Separating Online Scheduling Algorithms with the Relative Worst Order Ratio

Transcription

1 Separating Online Scheduling Algorith with the Relative Wort Order Ratio Leah Eptein 1, Lene M Favrholdt 2, and Jen S Kohrt 2 1 Departent of Matheatic, Univerity of Haifa, Irael, lea@athhaifaacil 2 Departent of Matheatic and Coputer Science, Univerity of Southern Denark, Odene, Denark, {lene,valle}@iadadudk Atract The relative wort order ratio i a eaure for the quality of online algorith Unlike the copetitive ratio, it copare algorith directly without involving an optial offline algorith The eaure ha een uccefully applied to prole like paging and in packing In thi paper, we apply it to achine cheduling We how that for preeptive cheduling, the eaure eparate ultiple pair of algorith which have the ae copetitive ratio; with the relative wort order ratio, the algorith which i intuitively etter" i alo provaly etter Moreover, we how one uch exaple for non-preeptive cheduling 1 Introduction The relative wort order ratio i a relatively new quality eaure for online algorith, inpired y the Max/Max ratio Ben-David and Borodin, 1994 and the rando order ratio Kenyon, 1996 and defined in Boyar and Favrholdt, 2003 Since it copare two algorith directly, it oetie give ore detailed inforation than the copetitive ratio So far, the reult on the relative wort order ratio have een either conitent with copetitive analyi or cloer to epirical reult and/or intuition than reult on the copetitive ratio In thi paper we analyze everal cheduling prole where the copetitive ratio doe not give the right eparation of algorith, in the hope that the relative wort order ratio would do etter In ot of the cae conidered, the relative wort order ratio prefer the intuitively et algorith wherea the copetitive ratio doe not ditinguih the algorith Analyzing the relative wort order ratio i often ore traightforward than copetitive analyi Unlike the optial offline algorith, which i generally not known, we know exactly what the two copared algorith do on a given input In any cae, thi can iplify the analyi We firt define the two quality eaure The Quality Meaure For any algorith A and any equence σ of jo, let Aσ e the akepan otained y A when cheduling σ Siilarly, let OTσ e the akepan otained y an optial offline algorith The general definition of the copetitive ratio of an algorith A i CR A inf {c : σ : Aσ c OTσ + } However, cheduling prole are typically calale Thi ean that all jo length of an adverarial intance can e caled y any factor, and thu the additive contant ha no effect Hence, for the prole conidered in thi paper, the definition reduce to CR A up σ Aσ OTσ

2 2 L Eptein, LM Favrholdt, and JS Kohrt For the prole conidered in thi paper, the relative wort order ratio i defined in the following way a general, and thu lightly ore involved, definition can e found in Boyar et al, 2005 For any algorith A and any input equence σ, let A W σ e the akepan of A on it wort perutation of σ, ie, A W σ ax p Apσ, where p i a perutation on σ eleent If A W σ B W σ for every equence σ, we ay that the two algorith are coparale, and the relative wort order ratio of A to B i WR A,B up σ A W σ B W σ For oe pair of algorith, there are equence σ uch that A W σ > B W σ and other equence σ uch that A W σ < B W σ In thi cae we ay that the two algorith are incoparale For coparale algorith, the eaure i tranitive Specifically, it i hown in Boyar and Favrholdt, 2003 that given three algorith A, B, C uch that WR C,B 1 and WR B,A 1, then WR C,A 1 and oreover in{wr C,B, WR B,A } WR C,A WR C,B WR B,A The relative wort order ratio ha previouly een applied to in packing Boyar and Favrholdt, 2003, paging Boyar et al, 2005, eat reervation Boyar and Medvedev, 2004, and in coloring Kohrt, 2004 In thi paper, the eaure i applied to cheduling The Scheduling role In the aic cheduling prole, we are given achine and a equence of jo, each characterized y the tie it take to execute it on a unit peed achine For prole where the achine have different peed, a jo of ize p require tie p when run on a achine of peed The load of a achine i the total ize of jo or part of jo cheduled on thi achine The goal i to iniize the akepan, ie, the tie when all jo are copleted The jo arrive one y one Each jo ut e cheduled at arrival, and thi chedule cannot e changed afterward In the non-preeptive cae, a jo ha to run without interruption on a ingle achine In the preeptive cae, the algorith are allowed to preept the jo and run part of it on different achine, a long a two part of a jo are never run at the ae tie For preeptive algorith it ay ake ene to ue idle tie However, the algorith tated and defined in thi paper do not ue idle tie We tudy four cheduling prole, preeptive and non-preeptive cheduling on identical achine and on two uniforly related achine Reult We firt conider the preeptive prole For identical achine, we define a cla of algorith Thi cla generalize two previouly known algorith, thoe of Seiden, 2001 and of Chen et al, 1995 For any pair of algorith in thi cla, one algorith A i etter than the other algorith B in the ene that A i never wore than B and on oe equence it i etter In contrat to the copetitive ratio which i the ae for all thee algorith, the relative wort order ratio how thi eparation The previouly known algorith are the two extree of thi cla, eing the et and the wort algorith in the cla For two uniforly related achine we again conider two previouly known algorith, of Wen and Du, 1998 and Eptein et al, 2001 We generalize the firt of the into a cla of algorith, all having the optial copetitive ratio All the new algorith in the cla turn out to e etter than the original one We copare all thee algorith to the econd one, and find that thi algorith i trictly etter than all the algorith in the cla Again, we how a clear eparation etween any two algorith in the preented cla For non-preeptive cheduling on two related achine, we again how a clear eparation etween two algorith having the ae copetitive ratio For non-preeptive cheduling on identical achine, the three algorith conidered are hown to e incoparale

