A New Upper Bound on 2D Online Bin Packing

50 A New Upper Bound 2.5545 on 2D Onlne Bn Packng XIN HAN, Dalan Unverty of Technology FRANCIS Y. L. CHIN and HING-FUNG TING, The Unverty of Hong Kong GUOCHUAN ZHANG, Zhejang Unverty YONG ZHANG, The Unverty of Hong Kong The 2D Onlne Bn Packng a fundamental problem n Computer Scence and the determnaton of t aymptotc compettve rato ha reearch attenton. In a long ere of paper, the lower bound of th rato ha been mproved from 1.808, 1.856 to 1.907 and t upper bound reduced from 3.25, 3.0625, 2.8596, 2.7834 to 2.66013. In th artcle, we rewrte the upper bound record to 2.5545. Our dea for the mprovement a follow. In 2002, Seden and van Stee [Seden and van Stee 2003] propoed an elegant algorthm called H C, compred of the Harmonc algorthm H and the Improved Harmonc algorthm C, for the two-dmenonal onlne bn packng problem and proved that the algorthm ha an aymptotc compettve rato of at mot 2.66013. Snce the bet known onlne algorthm for one-dmenonal bn packng the Super Harmonc algorthm [Seden 2002], a natural queton to ak : could a better upper bound be acheved by ung the Super Harmonc algorthm ntead of the Improved Harmonc algorthm? However, a mentoned n Seden and van Stee [2003], the prevou analy framework doe not work. In th artcle, we gve a potve anwer for th queton. A new upper bound of 2.5545 obtaned for 2-dmenonal onlne bn packng. The man dea to develop new weghtng functon for the Super Harmonc algorthm and propoe new technque to bound the total weght n a rectangular bn. Categore and Subject Decrptor: F.2.2 [Analy of Algorthm and Problem Complexty]: Nonnumercal Algorthm and Problem 2D bn packng General Term: Algorthm Addtonal Key Word and Phrae: Onlne algorthm, bn packng problem, compettve rato ACM Reference Format: Han, X., Chn, F. Y. L., Tng, H.-F., Zhang, G., and Zhang, Y. 2011. A new upper bound 2.5545 on 2D onlne bn packng. ACM Tran. Algor. 7, 4, Artcle 50 (September 2011), 18 page. DOI = 10.1145/2000807.2000818 http://do.acm.org/10.1145/2000807.2000818 1. INTRODUCTION In two-dmenonal bn packng, each tem (w, h ) a rectangle of wdth w 1and heght h 1. Gven a lt of uch rectangular tem, one aked to pack all of them nto a mnmum number of quare bn of de length one o that ther de are parallel to the de of the bn. Rotaton not allowed. The problem clearly trongly X. Han wa partally upported by the Fundamental Reearch Fund for the Central Unverte and NFSC (11101065). G. Zhang wa partally upported by NSFC (10971192). Author addree: X. Han, Software School, Dalan Unverty of Technology, Road 8, Economy and Technology Development Zone, Dalan, P.R. Chna, 116620; emal: hanxn.mal@gmal.com; F. Y. L. Chn, H.-F. Tng, and Y. Zhang, Department of Computer Scence, The Unverty of Hong Kong, Pokfulam, Hong Kong, Chna; emal: {chn, hftng, yzhang}@c.hku.hk; G. Zhang, College of Computer Scence, Zhejang Unverty, ZheDa Road 38, Hangzhou 310027, Chna; emal: zgc@zju.edu.cn. Permon to make dgtal or hard cope of part or all of th work for peronal or claroom ue granted wthout fee provded that cope are not made or dtrbuted for proft or commercal advantage and that cope how th notce on the frt page or ntal creen of a dplay along wth the full ctaton. Copyrght for component of th work owned by other than ACM mut be honored. Abtractng wth credt permtted. To copy otherwe, to republh, to pot on erver, to redtrbute to lt, or to ue any component of th work n other work requre pror pecfc permon and/or a fee. Permon may be requeted from Publcaton Dept., ACM, Inc., 2 Penn Plaza, Sute 701, New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or permon@acm.org. c 2011 ACM 1549-6325/2011/09-ART50 $10.00 DOI 10.1145/2000807.2000818 http://do.acm.org/10.1145/2000807.2000818

