforms, whle respectng users own selfsh objectves. A prelmnary analyss of the schedulng problem on homogeneous clusters was presented wth the target of mnmzng the makespan, resultng n a centralzed 3-approxmaton algorthm. Ths problem was then extended for relaxed local objectves n [9]. The noton of cooperaton between dfferent organzatons and the study of the mpact of users selfsh objectves are drectly related to Game Theory. The study of the Prce of Anarchy [10] on non-cooperatve games allows to analyze how far the socal costs results obtaned by selfsh decsons are from the socal optmum on dfferent problems. In selfsh load-balancng games (see [11] for more detals), selfsh agents am to allocate ther jobs on the machne wth the smallest load. In these games, the socal cost s usually defned as the completon tme of the last job to fnsh (makespan). Several works studed ths problem focusng on varous aspects, such as convergence tme to a Nash equlbrum [12], characterzaton of the worst-case equlbra [13], etc. We do not target here such game theoretcal approaches Contrbutons and road map As suggested n the prevous secton, the problem of schedulng n mult-organzaton clusters has been studed from several dfferent vew ponts. In ths paper, we propose a theoretcal analyss of the problem usng classcal combnatoral optmzaton approaches. Our man contrbuton s the extenson and analyss of the problem for the case n whch sequental jobs are submtted by selfsh organzatons that can handle dfferent local objectves (namely, makespan and average completon tmes). We ntroduce new restrctons to the schedule that take nto account the noton of selfsh organzatons,.e. organzatons that refuse to cooperate f ther objectves could be mproved just by executng earler one of ther jobs n one of ther own machnes. The formal descrpton of the problem and the notatons used n ths paper are descrbed n Secton 2. A farness study s conducted usng two dfferent farness metrcs. The objectve of ths study s to evaluate f the results obtaned by all organzatons equally mprove every organzaton s local objectve by constructng schedules that are consdered far accordng to these metrcs. Unfarness n the results obtaned by each organzaton could be consdered a reason to make an organzaton stop cooperaton wth the others. Secton 3 shows that any algorthm respectng our new selfshness restrctons cannot acheve approxmaton ratos better than 2 and that both problems are ntractable. 2-approxmaton algorthms for solvng the problem are presented n Secton 4. Those heurstcs are analyzed usng farness metrcs presented n Secton 5. Smulaton experments, dscussed n Secton 6, show

3 MULTI-ORGANIZATION SCHEDULING APPROXIMATION ALGORITHMS the good results obtaned by our algorthms n average and how each organzaton perceves the farness of the results obtaned. 2. PROBLEM DESCRIPTION AND NOTATIONS In ths paper, we are nterested n the schedulng problem n whch dfferent organzatons own a physcal cluster of dentcal machnes that are nterconnected. They share resources and exchange jobs wth each other n order to smultaneously maxmze the profts of the collectvty and ther own nterests. All organzatons ntent to mnmze the global makespan (.e. the maxmum completon tme n any local schedule) whle they ndvdually try to mnmze ther own objectves ether the makespan or the average completon tme of ther own jobs n a selfsh way. Although each organzaton accepts to cooperate wth others n order to mnmze the global makespan, ndvdually t behaves n a selfsh way. An organzaton can refuse to cooperate f n the fnal schedule one of ts mgrated jobs could be executed earler n one of the machnes owned by the organzaton. Formally, we defne our target platform as a grd computng system wth N dfferent organzatons nterconnected by a mddleware. Each organzaton O (k) (1 k N) hasm (k) dentcal machnes avalable that can be used to run jobs submtted by users from any organzaton. Each organzaton O (k) has n (k) jobs to execute, and the total number of jobs s denoted by n = k n(k). Each job J (k) (1 n (k) ) wll use one processor for exactly p (k) unts of tme. The sze of the largest job from organzaton O (k) s denoted as p max. (k) No preempton s allowed,.e. after ts actvaton, a job runs untl ts completon at tme C (k). We denote the makespan of a partcular organzaton k by C max (k) =max 1 n (k)(c (k) ) and ts sum of completon tmes as C (k). The global makespan for the entre grd computng system s defned as C max =max 1 k N (C (k) max) Local constrant The MOSP, as frst descrbed n [7] conssts n mnmzng the global makespan (C max )wthan addtonal local constrant: at the end, no organzaton can have ts makespan ncreased f compared wth the makespan that the organzaton could have by schedulng the jobs n ts own machnes ). More formally, we call MOSP(C max ) the followng optmzaton problem: (C (k)local max mnmze C max such that for all k(1 k N), C (k) max C(k)local max. In ths work, we also study the case where all organzatons are locally nterested n mnmzng ther average completon tme whle mnmzng the global makespan. As n MOSP(C max ), each organzaton mposes that the sum of completon tmes of ts jobs cannot be ncreased f compared wth what the organzaton could have obtaned usng only ts own machnes ( C (k)local ). We denote ths problem MOSP( C ) and the goal of ths optmzaton problem s to mnmze C max such that for all k(1 k N), C (k) C (k)local Selfshness In both MOSP(C max ) and MOSP( C ), whle the global schedule mght be computed by a central entty, the organzatons keep control on the way they execute the jobs n the end. Ths property means that, n theory, t s possble for organzatons to cheat the devsed global schedule by re-nsertng ther jobs earler n the local schedules. All machnes are dentcal,.e. every job wll be executed at the same speed rrespectve of the chosen machne.