3 2 reeptive cheduling to iniize akepan Separating Online Scheduling Algorith with the Relative Wort Order Ratio 3 We ue the following notation For any equence σ of n jo with ize p 1, p 2,, p n, we let and p ax denote the total and axiu ize of the jo in σ, repectively Siilarly, t and p t ax denote the total and axiu ize of the firt t jo J 1, J 2,, J t in σ For any achine i and any jo J t, L t i denote the load of achine i jut after cheduling J 1, J 2,, J t For any algorith A, Aσ denote the akepan of A chedule for σ In particular, OTσ denote the akepan of an optial offline algorith OT 21 Identical achine For the cheduling prole tudied in thi ection, we have 2 identical achine availale, and preeption i allowed We tudy two known algorith and a generalization of the two Two algorith with optial copetitive ratio Two online algorith with optial copetitive ratio have een uggeted for thi prole, REEMTIVE RE y Chen, van Vliet, and Woeginger 1995 and MODIFIED REEMTIVE MRE y Seiden 2001 In hort, they oth keep track of a axial allowale akepan M t, where M t i defined differently for the two algorith The algorith MRE i defined o that it ue at ot one preeption per jo The other algorith RE i defined to ue up to 1 preeption for each jo It i explained in Chen et al, 1995, however, that it i not difficult to adapt it o that it alo ue at ot one preeption per jo Moreover, with the ae definition of M t, the two algorith would aintain the ae et of load MRE chedule each jo J t in the following way Let e the currently ot loaded achine jut efore J t i cheduled The jo i cheduled copletely on, if thi give a akepan of at ot M t Otherwie, a uch a poile of the jo i cheduled on the leat loaded achine aong thoe on which the jo cannot finih earlier than M t, and the ret of it i cheduled on the ot loaded achine aong thoe that have load le than, giving a akepan of exactly M t Let 1,, e the equence of achine orted y non-decreaing load RE aign the t + 1 t jo J t+1 a follow Firt, a new axial allowale akepan M t+1 i coputed Then, on each achine j, the tie interval I j i reerved for jo J t+1, where I [L t, M t+1 ] and I j [L t j, L t j+1], for 1 j 1 Thoe interval are dijoint The total proceing tie that can e aigned on all interval i 1 L t j+1 L t j + M t+1 L t M t+1 L t 1 j1 To aign J t+1, go fro I to I 1, putting a part of the jo, a large a poile in each interval, until all the jo i aigned After the aignent there will e oe fully occupied interval I z+1,, I, oe epty interval I 1,, I z 1 and a partially or fully occupied interval I z Chen, van Vliet, and Woeginger 1995 prove that uing thi trategy alway reult in a feaile chedule, if M t RE ax {β t }, β pt ax, where β θ θ 1, θ 1

4 4 L Eptein, LM Favrholdt, and JS Kohrt i ued a M t Since the optial offline akepan i ax {, p ax} McNaughton, 1959, the copetitive ratio otained i β Even if randoization i allowed, thi i the optial copetitive ratio A approache e infinity, β approache e fro elow Note that β/ < 1, for 2 Seiden 2001 prove that uing M t MRE ax {β t, β 1 pt ax + β 1 } 1 t alo give a feaile chedule for any jo equence For any input equence, the akepan of MRE i never ore, and oetie le, than the akepan of RE, ince for p ax >, β 1 p ax + β 1 1 < β p ax In contrat to the copetitive ratio, the relative wort order ratio reflect the fact that MRE i never wore than RE and oetie etter Corollary 1 A generalized algorith We define a generalized algorith with a paraeter, 0 β 1 1, ADATED REEMTIVE ARE, and ue { M t ax β t }, β pt ax + t a M t In the analyi of the algorith, we aue that the algorith, like RE, ay ue everal preeption per jo ut, of coure, the reult are alo valid for the algorith uing at ot one preeption per jo, like MRE Note that, for 0, M t M t β 1 RE, wherea uing 1 lead to M t M t MRE Alo note that β i alway poitive We prove that ARE i well-defined and ha copetitive ratio β Theore 1 The relative wort order ratio how that a larger value of iplie a etter algorith Theore 2 even though all thee algorith have the ae copetitive ratio The algorith aintain the following three invariant Thee invariant are the ae a the invariant defined in Seiden, 2001, except the change of M into M t 1 At any tie t, L t 1 Lt 2 Lt 2 At any tie t, L t M t 3 At any tie t, for every 1 k, k i1 L t i θk 1 θ 1 t The firt two invariant follow fro the definition of the algorith The third i proved in Lea 3 Firt, we ue the invariant to how that it i alway poile to partition a jo aong it deignated interval Lea 2 Lea 2 ue the following lea, howing that which ter i axiu in the definition of M t depend only on t, p t ax, and ; not on Lea 1 After t jo, ARE ha M t β t if and only if t pt ax

5 Separating Online Scheduling Algorith with the Relative Wort Order Ratio 5 roof Thi follow fro β t β pt ax + t β t β p t ax t pt ax Lea 2 If the invariant are fulfilled at tep t, then the reerved interval are ufficient to aign J t+1 roof We conider two cae and how that the aignent i ucceful in oth cae Cae 1: p t+1 > t+1 The total ize of the reerved interval i M t+1 L t 1 β p t+1 ax + t+1 θ 1 β + p t+1 + Note that θ 1 θ 1 β 1 1 : θ 1 t, y Lea 1 M and the third invariant θ 1 θ t, ince p t+1 ax p t+1 and t+1 p t+1 + t 1 β 1 1 θ 1 θ 1 θ θ 1 1 θ 1 θ 1 1 θ θ 1 θ Uing t < 1p t+1 which follow fro p t+1 > t+1 M t+1 L t 1 β + p t+1 + β 1 1 Therefore p t+1 can e aigned into the interval Cae 2: p t+1 t+1 The total ize of the reerved interval i and t+1 p t+1 + t, we get 1p t+1 p t+1 M t+1 L t 1 β t+1 β 1 1 t β t β 1 + p t+1 1 t β p t+1 + β 1 t Uing t 1p t+1 we get, M t+1 L t 1 p t+1 Therefore p t+1 can e aigned into the interval in thi cae a well To coplete the proof that the algorith i well-defined, we need to how that all invariant are kept after an aignent of a new jo For the firt two invariant, thi i clear fro the definition of the algorith Since all load are initially zero, it uffice to prove the following lea for the lat invariant