50:2 X. Han et al. NP-hard nce t a generalzaton of the one-dmenonal bn packng problem [Coffman et al. 1987]. In th artcle, we wll conder the onlne veron of twodmenonal bn packng, n whch the tem are releaed one by one and we mut rrevocably pack the current tem nto a bn wthout any nformaton on the next tem. Before preentng the prevou reult and our work, we frt revew the tandard meaure for onlne bn packng algorthm. Aymptotc Compettve Rato. To evaluate an onlne algorthm for bn packng problem, we ue the aymptotc compettve rato defned a follow. Conder an onlne algorthm A. For any lt L of tem, let A(L) be the cot (number of bn ued) ncurred by algorthm A and let OPT(L) be the correpondng optmal value. Then, the aymptotc compettve rato for algorthm A R A = lm up max{a(l)/opt(l) OPT(L) = k}. k L Prevou Work. Bn packng ha been well-tuded. For the one-dmenonal cae, Johnon et al. [1974] howed that the Frt Ft algorthm (FF) ha an aymptotc compettve rato of 1.7. Yao [1980] mproved algorthm FF wth a better upper bound of 5/3. Lee and Lee [1985] ntroduced the cla of Harmonc algorthm, for whch an aymptotc compettve rato of 1.63597 wa acheved. Ramanan et al. [1989] further mproved the upper bound to 1.61217. The bet known upper bound o far from the Super Harmonc algorthm by Seden [2002] whoe aymptotc compettve rato at mot 1.58889. A for the negatve reult, Yao [1980] howed that no onlne algorthm ha aymptotc compettve rato le than 1.5. Brown [1979] and Lang [1980] ndependently provded a better lower bound of 1.53635. The bet known lower bound to date 1.54014 [van Vlet 1992]. A for two-dmenonal onlne bn packng, a lower bound of 1.6 wa gven by Galambo [1991]. The reult wa gradually mproved to 1.808 [Galambo and van Vlet 1994], 1.857 [van Vlet 1995], and 1.907 [Bltz et al. 1996]. Coppermth and Raghan [1989] gave the frt onlne algorthm wth aymptotc compettve rato 3.25. Crk et al. [1993] mproved the upper bound to 3.0625. Crk and van Vlet [1993] preented an algorthm for all d dmenon, where n partcular for two dmenon, they obtaned a rato of at mot 2.8596. Baed on the technque of the Improved Harmonc, Han et al. [2001] mproved the upper bound to 2.7834. The bet known onlne algorthm to date the one called A C preented by Seden and van Stee [2003], where A and C tand for two one-dmenonal onlne bn packng algorthm. Bacally, A and C are appled to one dmenon of the tem wth roundng ze. In th emnal paper, Seden and van Stee proved that the aymptotc compettve rato of H C at mot 2.66013, where H the Harmonc algorthm [Lee and Lee 1985] and C an ntance of the mproved Harmonc algorthm. It ha been open nce then to mprove the upper bound. A natural dea to ue an ntance of the Super Harmonc algorthm [Seden 2002] ntead of the mproved Harmonc algorthm. However, a mentoned n Seden and van Stee [2003], n that cae, the prevou analy framework cannot be extended to Super Harmonc. We alo brefly overvew the offlne reult on two-dmenonal bn packng. Chung et al. [1982] howed an approxmaton algorthm wth an aymptotc performance rato of 2.125. Caprara [2002] mproved the upper bound to 1.69103. Banal et al. [2009] derved a randomzed algorthm wth aymptotc performance rato of at mot 1.525. A for the negatve reult, Banal et al. [2006] howed that the two-dmenonal bn packng problem doe not admt an aymptotc polynomal-tme approxmaton cheme. For the pecal cae where tem are quare, there alo a large number of reult [Coppermth and Paghavan 1989; Seden and van Stee 2003; Myazawa and Wakabayah 2003; Epten and van Stee 2004, 2005b, 2005a; Han et al. 2006].