4 J. COHEN ET AL. In order to prevent such behavor, we defne a new restrcton on the schedule, called selfshness restrcton. The dea s that, n any schedule respectng ths restrcton, no sngle organzaton can mprove ts local schedule by cheatng. Gven a fxed schedule, let J (l) f frst dle tme f O (k) has no foregn job) and J (k) restrcton forbds any schedule where C (l) f be the frst foregn job scheduled to be executed n O (k) (or the any job belongng to O (k). Then, the selfshness p (l) f <C (k) p (k).inotherwords,o (k) refuses to cooperate f one of ts jobs could be executed earler n one of O (k) machnes even f ths leads to a larger global makespan. 3. COMPLEXITY ANALYSIS 3.1. Lower bounds Pascual et al. [7] showed wth an nstance composed of two organzatons and two machnes per organzaton that every algorthm that solves MOSP (for rgd, parallel jobs and C max as local objectves) has at least a 3 2-approxmaton rato when compared to the optmal makespan that could be obtaned wthout the local constrants. We show that the same bound apples asymptotcally even wth a larger number of organzatons. Take the nstance depcted n Fgure 1(a). O (1) ntally has two jobs of sze N and all the others ntally have N jobs of sze 1. All organzatons contrbute only wth one machne each. The optmal makespan for ths nstance s N +1 (Fgure 1(b)), nevertheless, t delays jobs from O (2) and, as a consequence, does not respect MOSP s local constrants. The best possble makespan that respects the local constrants (whenever the local objectve s the makespan or the average completon tme) s 3N/2, as shown n Fgure 1(c) Selfshness and lower bounds Although all organzatons wll lkely cooperate wth each other to acheve the best global makespan possble, ther selfsh behavor wll certanly mpact the qualty of the best attanable global makespan. We study here the mpact of new selfshness restrctons on the qualty of the achevable schedules. We show that these restrctons mpact MOSP(C max ) and MOSP( C ) as compared wth unrestrcted schedules and, moreover, that MOSP(C max ) wth selfshness restrctons suffers from lmted performances compared to MOSP(C max ) wth local constrants. Proposton 1 No approxmaton algorthm for both MOSP(C max ) and MOSP( C ) has a rato asymptotcally better than 2 regardng the optmal makespan wthout constrants f all organzatons behave selfshly. (a) (b) (c) Fgure 1. Rato between global optmum makespan and the optmum makespan that can be obtaned for both MOSP(C max ) and MOSP( C ). Jobs owned by organzaton O (2) are hghlghted. (a) Intal nstance C max =2N. (b) Global optmum wthout constrants C max = N +1. (c) Optmum wth MOSP constrants C max =3N/2.

5 MULTI-ORGANIZATION SCHEDULING APPROXIMATION ALGORITHMS (a) (b) Fgure 2. Rato between global optmum makespan wth MOSP constrants and the makespan that can be obtaned by MOSP(C max ) wth selfsh organzatons: (a) ntal nstance C max =2N 2 and (b) global optmum wth MOSP constrants C max = N. Proof We prove ths result by usng the example descrbed n Fgure 1. It s clear from Fgure 1(b) that an optmal soluton for a schedule wthout local constrants can be acheved n N +1. However, wth added selfshness restrctons, Fgure 1(a) (wth a makespan of 2N) represents the only possble vald schedule. We can, therefore, conclude that local constrants combned wth selfshness restrctons mply that no algorthm can provde an approxmaton rato asymptotcally better than 2 when compared wth the problem wthout constrants. Proposton 1 gves a rato regardng the optmal makespan wthout the local constrants mposed by MOSP. We can show that the same approxmaton rato of 2 also apples for MOSP(C max ) regardng the optmal makespan even f MOSP constrants are respected. Proposton 2 Any approxmaton algorthm for MOSP(C max ) has a rato greater than or equal to 2 (2/N) regardng the optmal makespan wth local constrants f all organzatons behave selfshly. Proof Take the nstance depcted n Fgure 2(a). O (1) ntally has N jobs of sze 1 and O (N) has two jobs of sze N 1. The optmal soluton that respects MOSP local constrants s gven n Fgure 2(b) and has C max equal to N. Nevertheless, the best soluton that respects the selfshness restrctons s the ntal nstance wth a C max equal to 2N 2. Thus, the rato of the optmal soluton wth the selfshness restrctons to the optmal