Author name used in this publication: T.C.E. ChengAuthor name used in this publication: C. T. Ng2005-2006 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

Cheng, TCE

Kotov, V

Ng, CTD

Information Processing Letters

Author name used in this publication: T.C.E. ChengAuthor name used in this publication: C. T. Ng2005-2006 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishedGreen (AAM

PolyU Institutional Repository

A new algorithm for online uniform-machinescheduling to minimize the makespanT.C.E. Cheng∗† C.T. Ng‡ Vladimir Kotov§AbstractWe consider the online scheduling problem with m − 1, m ≥ 2,uniform machines each with a processing speed of 1, and one machinewith a speed of s, 1 ≤ s ≤ 2, to minimize the makespan. The well-known list scheduling (LS) algorithm has a worst-case bound of 3m−1m+1[1]. An algorithm with a better competitive ratio was proposed in [3].It has a worst-case bound of 2.8795 for a big m and s = 2. In thisnote we present a 2.45-competitive algorithm for m ≥ 4 and any s,1 ≤ s ≤ 2.Keywords: Online algorithms; competitive ratio; multi-machine schedul-ing; uniform machines1 IntroductionIn the classical uniform-machine scheduling problem, there are m ≥ 2 ma-chines (M1,M2, . . . ,Mm) with speeds (s1, s2, . . . , sm) associated with themachines, respectively. A list of n independent jobs with nonnegative process-ing times p1, . . . , pn has to be scheduled nonpreemptively on these machineswith the objective of minimizing the makespan. In the online version of themulti-machine scheduling problem, each job must be immediately and irrev-ocably assigned to one of the machines without any knowledge of the future∗Department of Logistics, The Hong Kong Polytechnic University, Kowloon, HongKong, lgtcheng@inet.polyu.edu.hk†corresponding author‡Department of Logistics, The Hong Kong Polytechnic University, Kowloon, HongKong, lgtctng@inet.polyu.edu.hk§Faculty of Applied Mathematics and Computer Science, Belarusian State University,Skarina ave. 4, Minsk, 220050, Belarus, kotovVM@bsu.by1jobs. The performance of an online algorithm is measured by its competitiveratio, i.e., the worst-case ratio with respect to the optimal solution of thecorresponding oﬄine problem.The online multi-machine scheduling problem was first investigated by Gra-ham, who showed that the list scheduling (LS) algorithm has a competitiveratio of exactly 2− 1/m [2]. Cho and Sahni [1] proved that for the uniform-machine system, the LS algorithm has a worst-case bound of 3m−1m+1 form ≥ 3.When si = 1 (i = 1, . . . ,m−1) and sm > 1, Cho and Sahni also showed thatthe LS algorithm has a worst-case bound of 1+ (m−1)m+s−1 min(2, s) ≤ 3− 4m+1 ,and the bound 3− 4m+1 is achieved when s = 2.Li and Shi [3] proved that the LS algorithm is the best possible one form ≤ 3, and proposed an algorithm that is significantly better than the LSalgorithm when si = 1 (i = 1, . . . ,m−1) and sm = 2, m ≥ 4. The algorithmhas a worst-case bound of 2.8795 for a big m.In this note we present a 2.45-competitive algorithm for any m ≥ 4 and anysm, 1 ≤ sm = s ≤ 2. This is a marked improvement over the algorithm of Liand Shi because our algorithm yields a considerably lower competitive ratiowhile our results hold under more general conditions. The construction ofthe algorithm is based on simple logical reasoning, which is different fromthe traditional approach applied to construct the LS algorithm.2 A 2.45-competitive algorithm for an arbitrarynumber of machinesBefore presenting the main results, we introduce some notation.m = the number of machines;Vi,j = the current load of the ith machine after assigning pj in step j;Vj = the minimum current load of the machines 1, . . . ,m− 1 after assigningpj in step j;Lj = the lower bound for the makespan in step j; Lj = max(p1+...