6 6 L Eptein, LM Favrholdt, and JS Kohrt Lea 3 If the third invariant i fulfilled after tep t, then it i alo atified after tep t + 1 roof According to the definition of the algorith, there exit a achine z uch that for i < z, L t+1 L t i, for z < i, Lt+1 i L t i+1, and Lt z < L t+1 z L t z+1 for convenience let Lt +1 M t For k < z, k i1 L t+1 i k i1 For k z, it i ufficient to how the inequality L t i θk 1 θ 1 t θk 1 θ 1 t+1 i t+1 k i1 L t+1 i ik+1 L t+1 i θ θ k θ 1 t+1 t+1 θk 1 θ 1 t+1 Since k + 1 > z, L t+1 i ik+1 L t+1 i Hence, it uffice to how L t i+1, and the left hand ide i equal to k+1 ik+2 L t i θ θ k+1 θ 1 M t+1 Thi i proved uing that y definition, + M t+1 t i1 L t i t + M t+1 θ θ k+1 θ 1 + M t+1 θ θ k+1 θ 1 pt+1 ax + θk+1 θ k θ 1 t+1 t+1 p t+1 ax + M t+1 M t+1 β t+1 and M t+1 β p t+1 ax + t+1 Let α θ θ k+1 θ 1β and note that 0 α 1 Multiplying the econd inequality i y α and the firt y 1 α and adding the two reulting inequalitie, we arrive at M t+1 θ θ k+1 θ 1 pt+1 ax + θ θ k+1 θ 1 pt+1 ax + θ θ k+1 θ 1 pt+1 ax θ θ k+1 θ 1 θ θ k+1 θ 1β t θ θ k+1 θ β t+1 + 1β t+1 + β t+1 θ θ k+1 θ 1 pt+1 ax θ θ k+1 + βθ 1 t+1 θ 1 θ θ k+1 θ 1 pt+1 ax + θk+1 t+1 θ 1 θ θ k+1 β t+1 θ 1β β t+1

7 Thu, we jut need to prove that Separating Online Scheduling Algorith with the Relative Wort Order Ratio 7 θ k+1 t+1 θ 1 θk+1 θ k θ 1 t+1, which i equivalent to Now, θ k+1 θk+1 θ k θ k+1 θk+1 θ k 1 Thi coplete the proof of the inequality M t+1 k+1 k+1 k θ θ k+1 θ 1 pt+1 ax + θk+1 θ k θ 1 t+1, concluding the cae k z We are now ready to prove Theore 1 Theore 1 For every 0 β 1 1, ARE i well-defined and ha copetitive ratio β roof By Lea 2 and 3 and the dicuion efore Lea 3, the algorith i well-defined The copetitive ratio i not etter than β, ince thi i the optial copetitive ratio For the upper ound on the copetitive ratio, conider any input equence σ If M σ β, then clearly the copetitive ratio i at ot β, ince OT ax{, p ax} Otherwie, y Lea 1, < p ax and thu, M σ < βp ax We can now find the relative wort order ratio of pair of algorith with optial copetitive ratio For thi purpoe, we prove the following lea Lea 4 Let σ e an input equence with n jo and let 0 β 1 1 A perutation, σ w, of σ where the jo appear in order of non-increaing ize i a wort order for ARE, and ARE σ w in {, M σ} roof Note that L t+1 in{l t + p t+1, M t+1 } Since L t + p t+1 a well a M t+1 are axiized when t+1 i axiized, we can prove y induction that the larget akepan after i jo i achieved if the firt i jo are the larget one Therefore, no order can e wore than a non-increaing order Thu, it i enough to how that a non-increaing order give ARE σ L n in{, M σ} Let p 1,, p n e the orted lit of jo ize We prove that if L t M t then Lt+1 M t+1 Aue that L t M t Then, the interval reerved for jo J t+1 on the ot loaded achine i M t+1 M t

8 8 L Eptein, LM Favrholdt, and JS Kohrt If M t+1 β t+1, then M t+1 M t β t+1 M t β t+1 β t β p t+1 < p t+1, ince β < 1 Thu, the interval i filled copletely, giving a akepan of M t+1 β t+1 Otherwie, M t+1 β p t+1 ax + t+1, and ince p t ax p t+1 ax p 1, M t+1 M t+1 t+1 t p t+1 < p t+1, ince < 1 Thu again, the interval i filled copletely, giving a akepan of M t+1 β p t+1 ax + t+1 We are now ready to prove the lea A long a jo are aigned o that the deignated interval on the ot loaded achine i not filled, there are no jo aigned to other achine, and the akepan i t at tie t Once thi interval i filled copletely, we howed that it will e filled in the next tep a well Thi prove the clai Lea 5 For every pair of value 0 1 < 2 β 1 1, ARE 1 and ARE 2 are coparale, and ARE 2 i never wore than ARE 1 roof Conider any input equence σ By Lea 4, it i ufficient to prove that M 2 σ M 1 σ, and y Lea 1, it i ufficient to conider the cae when < p ax The clai then iediately follow y the definition of M, ince M 2 σ M 1 σ p ax 2 1 < 0 Lea 6 For any pair of value 0 1 < 2 β 1 1, WR ARE 1,ARE 2 β 1 β roof By Lea 5, it i ufficient to find an input equence giving the tated ratio Let λ 1 1 1, λ 2 β Note that λ 1 > λ 2, ince > β a β < 2 Moreover, ince 1 < β 1 1, we have λ 2 > 0 Conider the input equence σ λ 1, λ 2 By Lea 4, it i ufficient to conider thi ordering of the two jo We have β 1, and p ax λ Note that, for any, we get the ae reult with ARE, if the lat jo of ize λ 2 i plit into aller jo of total ize λ 2 We have < ince β <, and thu < p ax Hence, y Lea 1, we ut have p ax β M i σ β i p ax + i Now, y Lea 4, ARE i σ M i σ if any only if β i p ax + i Thi i equivalent to, which hold for 1 with equality y the definition of λ 1 and p ax β i 1 i λ 2 Moreover, β i 1 i i a onotonically decreaing function of i, and hence β 2 Thu, 1 2 β ARE 1 σ ARE 2 σ β 1λ λ 1 + λ 2 β 2 λ λ 1 + λ 2 β 1 β β β 1 β β 1