soluton wth MOSP constrants s 2 (2/N) Computatonal complexty Ths secton studes how dffcult t s to fnd optmal solutons for the MOSP problem. We consder the decson verson of the MOSP defned as follows: Instance: AsetofN organzatons (for 1 k N, organzaton O (k) has n (k) jobs, m (k) dentcal machnes, and makespan as the local objectve) and an nteger l. Queston: Does there exst a schedule wth a makespan less than l that respects the local constrants? Theorem 1 MOSP(C max ) and MOSP( C ) are strongly NP-complete. A reducton from the classcal 3-PARTITION problem [14] (see defnton below) to MOSP s straghtforward and t s schematzed n Fgure 3. Ths reducton, however, represents a degenerate case of the MOSP problem. The nstance presents a confguraton where ntally one organzaton s very loaded and some organzatons do not contrbute wth any work. We beleve that ths confguraton msrepresents the concept of cooperaton ntroduced by the MOSP problem. We present n ths secton a slghtly more complex proof for the complexty of MOSP(C max ) and MOSP( C ) wth an nstance where any organzaton could potentally have ts makespan

6 J. COHEN ET AL. (a) (b) Fgure 3. Drect reducton of MOSP from 3-PARTITION: (a) ntal nstance and (b) optmum. Fgure 4. Reducton of MOSP(C max ) and MOSP( C ) from 3-PARTITION. mproved by cooperatng wth others. We show that the problem remans dffcult even for nstances lmted to two jobs per organzaton. Theorem 2 MOSP(C max ) and MOSP( C ) are strongly NP-complete even f each organzaton has exactly two jobs per organzaton. Proof It s straghtforward to see that MOSP(C max ) NP and MOSP( C ) NP. Our proof s based on a reducton from the well-known 3-PARTITION problem [14]: Instance: A bound B Z + and a fnte set A of 3m ntegers {a 1,...,a 3m }, such that every element of A s strctly between B/4 andb/2 and such that 3m =1 a =mb. Queston: CanA be parttoned nto m dsjont sets A 1, A 2,..., A m such that, for all 1 m, a A a = B and A s composed of exactly three elements? Gven an nstance of 3-PARTITION, we construct an nstance of MOSP where, for 1 k 3m, organzaton O (k) ntally has two jobs J (k) 1 and J (k) 2 wth p (k) 1 =(m +1)B +11 and p(k) 2 =(m + 1)a k +3, and all other organzatons have two jobs wth processng tme equal to 1. We then set l to be equal to (m +1)B +11. Fgure 4 depcts the descrbed nstance. Ths constructon s performed n polynomal tme. Now, we prove that A can be splt nto m dsjont subsets A 1,..., A m, each one summng up to B, f and only f ths nstance of MOSP has a soluton wth C max (m +1)B +11. Assume that A ={a 1,...,a 3m } can be parttoned nto m dsjont subsets A 1,..., A m, each one summng up to B. In ths case, we can buld an optmal schedule for the nstance as follows: for 1 k 3m, J (k) 1 s scheduled on machne k; for 3m +1 k 4m, J (k) 1 and J (k) 2 are scheduled on machne k; for 1 m, leta ={a 1,a 2,a 3 } A. The jobs J (a 1 ) 2, J (a 2 ) 2,andJ (a 3 ) 2 are scheduled on machne 3m +. Ths constructon provdes a schedule where the global C max s the optmal one. Also, t s easy to see that the local constrants for MOSP(C max ) and MOSP( C ) are respected: for MOSP(C max ),

7 MULTI-ORGANIZATION SCHEDULING APPROXIMATION ALGORITHMS t s suffcent to note that no organzaton has ts makespan ncreased; for MOSP( C ), note that each job ether remans n ts orgnal organzaton and schedule (lghter colored jobs n Fgure 4) or s mgrated to a dfferent organzaton at an earler tme (darker colored jobs n Fgure 4), strctly decreasng the average completon tme. Conversely, assume that MOSP has a soluton wth C max (m +1)B +11. The total work (W ) of the jobs that must be executed s W =3m ((m +1)B +11)+ 3m =1 ((m +1)B +a )+m 2=4m ((m +1)B +11). Snce we have exactly 4m organzatons, the soluton must be the optmal soluton and there are no dle tmes n the schedulng. Moreover, 3m machnes must execute only one job of sze (m +1)B +11. W.l.o.g, we can consder that for 3m +1 k 4m, machne k performs jobs of sze less than (m +1)B +11. To prove our proposton, we frst show two lemmas that characterze the structure of the soluton for the MOSP problem: Lemma 1 For all 3m +1 k 4m, exactly three jobs of sze not equal to 1 must be scheduled on machne k f C (k) max =(m +1)B +11. Proof We prove ths lemma by contradcton. Frst, note that for all ntegers from the 3-PARTITION nstance, t holds that a >B/4. Assume that one machne has less than three jobs of sze not equal to 1. 