A New Upper Bound 2.5545 on 2D Onlne Bn Packng 50:3 Epecally for bounded pace onlne algorthm, Epten and van Stee [2005b] gave an optmal onlne algorthm. Our Contrbuton. There are two man contrbuton n th artcle. We revt the 1D onlne bn-packng algorthm: Super Harmonc, and gve new weghtng functon for t, whch are much mpler than the one ntroduced n Seden [2002], and the new weghtng functon have nteret n t own. We generalze the prevou analy framework for 2D onlne bn-packng algorthm ued n Seden and van Stee [2003], and how that the new analy framework very ueful n analyzng 2D or multdmenonal onlne bn-packng problem. By combnng the new weghtng functon wth the new analy framework, we degn a new 2D onlne bn-packng algorthm wth a compettve rato 2.5545, whch mprove the prevou bound of 2.66013 n 2002 [Seden and van Stee 2003]. A mentoned n Seden and van Stee [2003], the old analy framework doe not work well wth the old weghtng functon n Seden [2002], that, the old approach doe not guarantee an upper bound better than 2.66013. Th tetfed n the followng way: conder our algorthm, f we ue old weghtng functon wth the old framework to analyze t, the compettve rato at leat 3.04, and f we ue the old weghtng functon wth the new framework, the compettve rato at leat 2.79. Organzaton of Th Artcle. Secton 2 wll revew the Harmonc and Super Harmonc algorthm a prelmnare. Secton 3 defne the weghtng functon for Super Harmonc. Secton 4 decrbe and analyze the two-dmenonal onlne bn-packng algorthm H SH+. Secton 5 conclude. 2. PRELIMINARIES We frt revew two onlne algorthm for one-dmenonal bn packng, Harmonc and Super Harmonc, whch are employed n degnng onlne algorthm for twodmenonal bn packng. 2.1. The Harmonc Algorthm The Harmonc algorthm a fundamental bn packng algorthm wth a mple and nce tructure, that wa ntroduced by Lee and Lee [1985]. The algorthm work a follow. Gven a potve nteger k, each tem mmedately clafed nto one of k type accordng to t ze upon t arrval. In partcular, f an tem ha a ze n nterval (1/( + 1), 1 ] for ome nteger, where 1 < k, then t a type tem; otherwe, t of type k. The type tem then packed, ung the mple Next Ft (NF) algorthm, nto the open (not fully packed) bn degnated to contan type tem excluvely; new bn are opened when neceary. At any tme, there at mot one open bn for each type and any cloed (fully packed) bn for type packed exactly wth tem of type for 1 < k. For an tem of ze x, we defne a weghtng functon W H (x) for the Harmonc algorthm a follow: { 1 W H (x) =, f 1 +1 < x 1 wth 1 < k. k k 1 x f 0 < x 1 k. The followng lemma drectly from Lee and Lee [1985]. LEMMA 2.1. For any lt L, we have H(L) p L W H (p) + O(1), where H(L) the number of bn ued by the Harmonc algorthm for lt L.

50:4 X. Han et al. 2.2. The Super Harmonc Algorthm The Super Harmonc algorthm [Seden 2002] a generalzaton of the Improved Harmonc algorthm and the Harmonc algorthm. Super Harmonc frt clafe each tem nto one of k+ 1 type, where k a potve nteger, and then agn to the tem a color of ether blue or red. It allow tem of up to two dfferent type to hare the ame bn. In any one bn, all tem of the ame type have ame color and tem of dfferent type have dfferent color. For tem of type ( k), the algorthm mantan two parameter β and γ to bound repectvely the number of blue tem and the number of red tem n a bn. More detal are gven n th ecton. Clafcaton nto Type. Let t 1 = 1 > t 2 > > t k > t k+1 = ɛ>t k+2 = 0 be real number. An nterval I defned to be (t +1, t ], for = 1,...,k + 1. An tem wth ze x of type f x I. Colorng Red or Blue. Each type tem alo agned a color, ether red or blue, for k. The algorthm ue two et of counter, e 1,...,e k and 1,..., k, all of whch are ntally zero. The total number of type tem denoted by, whle the number of type red tem denoted by e.for1 k, durng the packng proce, the fracton of type tem that are red mantaned, that, e = α, where α 1,...,α k [0,1] are contant. Maxmal Number of Blue Item. Let β = 1/t for 1 k, that the maxmal number of blue tem of type that can be accepted n a ngle bn. Space Left for Red Item. Let δ = 1 t β, whch the lower bound of the pace left when a bn cont of β blue tem of type. If poble, we want to ue the pace left for mall red tem. Note that, n the algorthm, n order to mplfy the analy, ntead of ung δ, le pace ued, namely D ={ 0, 1,..., K }, a the pace nto whch red tem can be packed, where 0 = 0 < 1 < < K < 1/2 andk k. Let φ() be the pace to be ued to accommodate red tem n a bn that hold β blue tem of type, where functon φ defned a {1,...,k} {0,...,K} uch that φ() the maxmum number uch that φ() δ.ifφ() = 0, then no red tem are accepted. We et α = 0ft > K. For convenent ue n our analy n the next ecton, we ntroduce a functon called ϕ(), whch gve the ndex of the mallet pace n D nto whch a red tem of type can be placed: ϕ() = mn{ j t j, 1 j K}. Maxmal Number of Red Item. Now we defne γ.letγ = 0ft > K ; otherwe, γ = max{1, 1 /t }, that,f 1 < t K,weetγ = 1, otherwe, γ = 1 /t. Namng Bn. It alo convenent to name the bn by group a follow. {() φ = 0, 1 k}. {(,?) φ 0, 1 k}. {(?, j) α j 0, 1 j k}. {(, j) φ 0,α j 0,γ j t j φ(), 1, j k}. Group () cont of bn that hold only blue tem of type. Group (, j) cont of bn that contan blue tem of type and red tem of type j. Blue group (,?) and red group (?, j) are ndetermnate bn currently contanng only blue tem of type or red tem of type j repectvely. Durng packng, red tem or blue tem wll be packed nto ndetermnate bn f neceary, that, ndetermnate bn wll be changed nto (, j).