9 Separating Online Scheduling Algorith with the Relative Wort Order Ratio 9 The following lea give a atching upper ound on the relative wort order ratio Lea 7 For any pair of value 0 1 < 2 β 1 1, WR ARE 1,ARE 2 roof Conider any input equence σ By Lea 4, there are three poile cae: β 1 β If ARE 2 σ, then ARE 1 σ ARE 2 σ If ARE 2 σ β, then y Lea 1, p ax, and ARE 1 σ ARE 2 σ Finally, if ARE 2 σ β 2 p ax + 2, then y Lea 1, p ax By the ae lea we get M t 1 β 1 p ax + 1, and thu ARE 1 σ in{, β 1 p ax + 1 } If ARE 1 σ, we have Conequently, ARE 1 σ ARE 2 σ p ax β β 2 p ax + 2 β β β 1 β Otherwie, p ax β and ARE 1 σ β 1 p ax + 1 We have, ARE 1 σ ARE 2 σ β 1 p ax + 1 Thi i a function which i onotonically decreaing in find it axiu, which i again p ax p ax β 2 p ax + 2 β β β 1 β p ax Thu, we can utitute A the ratio proved in the two lea atch, we arrive at the following theore Theore 2 For any pair of value 0 1 < 2 β 1 1, where β θ θ 1 and θ 1 WR ARE1,ARE 2 β 1 β > 1, p ax β roof The ratio follow directly fro Lea 6 and 7 It i eaily checked that the ratio i greater than 1: β 1 β > 1 β 1 > β , ince β > > β 2 1 Sutituting 1 0 and 2 β 1 1 in β 1 β , we get the following corollary Corollary 1 WR RE,MRE β 1 +β 2 2β, where β θ θ 1 and θ 1 For 2, 3, and 4, the ratio WR RE,MRE i , , and , repectively A approache infinity, the ratio approache β, the copetitive ratio of the two algorith Note that for any other pair of algorith conidered in thi ection, the relative wort order ratio i aller to

10 10 L Eptein, LM Favrholdt, and JS Kohrt 22 Two related achine We now turn to the cae of two uniforly related achine, 1 of peed 1 and 2 of peed 1 A in the previou ection, the goal i to iniize the akepan, and preeption i allowed For any input equence σ, a traightforward optial offline algorith wa found y Gonzalez and Sahni 1978 The akepan found y thi algorith i { OTσ ax + 1, p } ax Two algorith with optial copetitive ratio Two lightly different deterinitic online algorith found independently y Wen and Du 1998, and y Eptein et al 2001 oth have an optial copetitive ratio of + 12 CR α Thi i 4 3 for 1, and decreae for increaing The ratio i optial even if randoization i allowed The algorith of Wen and Du 1998 [ work iilarly to the algorith RE Chen et al, 1995 For jo L t J t+1, it reerve the tie interval 2, α OT t+1] [ ] on the fat achine, and the interval L t 1, Lt 2 on the low achine Each jo i aigned firt to the reerved interval on the fat achine, and the reainder if any, to the other reerved interval The algorith of Eptein et al 2001 i different in the ene that on aignent of a jo, it alway aign to the low achine a uch a poile, ut not ore than t , and not ore than Lt 2 to avoid overlap The reainder of the jo i aigned to the fat achine A cla of algorith with optial copetitive ratio We define a cla of algorith called TWO-REEMTIVE c TRE c which ue a paraeter c, uch that 0 c c ax, c ax Whenever a new jo J t+1 arrive, chedule a large a fraction a poile on the fat achine within the tie interval [ 1 Lt 2, 1 M c t+1 ], where M t+1 c { ax α + 1 t+1, p t+1 ax 1 + c + } t+1 p t+1 ax cax c [ Schedule the reaining part of J t+1 on the low achine, within the tie interval L t 1, Lt 2 ] In oe cae, TRE c achieve a akepan which i etter than that of the algorith of Wen and Du 1998 Later, we alo copare it to the algorith of Eptein et al 2001 Firt, we prove that the algorith i well-defined and ha an optial copetitive ratio of α Theore 3 For that we need the following lea Lea 8 After t jo, TRE c ha M t c α +1 t if and only if t +1 p t ax

11 Separating Online Scheduling Algorith with the Relative Wort Order Ratio 11 roof Thi follow fro t + c t p t ax 1 + c + t p t 2 ax c c t + 1 t + 1 p t ax p t ax + c p t ax c Theore 3 For any c, 0 c c ax, TRE c i well-defined and ha copetitive ratio α roof Firt, we prove that the algorith i well-defined, ie, the reerved tie interval are alway ufficiently long To thi end we how that the algorith aintain the following two invariant 1 L t 2 M c t 2 L t t We conider the jo to e cheduled one at a tie, and how that each jo can e cheduled oerving the two invariant For J 1, the invariant clearly hold, ince the jo i copletely cheduled on the fat achine, and Mc 1 p c p 1 Now conider J t+1, t 1, and aue that the invariant hold jut efore J t+1 i aigned By the definition of the algorith, the firt invariant till hold after aigning J t+1 A for the econd invariant, if 1 L t 1 then it i clearly aintained Otherwie Lt+1 1 t+1 Mc t+1 t+1 α Thi i exactly the econd invariant The aount of tie availale for the new jo i L t+1 M t+1 c L t L t Lt 1 M t+1 c + Lt 2 t t+1 +1 t To coplete the proof that the algorith i well-defined, we jut need to prove that thi aount i at leat p t+1, or equivalently, that M t+1 c Lt 2 + t + p t+1 Lt 1 t + t + p t+1 Lt 1 + t+1 t We conider two cae depending on which of the two ter in the definition of M t+1 c If Mc t+1 t+1 α +1 then By Lea 8, t+1 +1 p t+1 ax Thu, i axiu t t+1 p t+1 t+1 p t+1 ax 1 pt+1 ax 1 p t+1

12 12 L Eptein, LM Favrholdt, and JS Kohrt Further, y uing the invariant L t 1 t it uffice to prove Mc t+1 t+1 α t t 2 + p t t+1 t By rearranging the ter and uing t+1 t + p t+1, we get t p t+1, which follow directly fro t 1 p t+1 In the econd cae, y the definition of Mc t+1 we need to how +1 Mc t+1 p t+1 ax 1 + c Rearranging we get c By Lea 8, p t+1 ax + t+1 p t+1 2 ax c Lt 1 + t+1 t p t+1 ax Lt c t+1 t +1 t+1 Sutituting +1 t+1 for p t+1 ax on the left hand ide, we get L t 1, which i iplied iediately y the econd invariant t Next, we how that TRE c σ α OTσ Thi i clear in the firt cae In the econd cae, p ax, ie, p ax pax Thu, TRE c σ p ax 1 + c + p ax c p ax 1 + c c α p ax α OTσ We firt etalih the relative wort order ratio etween pair of algorith in the cla TRE c Siilarly to Lea 4, we can how the following lea in which we identify the propertie of the output of the algorith defined aove and find an order which i alway a wort ordering Lea 9 For TRE c and for every c, and any input equence σ, a non-increaing order i alway a wort order The akepan for a wort perutation, σ w, of σ i { TRE c σ w in, 1 } M cσ roof Note that no order can give a larger akepan, o it i enough to how that a non-increaing order actually give thi akepan Let σ e any input equence, where the jo ize are given in non-increaing order, ie, p 1 p ax and p i 1 p i for i 2 Note that p t ax p ax p 1 for all t Firt, we prove the following clai If the akepan of TRE c after jo J t i aigned i 1 M c, t then the akepan after J t+1 i aigned ut e 1 M c t+1 We have two cae:

13 t+1 Separating Online Scheduling Algorith with the Relative Wort Order Ratio 13 If Mc t+1 α +1, then uing M c t α t +1, we get that the total load that can e cheduled within the deignated interval on the fat achine for J t+1 i at ot Mc t+1 Mc t t+1 α + 1 α t + 1 α + 1 p t p t+1 < p t+1, ie, the interval i filled copletely If Mc t+1 p t+1 ax 1 + c + t+1 p t+1 ax cax c, then the total load that can e cheduled within the deignated interval on the fat achine for J t+1 i M t+1 c M t c p t+1 c ax c < p t+1, ie, the interval i again filled copletely The clai i therey proved We are now ready to prove the lea A long a jo are aigned o that the deignated interval on the fat achine i not filled, there are no jo aigned to the low achine, and the akepan i t after jo J t Once thi interval i filled copletely, we howed that it will e filled in the next tep a well Thi prove the lea Lea 10 For every pair of value 0 c 1 < c 2 c ax, TRE c1 and TRE c2 are coparale, and TRE c1 i never wore than TRE c2 roof Conider any input equence σ By Lea 9, we only need to check the relation etween the akepan of the two algorith for a non-increaing orted order of jo with akepan a hown in the Lea We clai that TRE c2 i never etter than TRE c1 according to the relative wort order ratio By Lea 9, there are three cae: If TRE c2 σ, then clearly TRE c 1 σ TRE c2 σ If TRE c2 σ α +1, then y Lea 8, p ax +1 and thu, TRE c 1 σ Mcσ TRE c2 σ α +1 If the akepan of TRE c2 i given y the econd ter in the axiu, then y Lea8, p ax +1 Therefore the akepan of TRE c1 can either e given y the econd ter in the axiu, or it can e le ie, equal to The econd ter in the axiu i a function which i onotonically nondecreaing a a function of c for the cae p ax +1, therefore the akepan of TRE c 1 i not larger than the one of TRE c2 Lea 11 For any pair of value 0 c 1 < c 2 c ax, WR TRE c 2,TREc 1 c c c 1 + roof Conider the input equence σ c 2, c 2 By Lea 9, it i enough to conider thi perutation of the equence

14 14 L Eptein, LM Favrholdt, and JS Kohrt We have c , p ax c , and +1 p ax c > Thu, y Lea 8, M c2 p ax 1 + c 2 + p ax c c c c c 2 2 c c Thi function i onotonically increaing a a function of c 2, hence M c1 < M c2 Thu, a lower ound on WR TRE c 2,TREc 1 i given y Mc 2 M c1 We have M c1 p ax 1 + c 1 + p ax c c c c c 2 2 c 1 c c c c c 1 Thi give the tated lower ound Lea 12 For any pair of value 0 c 1 < c 2 c ax, WR TRE c 2,TREc 1 c c c 1 + roof Conider any input equence σ By Lea 9, we only need to conider the equence in nonincreaing ize order By Lea 9, there are three poile cae: If TRE c1 σ, then TRE c 2 σ TRE c1 σ and thu, y Lea 10, TRE c2 σ TRE c1 σ If TRE c1 σ α +1, then y Lea 8, p ax +1, and TRE c 2 σ TRE c1 σ Finally, if the akepan of TRE c1 i given y the econd ter in the axiu with the paraeter c 1, y Lea 8, +1 p ax, and thu y the ae lea, the akepan of TRE c2 i either given y the econd ter in the axiu with the paraeter c 2 or equal to If TRE c2 σ, we get WR TRE c 2,TREc 1 p ax 1 + c 1 + pax c ax c 1 p ax 1 + c 1 + p ax 1 c ax c 1 Thi function i onotonically non-decreaing in the ratio p, and thu we calculate the axiu ax ratio Since TRE c2 σ, p ax 1 + c 2 + pax c 2, which i equivalent to c p ax c

15 Separating Online Scheduling Algorith with the Relative Wort Order Ratio 15 Thu, p ax c c c c x + +1 x +, where x c Sutituting thi in the upper ound on WR TRE c 2,TREc 1 we get WR TRE c 2,TREc c 1 + x+ x+ +1 x+ +1 x+ +1 x+ +1 x+ x c 1 +1 x + + x c ax c 1 +1 x c ax c 1 x + Nx, where Nx x x + 1 x x + + c 1 + x c ax + 1 c x c ax + 1 x 1 c ax + + c 1 + x c ax 1 c c 1 + c 2 2 Thi give the tated upper ound If TRE c2 σ <, we get x c 1 + x + 1 x c ax + + c 1 + c WR TRE c 2,TREc 1 p ax 1 + c 2 + pax c ax c 2 p ax 1 + c 1 + pax c ax c c 2 + p 1 c ax c 2 ax 1 + c 1 + p 1 c ax c 1 ax Thi function i onotonically non-increaing a a function of the ratio p, and thu we calculate ax the axiu ratio Since TRE c2 σ <, we get p ax 1+ c 2 + pax c 2 Thu, iilarly to the previou cae we get c p ax c c c ,