3m jobs of ths knd must be assgned to the last m organzatons (snce the frst 3m organzatons executes each one large job of sze (m +1)B +11). If one machne has less than three jobs, then we must have at least one other machne wth four or more jobs of ths knd assgned to t. The makespan on ths other machne must be of at least 4 ((m +1)a +3)>4 ((m +1)B/4+3)=(m +1)B +12. Ths contradcts the fact that C max (k) =(m +1)B +11 and shows that exactly three jobs of sze not equal to 1 must be scheduled on each machne. Lemma 2 For all 3m +1 k 4m, exactly two jobs of sze 1 are scheduled on each machne f C (k) (m +1)B +11. max = Proof Snce C max (k) =(m +1)B +11, each large job of sze (m +1)B +11 must reman n ts orgnal organzaton. Consequently, the 2m jobs of sze 1 must be scheduled n one of the m organzatons not havng a large job assgned to t. However, each of these organzatons have C max (k)local =2, whch means that any mgraton that results n a schedule wth one organzaton not havng two jobs of sze 1 wll result n a schedule that does not respect both MOSP(C max ) and MOSP( C ) local constrants. From the soluton for the MOSP problem, we construct m dsjont subsets A 1, A 2,..., A m of A as follows: for all 1 m, a j s n A f the job wth sze (m +1)a j +3 s scheduled on machne 3m +. Note that all elements of A belong to one and only one set n {A 1,..., A m }. We prove that A s a partton wth the desred propertes. We focus on organzaton O (3m+),whereA was assgned. Lemmas 1 and 2 show how the jobs wll be assgned to ths organzaton. Frst, Lemma 1 shows that A s composed of exactly three jobs that wll be executed on O (3m+). Second, Lemma 2 shows that exactly two jobs of sze 1 wll be scheduled at the begnnng of the schedule. Snce all organzatons must have a local C max exactly equal to (m +1)B +11, we have that C (3m+) max =(m +1)B +11=1+1+ a j A ((m +1)a j +3), whch mples that: 2+ a j A ((m +1)a j +3)=(m +1)B +11 a j A (m +1)a j =(m +1)B a j A a j = B.

8 J. COHEN ET AL. Thus, we can deduce that A s composed of exactly three elements and a j A a j = B. Inother words, {A,..., A m } s a soluton for the 3-PARTITION problem. 4. ALGORITHMS In ths secton, we present four dfferent heurstcs to solve MOSP(C max ) and MOSP( C ). All algorthms present the addtonal property of respectng selfshness restrctons Iteratve load balancng algorthm (ILBA) ILBA [8] s a heurstc that redstrbutes the load from the most loaded organzatons. The dea s to ncrementally rebalance the load wthout delayng any job. Frst, the less loaded organzatons are rebalanced. Then, one-by-one, each organzaton has ts load rebalanced. The heurstc works as follows. Frst, each organzaton schedules ts own jobs locally and the organzatons are enumerated by non-decreasng makespans,.e. C max (1) C max (2) C max. (N) For k =2 untl N, jobs from O (k) are rescheduled sequentally, and assgned to the less loaded of organzatons O (1)... O (k). Each job s rescheduled by ILBA ether earler or at the same tme that the job was scheduled before the mgraton. In other words, no job s delayed by ILBA, whch guarantees that the local constrant s respected for MOSP(C max ) and MOSP( C ). ILBA has the same tme complexty as any lst schedulng algorthm for the classcal problem of mnmzng the makespan. Its tme complexty s O(n logn). We recall that n s the total number of jobs LPT-LPT and SPT-LPT heurstcs We developed and evaluated (see Secton 6) two new heurstcs based on the classcal LPT (Longest Processng Tme Frst [15]) and SPT (Smallest Processng Tme Frst [16]) algorthms for solvng MOSP(C max ) and MOSP( C ), respectvely. Both heurstcs work n two phases. Durng the frst phase, all organzatons mnmze ther own local objectves. Each organzaton starts applyng LPT for ts own jobs f the organzaton s nterested n mnmzng ts own makespan, or starts applyng SPT f the organzaton s nterested n ts own average completon tme. The second phase s when all organzatons cooperatvely mnmze the makespan of the entre grd computng system wthout worsenng any local objectve. Ths phase works as follows: each tme an organzaton becomes dle,.e. t fnshes the executon of all jobs assgned to t, the longest job that has not started yet s mgrated