A New Upper Bound 2.5545 on 2D Onlne Bn Packng 50:5 The Super Harmonc algorthm outlned a follow. Super Harmonc Algorthm (1) For each tem p : type of p, (a) f = k + 1 then ue NF algorthm, (b) ele + 1; f e < α then e e + 1; { color p red }. If there a bn n group (?, ) wth fewer than γ type tem, then place p n t. Ele f, for any j, there a bn n group ( j, ) wth fewer than γ type tem then place p n t.. Ele f there ome bn n group ( j,?) uch that φ( j) γ t, then pack p n t and change the bn nto ( j, ).. Otherwe, open a bn (?, ), pack p n t. (c) ele {color p blue}:. f φ = 0 then f there a bn n group wth fewer than β tem then pack p n t, ele open a new group bn, then pack p n t.. Ele: A. f, for any j, there a bn n group (, j) or(,?) wth fewer than β type tem, then pack p n t. B. Ele f there a bn n group (?, j) uchthat φ() γ j t j then pack p n t, and change the group of th bn nto (, j). C. Otherwe, open a new bn (,?) and pack p n t. 3. NEW WEIGHTING FUNCTIONS FOR SUPER HARMONIC In th ecton, we develop new weghtng functon for Super Harmonc that are mpler than the weghtng ytem n Seden [2002]. The weghtng functon wll be ueful n analyzng the propoed onlne algorthm a we hall ee n the next ecton. 3.1. Intuton for Defnng Weght Weghtng functon are wdely ued n analyzng onlne bn-packng problem. Roughly peakng, for an tem, the value by one of weght functon the fracton of a bn occuped by the tem n the onlne algorthm. There a contrant n defnng weght for tem for an onlne algorthm, whch wll be gven later. We wll ue K +1 weghtng functon. Let W (p) betheth weght of an tem p, where 1 K + 1. For any nput L, the contrant { } A(L) max W (p) + O(1), (1) 1 K+1 p L where A(L) the number of bn ued by algorthm A. Conder the Super Harmonc algorthm. For 1 k, let be the number of type pece. For 1, k, letb (), B (,), B (,?), B (?,) be the number of bn n group (), (, ), (,?), and (?, ). Then we have { B () + } B (,) + B (,?) = (1 α ) + O(1) (2) β and { B (?,) + } B (,) = α + O(1). (3) γ So, for each tem wth ze x I, where k, f we defne t weght a: 1 α β + α γ,

50:6 X. Han et al. then t not dffcult to ee that the contrant (1) hold. However, th weghtng functon not good enough, that, t alway lead to a compettve rato of at leat 1.69103 f we ue th weghtng functon to analyze the Super Harmonc algorthm. The man reaon that for each bn n group (, ) we account t twce, where 1, k. Next, we gve ome ntuton for mprovng th weghtng functon. By (2) and (3), oberve that B (,) (1 α ) β + O(1) (4) and So, we have B (,) B (,) = B (,) + B (,) 2 (1 α ) + α + O(1) 2β 2γ ( 1 α = + α ) + O(1). 2β 2γ α γ + O(1). (5) Hence, for an tem wth ze x I, after packng, f there a bn n group (, ) and alo a bn n group (, ), then we can defne t weght a below: 1 α 2β + α 2γ. Th the man ntuton for defnng our new weghtng functon, whch wll be gven n the next ubecton. 3.2. New Weghtng Functon Remember that n Super Harmonc, there a et D ={ 0, 1,..., K } repreentng the free pace reerved for red tem. Recall the two functon φ() and ϕ() are related to free pace and have the followng meanng: φ() = j mple that free pace j reerved for red tem n a bn contng of β blue tem of type, andϕ() = j ndcate that a red tem of type could be packed n free pace j. We are now ready to defne new weghtng functon. Item wth ze larger than ɛ wll be frt condered. Let E be the number of ndetermnate red group bn (?, ) when the whole packng done. If E = 0, that, every red tem placed n a bn wth one or more blue tem, then we defne the weghtng functon a: W 1 (x) = 1 α β f x I. (6) Otherwe, E > 0 mple that for ome, an ndetermnate red group bn (?, ) ext after packng. Let r be the type of the mallet red tem n the ndetermnate red group