16 16 L Eptein, LM Favrholdt, and JS Kohrt Sutituting thi in the upper ound on WR TRE c 2,TREc 1 iple algera, we get uing the definition of c ax and WR TRE c 2,TREc 1 + c 2 c c c ax c 2 + c 1 c c c ax c 1 c c c 2 2 c c c 2 2 c c c c c c 1 + Theore 4 For any pair of value 0 c 1 < c 2 c ax, WR TRE c 2,TREc 1 By utituting c 1 0 and c 2 c ax we get the following corollary Corollary 2 WR TRE cax,tre c c c 1 + Next, we would like to copare the aove cla of algorith to the algorith of Eptein et al 2001 called BEST REEMTIVE BRE We firt how that the et algorith in the cla, TRE 0, i wore than BRE Due to the tranitivity of the relative wort order ratio, thi iplie that BRE i etter than all algorith TRE c for any c The algorith BRE i different fro the aove algorith in the ene that it, except for the firt jo which i cheduled copletely on the fat achine, BRE alway chedule a large a part on the low achine a poile while aintaining the following two invariant: 1 L t 1 Lt If L t 1 < Lt 2 2 +, then alo Lt 2 α pt ax Note that the firt invariant iplie that L t 1 Lt 2, ie, the akepan of BRE i alway Lt 2 Alo note that y the ae invariant, t L t 1 + Lt L t Lt 2, ie, t 1 Lt 2 1 We prove that TRE 0 can never have a aller akepan than BRE, on any equence Note that BRE doe not necearily have the wort akepan in the cae that the equence i orted y non-increaing jo ize A an exaple, conider the cae 2 and the jo 2, 12 In the order 12, 2, the jo 12 i cheduled on the fat achine The jo 2 fit perfectly on the low achine, and the akepan i However, if the jo 2 i aigned firt, then the low achine can receive at ot ize 1 in the next tep, ince otherwie the two part of the econd jo cheduled on the low and the fat achine, repectively, will overlap in tie Therefore, the akepan i

17 Lea 13 For any input equence σ, BREσ TRE 0 σ Separating Online Scheduling Algorith with the Relative Wort Order Ratio 17 Note that the clai of the lea i tated for any equence without any reordering roof Given a equence of jo, σ, we how that the lea hold at every tep uing induction Clearly, the lea hold efore any jo i aigned Next, aue that the lea hold for the previou part of the input equence, and conider the next jo, J t+1 We now have everal cae: If the jo i aigned uch that L t+1 2 of BRE doe not change, then the lea hold y induction If TRE 0 aign the jo copletely to the fat achine, then ince BRE ay aign at ot that uch to the fat achine, and ha a akepan which i not larger than the one of TRE 0 efore the aignent, thi ituation reain after the aignent Invariant 2 for TRE 0, L t t+1, can e rewritten to L t Lt+1 2 Hence, if BRE aign the jo uch that L t , TRE 0 ut have at leat the ae value of L t+1 2 a BRE, 2 + Lt+1 ie, the akepan of TRE 0 i at leat the ae a the akepan of BRE Finally, the lat cae i when for BRE L t+1 1 < Lt+1 2 and it wa ipoile for BRE to put everything on the low achine, ie, L t Lt 2 utting thi together we otain Lt 2 < 1 +1 Lt+1 2, and uequently t+1 L t L t+1 1 > + 1L t Lt L t 2 Uing thi we get that p t+1 t+1 t > L t 2 t + 1 t t t, where the econd inequality hold y 1 We conclude that p t+1 p t+1 ax By Lea 8 and 9 and ince we know that TRE 0 did not chedule jo p t+1 on the fat achine only, the akepan of TRE 0 ut e M0 t+1 σ 1 p t t For BRE y the Invariant 1, the akepan i at ot L t+1 2 t + p t+1 1 Lt 2 Therefore, the clai i proved t + p t t p t+1 + t By Lea 9 and ince M c σ i non-decreaing a a function of c, we iediately otain the following corollary Corollary 3 For every c and any input equence σ, BREσ TRE c σ Next, we etalih the relative wort order ratio etween BRE and each algorith TRE c We prove the following theore Theore 5 The relative wort order ratio etween BRE and TRE c i { c WR TRE c,bre ax c , c + 4 } For 1 and c c ax , thi value i α 4 3 For > 1, thi value i aller than α For all value of the relative wort order ratio i a trictly onotonically increaing function of roof To prove the lower ound, conider two input equence

18 18 L Eptein, LM Favrholdt, and JS Kohrt The firt equence conit of two jo of ize and Here and p ax Since M c σ, and ince +1 p ax, and uing Lea 8, the akepan of TRE c i Mcσ c c Next, we clai that the akepan of BRE i + 1 for oth poile perutation of the jo in the equence Both cae reult in the ae akepan ince in oth cae we have a iilar ituation a follow The firt jo i aigned to the fat achine, and out of the econd jo, a part of ize 1 i aigned to the low achine, to get a alance etween the load of the two achine a the the definition of the algorith tate In total, we get the ratio c The econd equence i defined for c > 0 It conit of the two jo c and c Here c and p ax c Since M c σ, uing Lea 8 we need to conider the econd ter in the axiu to find M c σ We have M c σ c Therefore, the akepan of TRE c i M cσ c For BRE, if the aller jo i aigned firt, then the part of the econd jo that can e aigned on the low achine i at ot c which i conitent with Invariant 1 If the larger jo i aigned firt, the part aigned to the low achine can only e larger Thu we get that the akepan of BRE i at ot L 2 c c By dividing the two, we get the ratio c c To prove the upper ound, we tart with giving lower ound on the akepan of BRE Given an input equence σ let R p ax, ie, the u of all jo except the larget By Invariant 1, we have BRE W σ R + p ax Now conider an ordering of the jo, uch that p ax i lat Let µ e the akepan efore p ax i aigned After p ax i aigned the akepan i at leat µ plu the tie to run p ax inu the aount that can e cheduled on the low achine, which i at ot µ R µ, ie, the akepan i at leat µ + p ax µ+r µ p ax µ+r Since µ R, we get at leat BRE Wσ p ax+r R 2 The akepan of TRE c for any poile order never exceed { ax α + 1, p ax 1 + c 2 + R c } If the firt option i the axiu then BRE ha at leat the ae akepan y 1 For the econd option, we conider everal cae If p ax R + 1, we ue the firt lower ound, and get a ratio etween the two algorith of at ot p 1 ax + c + R c R + p ax