and executed by the dle organzaton. Ths greedy algorthm works lke a global LPT, always choosng the longest job yet to be executed among jobs from all organzatons. The frst and second phases correspond to a lst schedulng algorthm, hence the total runnng tme of both algorthms s O(n logn) Less Helped Frst heurstc Lke the prevous algorthms, Less Helped Frst works n two phases. Frst, each organzaton O (k) s classfed accordng to the total work orgnally submtted by a user n O (k) that was actually executed by a dfferent organzaton. Locally, the order of the jobs s arbtrary (t could be the submsson order, for nstance). Second, each tme a processor becomes dle, the algorthm wll prortze the organzaton that receved less help from the others,.e. that had less work executed by other organzatons. The algorthm wll mgrate any job from the less helped organzaton that has not started yet to be executed by the dle processor. The runnng tme of the frst phase s O(n logn). Durng the second phase, after each job mgraton, the classfcaton of the organzatons must be updated. Ths update can be done n

9 MULTI-ORGANIZATION SCHEDULING APPROXIMATION ALGORITHMS O(logn) and there are at most n mgratons (one per job). Snce the number of organzatons s neglgble compared to the number of jobs, the total runnng tme of the algorthm s O(n logn) Analyss ILBA, LPT-LPT, SPT-LPT, andless Helped Frst do not delay any of the jobs when compared to the ntal local schedule. Durng the rebalancng phase, all jobs ether reman n ther orgnal organzaton or are mgrated to an organzaton that became dle at a precedng tme. The mplcatons are the selfshness restrcton s respected f a job s mgrated, t wll start before any foregn jobs that may have been assgned to ts orgnal organzaton; f organzatons local objectve s to mnmze the makespan, mgratng a job to a prevous moment n tme wll decrease the job s completon tme and, as a consequence, t wll not ncrease the ntal makespan of the organzaton; f organzatons local objectve s to mnmze the average completon tme, mgratng a job from the ntal organzaton to another that became dle at a prevous moment n tme wll not ncrease the completon tme of any job. Ths means that the C of the jobs from the ntal organzaton s ether decreased or remans constant; the rebalancng phase of all four algorthms works as the lst schedulng algorthms. Graham s classcal approxmaton rato 2 (1/N) of lst schedulng algorthms [15] holds for all of them. Proposton 1 (Secton 3.2) states that no algorthm respectng selfshness restrctons can acheve an approxmaton rato for MOSP(C max ) better than 2 regardng the optmal makespan wthout constrants. Snce all our algorthms reach an approxmaton rato of 2, no further enhancements are possble wthout removng selfshness restrctons. 5. FAIRNESS Another form of encouragement of cooperaton between the organzatons s provdng algorthms that equally mprove ther local objectves. Organzatons that are not nvdous of other organzatons are more lkely to share ther resources to the platform. Selfsh organzatons could potentally become nvdous f the results obtaned when solvng the MOSP problem mprove the local objectves of the organzatons n an unfar manner. Some farness metrcs can be found n the lterature Varance, Coeffcent of Varance (Varance/Mean), Mn-Max rato, etc. The Jan Index [17] s nowadays one of the most popular metrcs used to measure farness [18]. We also studed the Stretch as an ndex of farness, as we wll show n the subsequent sectons. In ths secton we study the farness of the results produced by the algorthms presented n Secton 4. Usng the same workload generated for the expermental results shown n Secton 6, we evaluate the farness of these algorthms for two mportant metrcs: Stretch and the Jan Index Stretch Stretch s a performance metrc that was extensvely studed n the schedulng lterature. Ths performance metrc can also be consdered as a farness metrc. We adapted the noton of Stretch to the MOSP problem, to measure the Stretch of an entre organzaton. We defne Stretch as the rato of the local C max obtaned by the organzaton to the C max that the organzaton could have obtaned f t had all the machnes avalable only for ts jobs (denoted by C max (k)alone ). Formally, the Stretch s calculated as follows: Stretch=max k C (k) max C (k)alone max.