A New Upper Bound 2.5545 on 2D Onlne Bn Packng 50:7 bn. If 2 ϕ(r) K, then we defne the correpondng weghtng functon a follow: 1 α β + α 2γ f x I,φ() <ϕ(r) andϕ() <ϕ(r), 1 α W K+2 ϕ(r) β + α γ f x I,φ() <ϕ(r) andϕ() ϕ(r), (x) = 1 α 2β + α γ f x I,φ() ϕ(r) andϕ() ϕ(r), f x I,φ() ϕ(r) andϕ() <ϕ(r). 1 α 2β + α 2γ If ϕ(r) = 1, we defne W K+1 (x) = 1 α β f x I,φ() = 0andϕ() = 0, 1 α β + α γ f x I,φ() = 0andϕ() > 0, 0 f x I,φ() > 0andϕ() = 0, α γ f x I,φ() > 0andϕ() > 0. Note that n thee defnton, f γ = 0 then we replace α γ ze x I k+1, we alway defne W j (x) = x/(1 ɛ) for all j. wth zero. For an tem wth THEOREM 3.1. For any lt L, we have A(L) max W (p) 1 K+1 + O(1), p L where A(L) the number of bn ued by Super Harmonc for lt L. PROOF. Fx a lt L.LetDbe the um of the ze of the tem of type (k+1). By NEXT FIT, we know that the number of bn ued for type (k + 1) at mot D/(1 ɛ) + 1. Agan, we ue E to denote the number of ndetermnate red group bn when all the packng done. For 1 k, let be the number of type pece. Let B (), B (,), B (,?), B (?,) be the number of bn n group (), (, ), (,?) and (?, ). To prove th theorem, we conder two cae. Cae 1. If E = 0, that, B (?,) = 0, every red tem packed n a bn wth one or more blue tem. Therefore, we jut need to count bn contanng blue tem: A(L) D 1 ɛ + { B () + } B (,) + B (,?) + O(1) W 1 (x) + (1 α ) + O(1) by (2) β x I k+1,x L = W 1 (x) + W 1 (x) + O(1). x I k+1,x L x / I k+1,x L Cae 2. E > 0. We frt bound the number of bn of B (?,) and B (,?) for all, then conder two ubcae to bound the total number of bn ued. Aume that ϕ(r) = j 1 (by the defnton of functon ϕ). Let e be the mallet red tem n ndetermnate red group bn and r be t type. Then, every type red tem wth ϕ() < j placed n a fnal group bn (, ); otherwe, tem e would not be the mallet

50:8 X. Han et al. red tem. Hence, we have ϕ()< j B (?,) = 0. (7) On the other hand, we know a bn B (?,r) and a bn B (,?) cannot ext at the ame tme by the rule of Super Harmonc f φ() j. Hence, we have B (,?) = 0. (8) φ() j In accordance wth the Super Harmonc algorthm, for any type bn B (),wehave Defne φ() = 0. (9) X = φ() j ϕ()< j B (,), whch the total number of all the bn n group (, ) uch that φ() j and ϕ() < j. Then, we have A(L) D 1 ɛ + ( ) B() + B (,?) + B (?,) + B (,) + O(1) = D 1 ɛ + ( ) B() + B (,?) + B (?,) + X + B (,) φ()< j + B (,) + O(1) ϕ() j = D 1 ɛ + + ϕ() j φ()< j ( B () + B (,?) + ( B (?,) + B (,) ) B (,) ) + X + O(1). (10) The lat nequalty hold from equalte B () = φ()< j B () φ()< j B (,?) by (8) and B (?,) = ϕ() j B (?,) by (7). by (9), B (,?) = Cae 2.1. j 2. By the defnton of varable X, wehave X It not dffcult to ee, X j φ() K j φ() K B (,) and X B (,) + 1 ϕ() j 1 1 ϕ() j 1 B (,). B (,) / 2. (11)