+pjm+s−1 ,pmaxs , pmax2),where pmax = max(p1, . . . , pj), and pmax2 = max(p1, . . . , pmax−pmax, . . . , pj).Property 1 L1 ≤ L2 ≤ . . . ≤ Lj.Algorithm S.Let α = 1.45. In step j we assign pj to the most loaded machine i < m withthe property Vi,j−1 + pj ≤ (1 + α)Lj if such a machine exists; otherwise, weassign pj to the mth machine.2Lemma 1 In step j of algorithm S, if Vm,j−1 < Vm,j holds, then pj > pmax2.Proof: Let pj ≤ pmax2. The algorithm assigns pj to machine m only whenVt,j−1 + pj > (1 + α)Lj for all t = 1, . . . ,m − 1. From this, it followsthat Vt,j−1 > (1 + α)Lj − pj ≥ αLj for all t = 1, . . . ,m − 1. Therefore,Lj > Lj(m−1)α+1m+s−1 . But this is not possible for m ≥ 4.Lemma 2 In step j of algorithm S, if Vm,j−1 = Vm,j > Lj holds, thenVj−1 < Vj only if pj > αLj.Proof: W.l.o.g. let Vj−1 be the load of the (m− 1)th machine, i.e., Vj−1 =Vm−1,j−1. It is easy to see that if pj ≤ αLj , then the algorithm assigns pj tomachine m−1 only when Vt,j−1+pj > (1+α)Lj for all t = 1, . . . ,m−2. Butthis is possible in case Vt,j−1 > Lj for all t = 1, . . . ,m− 3 and Vt,j−1 + pj >(1 + α)Lj for t = m− 2.From this and Vm,j−1 = Vm,j > Lj , it follows that the total processing timeof the jobs is larger than (m− 3 + 1 + α+ s)Lj .Hence Lj > Lj m−2+α+sm+s−1 . But this is not possible for m ≥ 4.Lemma 3 In step j of algorithm S, if Vm,j−1 < Vm,j holds, then pj >(1 + α)Lj − Vj−1.Proof: In step j, algorithm S assigns pj to machine m because Vj−1+ pj >(1 + α)Lj .Lemma 4 Let f1, f2, . . . , fq be the jobs assigned to machine m, then fl−1 <αfl for l = 2, . . . , q.Proof: In step j of algorithm S, let Vm,j−1 < Vm,j and pj < αfq hold. Itis easy to see that if pj ≤ αLj , then the algorithm assigns pj to machinem only when Vt,j−1 + pj > (1 + α)Lj for all t = 1, . . . ,m − 1. From this,it follows that Lj >Lj(m−1+α)+fqm+s−1 . But this is possible only when Lj >fq/(s− α) > pj/(s− α)α ≥ Lj/(2− α)α, a contradiction.Theorem 2 For α = 1.45, in any step j of algorithm S, either Vj−1+ pj ≤(1 + α)Lj or Vm,j−1 +pjs ≤ (1 + α)Lj.3Proof: In step n of algorithm S, letVn−1 + pn > (1 + α)Ln, Vm,n−1 +pns> (1 + α)Ln. (1)This means that Vm,n−1 > αLn; otherwise, we have a contradiction withpns ≤ Ln and Vm,n−1 + pns > (1 + α)Ln. Therefore, we can determine themaximum index k such that Vm,k−1 ≤ Ln and Vm,k > Ln. This means thatin any current step j, j = k + 1, . . . , n, we can apply Lemma 2.Let Z = pk.Let β = 1 + α− 1α and let γ = 1 + α− 1αβ .Let Z1, Z2, . . . , Zq = pn be the jobs assigned after job Z = pk to machine m,q ≥ 1. Let Y1, Y2, . . . , Yt be the jobs assigned to machine m before Z, andafter this assignment, let the current load become bigger than Vk. W.l.o.g.let Yj be assigned before Yj+1, j = 1, . . . , t− 1.First, we show thatβZ < Z1, βZ1 < Z2, . . . , βZq−1 < Zq. (2)Indeed, during the current step j, j = k+1, . . . , n, algorithm S either changesor does not change Vj−1. Recall that Vj < Vm,k. In the former case, let jcorrespond to the step of algorithm S after assigning Zf and before assigningZf+1. From Lemma 2, it follows that pj > αZf . Therefore, for Zf+1, wehave Zf+1 > αpj . Hence, Zf+1 > α2Zf > (1 + α − 1α)Zf for α ≥ 1.45. Inthe latter case, the result follows from Lemma 3, taking into considerationVj−1 = Vk−1 < Z/α.Therefore,Z + Z1 + . . .+ Zq < Zq(1 +1β+1β2+ . . .+1βq). (3)It should be mentioned that if job Z2 was assigned without changing Vk−1,then from Lemma 3, taking into consideration Vj−1 = Vk−1 < Z1αβ , it followsthat Z2 > γZ1.From Lemma 4, we haveY1 + . . .