10 J. COHEN ET AL. Snce C max (k)alone s bounded by the theoretcal lower bound p(k) /N =C max (k)local /N and snce MOSP(C max ) constrant guarantees that C (k),wehave Stretch=max k max C max (k)local C (k) max C (k)alone max C(k)local max C max (k)local N. N Note that ths upper bound for the Stretch metrc s tght. Consder the nstance wth N organzatons, each one havng N jobs wth processng tmes equal to 1. C max (k)alone s equal to 1 for all organzatons k, andc max (k)local = N. There s only one schedulng that respects MOSP(C max ) local constrant, the one that schedules all jobs n ther orgnal organzatons Jan Index Orgnally, the Jan Index was ntroduced for the study of resource sharng and allocaton problems. TheJanIndexofasetofn users recevng an allocaton x [17] s defned as ( n=1 ) 2 x f (x)= n n =1 x 2, x 0. (1) If the values of x follow a certan random dstrbuton, the ndex s the rato of the frst moment (the mean) to the second moment of the dstrbuton. The Jan Index has the followng useful propertes: t s ndependent of the sze of the populaton; t s ndependent of the scale (or metrc unt) used; the values of the ndex are bound between 1/n and 1; t s contnuous. We evaluated the farness of the heurstcs presented n Secton 4 regardng the mprovements obtaned on the local C max by each organzaton. We defne ths mprovement as the rato of the obtaned C max to the theoretcal lower bound of each organzaton over all avalable machnes of the system. The lower bound of each organzaton s calculated by LB (k) =max( p (k) N, p(k) max). The mprovement for organzaton O (k) (the x k of Equaton 1) s then defned as: C max/lb (k) (k). Thus, the fnal Jan Index calculated to measure the farness of the mprovements can be stated as follows: Jan Index= ( N k=1 N N k=1 ( C max (k) ) 2 LB (k) C max (k) LB (k) ) EXPERIMENTS We conducted an extensve seres of smulatons comparng ILBA, LPT-LPT, SPT-LPT, andless Helped Frst under varous expermental settngs. The workload was randomly generated wth parameters matchng the typcal envronment found n academc grd computng systems [8]. We generated random workloads wth N =5, 10, and 20 organzatons. For each number of organzatons, we have generated nstances wth dfferent number of total jobs (from 100 to 1000), wth szes chosen unformly at random from 1 to For each combnaton of these parameters, we generated random nstances. In our tests, the number of ntal jobs n each organzaton

11 MULTI-ORGANIZATION SCHEDULING APPROXIMATION ALGORITHMS (a) (b) (c) Fgure 5. Mean and confdence nterval of the measured C (k) max obtaned by ILBA, LPT-LPT, SPT-LPT, and Less Helped Frst: (a)n =5; (b) N =10; and (c) N =20. follows a Zpf dstrbuton wth exponent equal to , whch best models vrtual organzatons n real-world grd computng systems [19]. All results are presented wth confdence level of 95% Global C max analyss We are nterested n the mprovement of the global C max provded by the dfferent algorthms. The results are evaluated wth a comparson to C max obtaned by the algorthms wth the well-known theoretcal lower bound for the schedulng problem wthout constrants LB=max( p (k),k, p m (k) max ). Our man concluson s that, despte the fact that the selfshness restrctons are respected by all heurstcs, ILBA and LPT-LPT obtaned near optmal results for most cases. Ths s not unusual, snce t follows the patterns of expermental behavor of standard lst schedulng algorthms, n whch t s easy to obtan a near optmal schedule when the number of jobs grows large. SPT-LPT and Less Helped Frst produce worse results due to the effect of the local order of the jobs. Fgure 5 shows the average global C max obtaned by the heurstcs for nstances wth dfferent number of organzatons and the total number of jobs. However, n some partcular cases, n whch the number of jobs s not much larger than the number of machnes avalable, the experments yeld more nterestng results. Fgure 6 shows the hstogram of a representatve nstance of such a partcular case. The hstograms show the frequency of the rato C max obtaned to the lower bound over dfferent nstances wth 20 organzatons and 100 jobs for ILBA, LPT-LPT, SPT-LPT, andless Helped Frst. Smlar results have been obtaned for many dfferent sets of parameters. LPT-LPT outperforms ILBA (and SPT-LPT) for most nstances and ts average rato to the lower bound s less than 1.3. Less Helped Frst s the heurstc that produces the worst results Local C max analyss We are also nterested n the mprovements obtaned by each organzaton that our four algorthms provde. In order to have a global vew of the total mprovement obtaned, we have evaluated the sum of the local makespans obtaned by all organzatons: k C(k) max. The sum of the local makespans s mportant because t shows the average results that each selfsh organzaton obtans. The experments show that ILBA produces results wth a lower makespan n average. LPT-LPT, SPT-LPT, andless Helped Frst produce smlar results n average. Recall from Secton 4.1 that ILBA rebalances the load of all organzatons, from the less loaded to the more loaded organzaton. Ths mechansm mnmzes the makespan of the organzaton ndvdually n a more aggressve way and, therefore, produces lower average local makespans. Fgure 7 shows the average local C max obtaned by all heurstcs. It s nterestng to note that the gap between the average result obtaned by ILBA and the other heurstcs ncreases as the number of organzatons and the total number of jobs ncreases.

12 J. COHEN ET AL. (a) (b) (c) (d) Fgure 6. Frequency of results obtaned by ILBA, LPT-LPT, SPT-LPT, andless Helped Frst for N =20 and n =100: (a) ILBA; (b)lpt-lpt; (c)spt-lpt; and(d)less Helped Frst. (a) (b) (c) Fgure 7. Mean and confdence nterval of the measured k C(k) max obtaned by ILBA, LPT-LPT, SPT-LPT, and Less Helped Frst: (a)n =5; (b) N =10; and (c) N = Farness analyss In the two prevous sectons we have analyzed the performance obtaned by the four algorthms presented n Secton 4 for the global makespan obtaned and for the local makespan obtaned by each organzaton. In ths secton we use the same testbed to study how each organzaton perceves the farness of these results. To study the farness, we use the Stretch and Jan Index metrcs presented n Secton Stretch. We have measured the worst.e. maxmum Stretch (as defned n Secton 5.1) obtaned durng the smulatons. Fgure 8 shows how the Stretch vares accordng to the total number of jobs of the system.