19 Separating Online Scheduling Algorith with the Relative Wort Order Ratio 19 Let ρ p ax R If R 0, we have p ax and thu there i only one jo In thi cae all algorith act in the ae way, therefore we do not conider thi option Dividing the nuerator and the denoinator y R and utituting we get the function ρ 1 + c c ρ ++1 Thi function i onotonically nondecreaing a a function of ρ Hence, we utitute p ax R + 1, ince thi correpond to the axiu value of ρ Thu, we find the axial value to e c If p ax R + 1, we ue the econd lower ound and get a ratio etween the two algorith of at ot p 1 ax + c + R c p ax + c + R c p ax +R R 2 Let ρ p 3 ρ+c+ ax R We get the function ρ+ 1 c: c p ax + R R We now have two cae depending on the value of For c, the function aove i onotonically non-increaing and otherwie onotonically non-decreaing Therefore in thi cae we can utitute p ax R + 1 or ρ to find the axiu, which turn out to e c For larger value of c, we conider firt the cae when ρ c Note that thi value i c trictly larger than for any value of c > 0 The akepan of TRE c i alo at ot p ax+r, p ax +R and the ratio can e ounded y p ax +R R p ax+r p 2 ax +R R Uing ρ p ax ρ+ R we get the function ρ+ 1 which i onotonically non-increaing a a function of ρ We utitute uing ρ c to find the axiu which give the value c c c into ρ+c+ 3 c Otherwie, if + 1 ρ c , we utitute ρ c c which i onotonically non-decreaing for the current value of c, to get the axiu Thi again give c c c ++1 ρ+ 1, To find the relative wort order ratio etween BRE and TRE cax, which are the algorith of Eptein et al 2001 and Wen and Du 1998, we utitute the value of c and get the following corollary Corollary 4 WR TRE cax,bre Thi value i aller than α for every > 1 3 Non-preeptive cheduling to iniize akepan For copletene, we conider non-preeptive algorith a well

20 20 L Eptein, LM Favrholdt, and JS Kohrt 31 Identical achine Firt, we conider the cheduling prole for identical achine where preeption i not allowed, ie, a jo cannot e interrupted and run on ore than one achine For thi prole, a claical reult wa preented y Graha 1966 Graha conider the natural greedy algorith LIST, which alway chedule a jo on the leat loaded achine By Graha, 1966, LIST i 2 1 -copetitive Thi i optial when 3 Faigle et al, 1989 For 4, thi reult wa later iproved Firt y Galao and Woeginger 1993 with an 2 1 ɛ -copetitive algorith RLS Unfortunately for approaching infinity, ɛ tend to 0, ie, for general, thi reult i not etter Later Bartal et al 1995 gave an algorith which i 1986-copetitive, ut only for at leat 70 achine Karger, hilip, and Torng 1996 generalized thi and gave a copetitive algorith CHASM α Aler 1999 iproved thi even further, and gave a 1923-copetitive algorith M2 The current et reult i y Fleicher and Wahl 2000 They preent an algorith MR with a copetitive ratio of ln 2 /2 < , ut only for 64 The currently et lower ound for the prole wa etalihed y Gorley et al 2000 at Thi i a light iproveent of the previou lower ound y Aler 1999 Wherea the firt algorith LIST keep the load of all achine a cloe a poile to the average load, eentially all the later algorith alway keep a certain fraction of the achine ufficiently elow the average load, uch that they can accept a large jo without violating the copetitive ratio In thi Section, we how that Graha algorith, LIST, and the two ot recent algorith, M2 and MR, are pairwie incoparale uing the relative wort order ratio Thi i done uing the following two input equence: σ 1 conit of 1 unit ized jo, and σ 2 conit of the ae jo a σ 1 with an additional large jo of ize Note that all the jo of σ 1 are the ae ize, hence all perutation are equal For σ 2, we only need to conider the location of the large jo in the equence The optial algorith ditriute the jo in σ 1 evenly aong the proceor and get a akepan of 1 For σ 2, the large jo i put on a achine y itelf, and all the unit-ized jo are put on the reaining 1 achine, yielding a akepan of LIST ditriute the jo in σ 1 iilar to OT with a akepan of 1 For σ 2, the large jo of ize i placed on one of the achine with a load of 1, ie, a total akepan of 2 1 Thi i the wort poile perutation for LIST For the lat two algorith, we only give a ketch of the proof and we only conider the cae for approaching infinity, ince thi iplifie the calculation If neceary, the calculation can e done for any pecific with the ae concluion a a reult, naely that the three algorith are pairwie incoparale M2 and MR oth divide the achine into two group: the leat loaded all achine 1, 2,, and the reaining large achine +1, +2,, We have: M c 2c 2 and 1 MR c + 1 2c2 4c c where c ln 2 2, the copetitive ratio of MR Depending on different condition the algorith chooe etween putting a new jo on the leat loaded all achine, 1, or the leat loaded large achine, +1 When only conidering unit-ized jo, it can e hown that at any tie the difference in load for the leat and ot loaded all achine i at ot one The ae reult hold for the large achine It can alo e hown that for oth algorith a wort ordering

21 Separating Online Scheduling Algorith with the Relative Wort Order Ratio 21 of σ 2 i when the large jo appear a the lat jo, and in thi cae thi jo i placed on the leat loaded achine, 1 For M2 and σ 1, the ratio etween L 1 and L approache α for approaching infinity, where α 0923 M M2 0923/ / Next, for σ 1, we get 1 M2 L σ M2L σ 1 2 α+1lσ 1, and hence L σ OTσ OTσ 1 For σ 2, the load of L can e found a L σ 2 L σ αlσ α+1 1 2α α OTσ For MR and σ 1, the ratio etween L 1 and L approache β for approaching infinity, where β 2c 3 2c c 2, 1+ln 2 with c 1 + 2, the copetitive ratio of MR Now, for σ 1, we have, 1 MR L σ MRL σ 1 MR β + MR L σ 1 + MR β 1 L σ 1 2c 2 4c c L σ 1 2c 2 2c 2 2c2 4c + 1 L σ 1 1 c 2c 2 2c 1 c L σ 1 c2c 2 Hence, L σ 1 1c2c 2 2c 1 c 2c2 2c 2c 1 1 2c2 2c 2c 1 OTσ OTσ 1 For σ 2, the load of L can e found a L σ 2 L σ βl σ 1 + 2c 3 2c 2 2c 1 + 2c 1 2c 1 2c 3c 2c 1 2c2 c 1 2c 1 2c2 c 1 2c c 3c 2c 1 OTσ OTσ c 3c 2c 1