+ Yt < Yt(1 +1α+1α2+ . . .+1αt−1) <αYtα− 1 . (4)Let job Yt be the jobs assigned to machine m during step t1.Case 1. pn = Z1, i.e., only one job pn is assigned to machine m after job pk.41.1. Lk ≥ Vm,k−1. From Lemma 3 and the definition of k, it follows thatZ > αLk. From Lemma 3, it follows that pn > αZ. Therefore, from (1), itfollows that1 + α < Vm,k−1+Zs+ pnsLn≤ Lk+Zs+ pnsmax(Z, pns)≤ 1α + 1α + 1.For α ≥ 1.45, this is not possible.1.2. Lk < Vm,k−1. This means that Vk < Vm,k−1.1.2.1. Vk < Vn−1. The algorithm changes Vk by adding at least one job pjto the machine that has the minimum load in step k. From Lemma 2, itfollows that pj > αZ. Taking into consideration Lemmas 1 and 3, we havepn > αpj > α2Z. Therefore, from Ln ≥ Vm,k−1 and (1), we obtain1 + α < Vm,k−1+Zs+ pnsmax( pns,Ln)< 2 + 1α2,which is not possible for α ≥ 1.45.1.2.2. Vk = Vn−1. From Lemma 3, if follows that Yt > αLt1. Taking intoconsideration Lemma 3 for k = t1, we have pk > Yt(1 + α − 1α) = βYt.Therefore, from (1) and (4), it follows that1 + α < Vm,k−1+Zs+ pnsLn≤ Vm,k−1+Zs+ pnsmax(Ln,pns)≤ αβγ(α−1) + 1γ + 1,or1 + α < Vm,k−1+Zs+ pnsLn≤ Vm,k−1+Zs+ pnsmax(Ln,pns)≤ 1αβ(α−1) + 1β + 1.The first possibility corresponds to the situation when Vk = Vt1.This is not possible for α ≥ 1.45.Case 2. pn = Zq and q ≥ 2.2.1. Lk ≥ Vm,k−1 From (1), (2) and (3), it follows that1 + α < Vm,k−1+Zs+Z1s+...+Zqsmax(Z1,Zqs)< 1αβq + 1 +1β +1β2+ . . .+ 1βq ,which is not possible for α ≥ 1.45.2.2. Lk < Vm,k−1.2.2.1. q = 2 and Vt1 = Vn−1.From (1), (4) and Lemma 3, it follows that1 + α < Vm,k−1+Zs+Z1s+Z2smax(Z1,Z2s)< α(α−1)β2γ +1βγ +1γ + 1,which is not possible for α ≥ 1.45.2.2.3. q = 2, Vt1 < Vn−1.From (1), (4), Lemmas 2 and 3, it follows that51 + α < Vm,k−1+Zs+Z1s+Z2smax(Z1,Z2s)< 1αβ2(α−1) +1α2β+ 1β + 1,or1 + α < Vm,k−1+Zs+Z1s+Z2smax(Z1,Z2s)< 1αβγ(α−1) +1βγ +1γ + 1,which is not possible for α ≥ 1.45.2.4. q > 2.From (3) and (4), it follows that1 + α < Vm,k−1+Zs+Z1s+...+Zqsmax(Zq−1,Zqs)< α(α−1)βq+1 + 1 +1β +1β2+ . . .+ 1βq ,which is not possible for α ≥ 1.45.It should be pointed out that we have only applied the basic relation (1)without taking into consideration the value of m. If m is a fixed number,we can rewrite the inequality in Lemma 3 aspj > αLj +pjm+ s− 1 (5)because the difference between Vj−1 and Lj is at leastpjm+s−1 . Therefore,we can use the inequality pn > αmax(Z,Ln)m+s−1m+s−2 .Hence, for a fixed m, we can repeat the above analysis and obtain an evenbetter worst-case performance bound for our proposed algorithm.3 ConclusionsIn this note we presented a simple algorithm that yields the best knowncompetitive ratio for online uniform-machine scheduling to minimize themakespan. In addition, we developed a new technique for analyzing theworst-case performance of our algorithm, which is quite different from thetraditional approach used to analyze the LS algorithm. It is an interestingopen question whether there exists such an algorithm for the case of s > 2.AcknowledgmentsThis research was supported in part by The Hong Kong Polytechnic Uni-versity under grant number G-T596. Vladimir Kotov was also supported inpart by INTAS-Belarus 03-50-5975.6References[1] Y. Cho and S. Sahni, Bounds for list schedules on uniform processors,SIAM J. Comput., 9 (1980) 91–103.[2] R.L. Graham, Bounds on multiprocessing timing anomalies, SIAM J.Appl. Math. 17 (1969) 263–269.[3] R. Li and L. Shi, An on-line algorithm for some uniform processorscheduling, SIAM J. Comput., 27 (1998) 414–422.7

A new algorithm for online uniform-machine scheduling to minimize the makespan

Abstract

Similar works

Full text

Available Versions

PolyU Institutional Repository