A New Upper Bound 2.5545 on 2D Onlne Bn Packng 50:9 Hence, by (10) and (11), we have A(L) D 1 ɛ + ( B () + B (,?) + φ()< j + B (,) 2 + = φ() j D 1 ɛ + φ()< j D 1 ɛ + + φ() j ϕ() j x I k+1,x L φ()< j ϕ()< j ϕ()< j ) B (,) + ( B (?,) + B (,) 2 + O(1) φ()< j ϕ() j ϕ() j B (,) ) (1 α ) + α + (1 α ) + α + O(1) β γ 2β 2γ ϕ() j φ() j ϕ()< j ( (1 α ) + α ) + ( (1 α ) + α ) β 2γ β γ ( (1 α ) 2β W K+2 j (x) + + α ) + γ φ() j x / I k+1,x L ϕ()< j ( (1 α ) 2β W K+2 j (x) + O(1), + α ) + O(1) 2γ where the econd nequalty follow drectly from (2) and (3) and the lat equalty hold from the new weghtng functon defned n Subecton 3.2. Cae 2.2. j = 1. In accordance wth the Super Harmonc algorthm, for any type of bn (, ), we have ϕ() 1, where 1, k and k a parameter defned n Super Harmonc. Hence, there no uch bn (, ) wthϕ() < 1, that, X =. Then, by (10), we have A(L) = D 1 ɛ + φ()<1 D 1 ɛ + φ()=0 D 1 ɛ + x I k+1,x L φ()=0 ϕ()=0 ( B () + B (,?) + ) B (,) + ( B (?,) + ϕ() 1 (1 α ) + α + O(1) β γ (1 α ) β W K+1 (x) + ϕ() 1 + x / I k+1,x L φ()=0 ϕ()>0 ( (1 α ) β W K+1 (x) + O(1), B (,) ) + O(1) + α ) + α + O(1) γ γ φ()>0 ϕ()>0 where the lat equalty hold from the new weghtng functon defned n Secton 3.2. Therefore, we have A(L) max 1 K+1 { p L W (p)}+o(1). 4. ALGORITHM H SH + AND ITS ANALYSIS In the ecton, we frt revew a cla of onlne algorthm for two dmenonal onlne bn packng, called H C [Seden and van Stee 2003]. Next we ntroduce a new ntance of algorthm H SH+, where H Harmonc and SH+ (Strange Harmonc+) an ntance of Super Harmonc. Then we propoe ome new technque on how to

50:10 X. Han et al. bound the total weght n a ngle bn, whch crucal to obtanng a better aymptotc compettve rato for the H C algorthm. Fnally, we apply new weghtng functon for SH+ to analyze the two-dmenonal onlne bn-packng algorthm H SH+ and how t compettve rato at mot 2.5545, whch mple that the new weghtng functon work very well wth the generalzed approach of boundng the total weght n a ngle bn. 4.1. Algorthm H C and H C Now we revew two-dmenonal onlne bn-packng algorthm H C and H C [Seden and van Stee 2003], where H Harmonc and C Super Harmonc. Gven an tem p = (w, h), H C operate a follow. (1) Packng Item nto Slce. Ifw ɛ then pack p nto a lce of heght 1 and wdth t by H (Harmonc algorthm), where t +1 <w t ; ele pack t nto a lce of heght 1 and wdth ɛ(1 δ) by H (Harmonc algorthm), where ɛ(1 δ) +1 <w ɛ(1 δ) and δ>0 arbtrarly mall. (2) Packng Slce nto Bn. When a new lce requred n the tep, we allocate t from a bn ung algorthm C. H C a randomzed algorthm, whch operate a follow: before proceng begn, we flp a far con. If the reult head, then we run H C; otherwe, we run C H, that, the role of heght and wdth are nterchanged. Note that t poble to derandomze H C wthout ncreang t performance rato. For detal, we refer to Seden and van Stee [2003]. THEOREM 4.1. If an onlne 1D bn-packng algorthm C ha weghtng functon W C (x) uch that C(L) max { x L W C (x)}+o(1), then the cot by algorthm H Cfor nput L at mot 1 max W H C 2(1 δ) (p) + max W C H (p) + O(1), p L p L and the aymptotc compettve rato of algorthm H C at mot 1 2(1 δ) max max X (x,y) X W H (x)w C (y) + max (x,y) X W H (y)w C (x),, where δ a parameter defned n H C algorthm and X any et of tem that ft n a ngle bn. 4.2. An ntance of Super Harmonc SH + A mentoned n Seden [2002], t a hard problem to fnd approprate parameter n degnng an ntance of Super Harmonc, epecally ettng t. The parameter n