22 22 L Eptein, LM Favrholdt, and JS Kohrt The reult for approaching infinity are uarized in the following tale Recall that ALG W σ denote the akepan of ALG on the wort perutation of σ Note that for any pair of the three algorith, the order of the two i different for σ 1 when copared to σ 2 ALG ALG Wσ 1 ALG W σ 2 OTσ 1 OTσ 2 LIST 1 2 M MR Theore 6 LIST, M2, and MR are pairwie incoparale It i not urpriing that LIST i incoparale to M2 and MR The two latter algorith are deigned to do lightly ad on oe input equence, like the equence with unit ized jo, in order to avoid even wore perforance on other input equence 32 Two related achine Aue now that we have two achine availale, and one achine i a factor of tie fater than the other, 1 reeption i till not allowed Let OST-GREEDY e the algorith that chedule each jo on the achine where it will finih firt By Cho and Sahni, 1980; Eptein and Sgall, 2000; Eptein et al, 2001, OST-GREEDY ha an optial copetitive raito of , if φ, and, if φ, where φ 1618 i the golden ratio It i eay to ee that the algorith FAST that iply chedule all jo on the fatet achine i +1 -copetitive Eptein et al, 2001 Hence, for φ, oth algorith have the optial copetitive ratio However, OST-GREEDY ee to e the ore reaonale algorith: it never give a larger akepan than FAST, and in any cae it even ha a uch aller akepan Thi i reflected y the relative wort order ratio: For any n 1, conider the input equence coniting of n + 1 jo of unit ize On thi input equence, FAST ha a akepan of n+1 and OST-GREEDY ha a akepan of at ot n Since n+1 approache +1 a n approache infinity, WR FAST,OST-GREEDY +1, and ince thi ratio cannot e larger than the copetitive ratio of FAST, the reult i tight Theore 7 WR FAST,OST-GREEDY +1 4 Concluion In thi work we have applied the relative wort order ratio to a few online prole For ot of the conidered cheduling prole, copetitive analyi doe not ditinguih etween different optial algorith, wherea uing the relative wort order ratio we are ale to ditinguih the algorith, and in all cae the ratio prefer the intuitively etter algorith For non-preeptive cheduling on identical achine, the conidered algorith are incoparale uing the relative wort order ratio, even when the algorith have different copetitive ratio The reaon for thi i that, except for the firt algorith LIST, the algorith have een pecially tailored to get good

23 Separating Online Scheduling Algorith with the Relative Wort Order Ratio 23 copetitive ratio, ie, to work well on wort-cae equence Thi i done at the expene of getting a ad akepan for any noral input equence, where the algorith with a wore copetitive ratio are etter The copetitive ratio eaure in thi cae prefer certain non-preeptive algorith wherea the relative wort order ratio allow u to ee that the algorith are incoparale The order of their relative perforance depend on the type of input equence given, and it i ipoile to ay that one algorith i generally etter than the other for all input equence In general, our reult how that, in any cae, the relative wort order ratio can otivate earching for etter algorith, even when an algorith with optial copetitive ratio ha een found We aw that in any cae, a very all change in the algorith, without changing the copetitive ratio, can e iediately een in the reulting relative wort order ratio Biliography S Aler Better ound for online cheduling SIAM J Coput, 292: , 1999 Y Bartal, A Fiat, H Karloff, and R Vohra New algorith for an ancient cheduling prole J Coput Syte Sci, 513: , 1995 S Ben-David and A Borodin A new eaure for the tudy of on-line algorith Algorithica, 111:73 91, 1994 J Boyar and LM Favrholdt The relative wort order ratio for on-line algorith In roc 5th Italian Conf on Algorith and Coplexity, vol 2653 of Lect Note Cop Sci, pp Springer-Verlag, 2003 J Boyar, LM Favrholdt, and KS Laren The relative wort order ratio applied to paging In roc 16th Annu ACM-SIAM Syp Dicrete Algorith, pp , 2005 J Boyar and Medvedev The Relative Wort Order Ratio Applied to Seat Reervation In roc of the 9th Scand Workhop on Algorith Theory, vol 3111 in Lect Note Cop Sci, pp , 2004 B Chen, A van Vliet, and GJ Woeginger An optial algorith for preeptive on-line cheduling Oper Re Lett, 183: , 1995 Y Cho and S Sahni Bound for lit chedule on unifor proceor SIAM J Coput, 91:91 103, 1980 L Eptein, J Noga, SS Seiden, J Sgall, and GJ Woeginger Randoized online cheduling on two unifor achine J Sched, 42:71 92, 2001 L Eptein and J Sgall A lower ound for on-line cheduling on uniforly related achine Oper Re Lett, 261:17 22, 2000 U Faigle, W Kern, and G Turán On the perforance of on-line algorith for partition prole Acta Cyernet, 92: , 1989 R Fleicher and M Wahl On-line cheduling reviited J Sched, 36: , 2000 G Galao and GJ Woeginger An on-line cheduling heuritic with etter wort cae ratio than Graha lit cheduling SIAM J Coput, 222: , 1993 T Gonzalez and S Sahni reeptive cheduling of unifor proceor yte J ACM, 251:92 101, January 1978 T Gorley, N Reingold, E Torng, and J Wetrook Generating adverarie for requet-anwer gae In roc 11th Annu ACM-SIAM Syp on Dicrete Algorith, pp , 2000 RL Graha Bound for certain ultiproceing anoalie Bell Syte Techn J, 45: , 1966

24 24 L Eptein, LM Favrholdt, and JS Kohrt DR Karger, SJ hilip, and E Torng A etter algorith for an ancient cheduling prole J Algorith, 202: , 1996 C Kenyon Bet-fit in-packing with rando order In roc 7th Annu ACM-SIAM Syp on Dicrete Algorith, pp , 1996 JS Kohrt Online algorith under new auption, p 78 hd thei, Dept Math and Cop Sci, Univ South Den, 2004 R McNaughton Scheduling with deadline and lo function Manag Sci, 61:1 12, 1959 SS Seiden reeptive ultiproceor cheduling with rejection Theoret Cop Sci, : , 2001 J Wen and D Du reeptive on-line cheduling for two unifor proceor Oper Re Lett, 23: , 1998