13 MULTI-ORGANIZATION SCHEDULING APPROXIMATION ALGORITHMS (a) (b) (c) Fgure 8. Mean and confdence nterval of the measured Stretch obtaned by ILBA, LPT-LPT, SPT-LPT, and Less Helped Frst: (a)n =5; (b) N =10; and (c) N =20. (a) (b) (c) Fgure 9. Mean and confdence nterval of the measured Jan Index obtaned by ILBA, LPT-LPT, SPT-LPT, and Less Helped Frst: (a)n =5; (b) N =10; and (c) N =20. The results show that the Stretch ncreases asymptotcally to the upper bound of the metrc (.e. the number of organzatons) as the total number of jobs on the system ncreases. The Stretch obtaned wth our algorthms s gven by the less loaded organzaton. Our heurstcs never mgrate jobs from the organzaton that has the lower local C max. For ths organzaton, the C max (k) wll always be equal to the C max (k)local and the Stretch wll be hgher. The Zpf dstrbuton used to randomly assgn the jobs to the organzatons dstrbutes more jobs to the less loaded organzaton when the total number of organzatons s smaller. For ths reason, the average Stretch grows faster for N =5. When the number of jobs s small, the performance of ILBA degrades more slowly because ILBA rebalances frst the less loaded organzatons, whle the others rebalance the organzatons n a dynamc order. LPT-LPT presented the worst values for Stretch. Bgger Stretch values ndcate that organzatons are more penalzed by the jobs from other organzatons. Recall from Secton 6 that LPT-LPT produces better results for the global C max. To acheve these results, LPT-LPT must penalze some organzatons, and Stretch measures exactly ths behavor Jan Index. Jan Index and Stretch metrcs measure two dfferent forms of farness. The former measures how each organzaton perceves the mprovements on ts own C max, whle the latter shows how C max of each ndvdual organzaton s affected by the jobs from other organzatons. When the number of organzatons N s 5, our experments showed that the four heurstcs provde far results, wth Jan Index hgher than Recall that the Jan Index s bounded from 1/N (unfar) to 1 (perfect farness). Fgure 9(a) ndcates that the ILBA heurstcs have two moments. In the frst, when the number of jobs s smaller, ILBA heurstc produces more far results than the other heurstcs. Its Jan Index vares from 0.92 up to 0.95.

14 J. COHEN ET AL. Durng the second moment, when the number of jobs s hgher, ILBA produces the less far results between all the four heurstcs. LPT-LPT, SPT-LPT, andless Helped Frst produce results wth smlar farness ndces, whle ILBA farness s clearly worse when compared to the others. Recall that the ndex s almost constant when N =5 and there are more than 300 jobs on the platform. The experments wth N =10 shows that ILBA produces results wth farness smlar to the results wth N =5. Its Jan Index s of about When the number of jobs s over 700, LPT-LPT, SPT-LPT, andless Helped Frst are farer. Ther Jan Index vares between 0.79 and For N =20, the dfference between the performance of ILBA and the others are hgher. ILBA has Jan Index of about 0.90 whle the farness of the other three heurstcs decreases from 0.94 to The number of jobs of the smulatons shown n Fgure 9(c) was ncreased up to 2000 to show that for N =20 the results ndcate that ILBA has also two moments, lke for the cases where N =5 andn =10. Even f ILBA provded farer results than the others wth less jobs on the system, all four algorthms mprove the local C max obtaned by each organzaton n a far way. 7. CONCLUDING REMARKS In ths paper, we have nvestgated the schedulng on mult-organzaton platforms. We presented the MOSP(C max ) problem from the lterature and extended t to a new related problem MOSP( C ) wth another local objectve. In each case we studed how to mprove the global makespan whle guaranteeng that no organzaton wll worsen ts own results. We showed frst that both versons MOSP(C max ) and MOSP( C ) of the problem are strongly NP-hard. Furthermore, we ntroduced the concept of selfshness n these problems whch corresponds to addtonal schedulng restrctons desgned to reduce the ncentve for the organzatons to cheat locally and dsrupt the global schedule. We proved that any algorthm respectng selfshness restrctons cannot acheve a better approxmaton rato than 2 for MOSP(C max ) regardng the optmal makespan wthout MOSP local constrants. Three new schedulng algorthms were proposed, namely LPT-LPT, SPT-LPT, andless