A New Upper Bound 2.5545 on 2D Onlne Bn Packng 50:11 SH+ are found through a tral-and-error way and are defned a follow: t α β δ φ() ϕ() γ 1 1 0 1 0 0 0 0 2 0.706 0 1 0.294 1 0 0 3 0.657 0 1 0.343 2 0 0 4 0.647 0 1 0.353 3 0 0 5 0.625 0 1 0.375 4 0 0 6 0.6 0 1 0.4 5 0 0 7 0.58 0 1 0.42 6 0 0 8 0.5 0 2 0 0 0 0 9 0.42 0.162 2 0.16 0 6 1 10 0.4 0.192 2 0.2 0 5 1 11 0.375 0.2346 2 0.25 0 4 1 12 0.353 0.3004 2 0.294 1 3 1 13 0.343 0.3077 2 0.314 1 2 1 14 1/3 0 3 0 0 0 0 15 0.294 0.0816 3 0.118 0 1 1 16 1/4 0.186 4 0 0 1 1 17 1/5 0.092 5 0 0 1 1 18 1/6 0.1456 6 0 0 1 1 19 0.147 0.2162 6 0.118 0 1 2 20 1/7 0.1525 7 0 0 1 2 21 49 1/( 13) ff() 13 0 0 1 1 /t 50 1/37 0 37 0 0 0 0 51 1/38 0 j = φ() j Red accepted 1 0.294 15..50 2 0.343 13, 15..50 3 0.353 12, 13, 15..50 4 0.375 11..13, 15..50 5 0.4 10..13, 15..50 6 0.42 9..13, 15..50 where ff() = 1.35(50 )/37( 12). Then, we have even weghtng functon for SH+, that,w C a defned n the lat ecton, where 1 7. 4.3. Prevou Framework for Calculatng Upper Bound In th ecton, we frt ntroduce the prevou framework for computng the upper bound of the compettve rato of H SH+, then menton that the prevou framework doe not work well wth the ntance n the lat ecton, that, the prevou framework doe not lead to a better upper bound. Let p = (x, y) be an tem. We defne the followng functon. W H C (p) = W H(x)W C (y), W C H (p) = W H(y)W C (x), and W, j (x, y) = W H(x)W C (y) + W j C (x)w H(y). 2 Then, we can obtan an upper bound on the compettve rato R of algorthm H SH+ a follow by Theorem 3.1 and 4.1, where X any et of tem that ft n a ngle bn. 1 R 2(1 δ) max max W H C X 1 7 (p) + max W C H 1 7 (p) 1 ( = max W H C (1 δ) 1, j 7,X (p) + W j C H (p)) /2 1 = max W, j (x, y) (1 δ) 1, j 7,X. (12)

50:12 X. Han et al. The value of R can be etmated by the followng approach. Defnton 4.2. Let f be a functon mappng from (0, 1] to R +. P( f )themathematcal program: maxmze x X f (x) ubject to x X 1, over all fnte et of real number X. WealoueP( f ) to denote the value of th mathematcal program. LEMMA 4.3 ([SEIDEN AND VAN STEE 2003]). Let f and g be functon mappng from (0, 1] to R +.LetF= P( f ) and G = P(g). Then, the maxmum of f (h(p))g(w(p)) over all fnte multet of tem X that ft n a ngle bn at mot FG, where p a rectangle and h(p) and w(p) are t heght and wdth, repectvely. In Seden and van Stee [2003], f and g are defned a below: f, j (y) = W H(y) + W C (y), 2 and g, j (x) = up 0<y 1 W, j (x, y). f, j (y) By thee defnton, we have W, j (x, y) f, j (y)g, j (x), (13) for all 0 x 1and0 y 1. 4.4. A New Framework for Calculatng Upper Bound In th ecton, we frt generalze the prevou analy framework by ntroducng a new lemma and developng new functon for f and g n order to bound the total weght n a ngle bn. Then we apply our new weghtng functon for Super Harmonc to algorthm H SH+ and obtan a new upper bound for two-dmenonal onlne bn packng. LEMMA 4.4. max X { W, j (x, y)} =max X { W j, (x, y)}, where 1, j 7 and X any feable et for one bn. PROOF. By defnton, oberve that for any 1, j 7, W, j (x, y) = W j, (y, x). (14) Let X ={p 1, p 2,...,p m } be a et of rectangle whch ft nto a ngle bn, where p = (x, y )the-th rectangle n X. If we exchange role of x and y of p to get new rectangle p = (y, x ) for all, then t not dffcult to ee that the new et X ={p 1, p 2,...,p m } alo a feable pattern, that, all tem can ft n a ngle bn. On the other hand, by Eq. (14), we have W, j (p) = W j, (p ), p X where 1, j 7. There a one-to-one mappng between X and X n all the feable pattern. Therefore, we have th lemma. New Functon f and g. We gve new functon f and g uch that () Lemma 4.3 can be appled to bound the weght n a ngle bn, and () the reultant bound not too looe. The new functon f and g are gven a follow: f, j (y) = λ, j W H (y) + (1 λ, j )W C (y),