Helped Frst n addton to ILBA from the lterature. All these algorthms are lst schedulng, and thus acheve a 2-approxmaton. We provded an n-depth analyss of these algorthms, showng that all of them respect the selfshness restrctons. Fnally, all these algorthms were mplemented and analyzed through expermental smulatons. The results show that when consderng the global makespan, our new LPT-LPT outperforms ILBA, and that all algorthms exhbt near optmal global performances when the number of jobs becomes large. Consderng the local objectves, ILBA acheves better average local makespans. ILBA also proved to present more far results accordng to the Stretch metrc. All the four algorthms presented good farness usng the Jan Index, showng that the local makespan of all organzatons s equally mproved. The future research drectons wll be more focused on game theory. We ntend to study schedules n the case where several organzatons secretly cooperate to cheat the central authorty. ACKNOWLEDGEMENTS Ths work was partally sponsored by a Google Research Award and partally supported by the blateral Brazlan-French CAPES-COFECUB program (Ma 660/10). REFERENCES 1. Cohen J, Cordero D, Trystram D, Wagner F. Analyss of mult-organzaton schedulng algorthms. Euro-Par 2010 Parallel Processng (Lecture Notes n Computer Scence, vol. 6272), D Ambra P, Guarracno M, Tala D (eds.). Sprnger: Berln/Hedelberg, 2010;

15 MULTI-ORGANIZATION SCHEDULING APPROXIMATION ALGORITHMS 2. Baker BS, Coffman EG Jr, Rvest RL. Orthogonal packngs n two dmensons. SIAM Journal on Computng 1980; 9(4): Zhuk SN. Approxmate algorthms to pack rectangles nto several strps. Dscrete Mathematcs and Applcatons 2006; 16(1): Ye D, Han X, Zhang G. On-lne multple-strp packng. Proceedngs of the Thrd Internatonal Conference on Combnatoral Optmzaton and Applcatons (Lecture Notes n Computer Scence, vol. 5573), Berln S. (ed.). Sprnger: Berln, 2009; Jansen K, Otte C. Approxmaton algorthms for multple strp packng. Proceedngs of Seventh Workshop on Approxmaton and Onlne Algorthms (WAOA), Copenhagen, Denmark, Schwegelshohn U, Tchernykh A, Yahyapour R. Onlne schedulng n grds. IEEE Internatonal Symposum on Parallel and Dstrbuted Processng (IPDPS), Mam, FL, U.S.A., 2008; Pascual F, Rzadca K, Trystram D. Cooperaton n mult-organzaton schedulng. Euro-Par 2007 Parallel Processng (Lecture Notes n Computer Scence, vol. 4641). Sprnger: Berln, 2007; DOI: 8. Pascual F, Rzadca K, Trystram D. Cooperaton n mult-organzaton schedulng. Concurrency and Computaton: Practce & Experence 2009; 21(7): DOI: 9. Ooshta F, Izum T. A generalzed mult-organzaton schedulng on unrelated parallel machnes. Internatonal Conference on Parallel and Dstrbuted Computng, Applcatons and Technologes (PDCAT). IEEE Computer Socety: Los Alamtos, CA, U.S.A., 2009; Koutsoupas E, Papadmtrou C. Worst-case equlbra. Proceedngs of 16th Annual Symposum on Theoretcal Aspects of Computer Scence (Lecture Notes n Computer Scence, vol. 1563). Sprnger: Berln, Trer, Germany, 1999; Nsam N, Roughgarden T, Tardos E, Vazran VV. Algorthmc Game Theory. Cambrdge Unversty Press: Cambrdge, Even-Dar E, Kesselman A, Mansour Y. Convergence tme to nash equlbra. ACM Transactons on Algorthms 2007; 3(3): Caraganns I, Flammn M, Kaklamans C, Kanellopoulos P, Moscardell L. Tght bounds for selfsh and greedy load balancng. Proceedngs of the 33rd Internatonal Colloquum on Automata, Languages and Programmng (Lecture Notes n Computer Scence, vol. 4051). Sprnger: Berln, 2006; Garey MR, Johnson DS. Computers and Intractablty: A Gude to the Theory of NP-Completeness. W. H. Freeman: New York, Graham RL. Bounds on multprocessng tmng anomales. SIAM Journal on Appled Mathematcs 1969; 17(2): Bruno JL, Coffman EG Jr, Seth R. Schedulng ndependent tasks to reduce mean fnshng tme. Communcatons of the ACM 1974; 17(7): DOI: Jan RK, Chu DMW, Hawe WR. A quanttatve measure of farness and dscrmnaton for resource allocaton n shared computer systems. Techncal Report, Dgtal Equpment Corporaton, September Legrand A, Touat C. How to measure effcency?. Proceedngs of the Frst Internatonal Workshop on Game theory for Communcaton networks (Game-Comm 07), Nantes, France, Iosup A, Dumtrescu C, Epema D, L H, Wolters L. How are real grds used? The analyss of four grd traces and ts mplcatons. Seventh IEEE/ACM Internatonal Conference on Grd Computng, Barcelona, Span, 2006;