A New Upper Bound 2.5545 on 2D Onlne Bn Packng 50:13 where 0 λ, j 1. By the ame approach ued n Seden and van Stee [2003], functon g defned here: g, j (x) = up 0<y 1 W, j (x, y). f, j (y) Note that n Seden and van Stee [2003], λ, j are 1/2 for all, j. It not dffcult to ee that (13) tll hold for all 0 x 1and0 y 1. New Approach of Calculatng P( f ). In order to ue Lemma 4.3 to obtan the upper bound on the compettve rato R of algorthm H C, we need to calculate P( f, j )and P(g, j ). Let f be one of f, j or g, j for 1, j 7. Seden [2002] wrote a programmng to enumerate all the feable pattern to get the bound for P( f ).Here,wegveample approach by callng LP olver drectly to etmate P( f ), whch can be modeled a the followng mxed nteger program (MIP): max. f = uch that 50 =1 50 =1 ( x w + 1 50 =1 x (t +1 + ɛ) 1, x (t +1 + ɛ) ) 1 1 t 51 (1) x 0, nteger. where x the number of type tem n a feable pattern, w the weght for an tem of type, whch decded by functon f,that,w = f (p) fp (t +1, t ]. Note t defned n Secton 4.2. We can olve the above MIP (1) by ettng ɛ = 0. However, th approach doe not gve a good approxmaton oluton nce ome nfeable oluton are nvolved and affect the optmal oluton too much. In order to reduce the error caued by ome bad nfeable oluton, we append ome contrant that wll help u to remove ome bad nfeable oluton and do not elmnate any feable oluton of MIP (1). For example, a type tem ha ze larger than 0.5 f 1 7, then at mot one of type can be accepted n one bn, that, x 1for1 7. Smlarly, we have the followng nequalte: x 2, for 8 13, x 3, for 14 15, x 12, for 16 17, x 18 + x 19 6, x 13, for 20 50. Moreover, we can add ome complcated contrant uch a 2x 7 + x 15 3.9, 3x 7 + 2x 13 + x 17 5.9, 4x 13 + 3x 15 + x 24 11.9, 5x 7 + 3.53x 11 + 1.47x 18 9, 12x 7 + 8x 13 + 3x 20 + x 36 23, 9x 7 + 6x 13 + 2x 21 + x 30 17. Th contrant do not elmnate any feable oluton of MIP (1). For example, conder the contrant 5x 7 + 3.53x 11 + 1.47x 18 9. Snce an tem of type 7 ha ze larger than 0.5, an tem of type 11 ha ze larger than 0.353 and an tem of type 18 ha ze

50:14 X. Han et al. larger than 0.147, we have 0.5x 7 + 0.353x 11 + 0.147x 18 < 1. Equvalently, we have 5x 7 + 3.53x 11 + 1.47x 18 < 10. It not dffcult to ee that the nequalty 5x 7 + 3.53x 11 + 1.47x 18 9 equvalent to 5x 7 + 3.53x 11 + 1.47x 18 < 10 when x 7, x 11 and x 18 are nonnegatve nteger. For other contrant, the argument are analogou. Therefore, we remodel th MIP a follow. ( ) 50 50 max. f = x w + 1 x t +1 1 (2) 1 t 51 ubject to =1 50 =1 x t +1 1, x 1, for 1 7, x 2, for 8 13, x 3, for 14 15, x 12, for 16 17, x 18 + x 19 6, x 13, for 20 50, 2x 7 + x 15 3.9, 3x 7 + 2x 13 + x 17 5.9, 4x 13 + 3x 15 + x 24 11.9, 5x 7 + 3.53x 11 + 1.47x 18 9, 12x 7 + 8x 13 + 3x 20 + x 36 23, 9x 7 + 6x 13 + 2x 21 + x 30 17, x 0, nteger. To olve MIP (2), we ue a tool for olvng lnear and nteger program called GLPK [GLP]. We wrte a program to calculate W, j (x, y), g, j (x) and f, j (y) for each (, j), and then call the API of GLPK to calculate P( f, j )andp(g, j ). The value of P( f, j )and P(g, j ) are hown n the table n Appendx. Note that when we ue Lemma 4.3 for the upper bound on the weght max X { W, j (x, y)}, for all par (, j), the calculaton are ndependent. For dfferent par (, j), λ, j may be dfferent. So, n order to get an upper bound near the true value of max X { W, j (x, y)}, we have to elect an approprate λ, j. Th can be done by a tral-and-error approach. THEOREM 4.5. 2.5545. =1 For all δ>0, the aymptotc compettve rato of H B at mot PROOF. In accordance wth the table n Appendx, by Lemma 4.4 and Lemma 4.3, we have max W 1,2 (p) X = max W 2,1 (p) X P( f 1,2 )P(g 1,2 ) 2.5539. max X max X max X W 1,6 (p) = max W 6,1 (p) X P( f 6,1 )P(g 6,1 ) 2.5545. W 2,5 (p) = max W 5,2 (p) X P( f 5,2 )P(g 5,2 ) 2.5340. W 2,6 (p) = max W 6,2 (p) X P( f 6,2 )P(g 6,2 ) 2.5364.

A New Upper Bound 2.5545 on 2D Onlne Bn Packng 50:15 For all the other (, j), by Lemma 4.3, we have max W, j (p) X P( f, j )P(g, j ) P( f 1,1 )P(g 1,1 ) 2.5545. Remark. If we ue the weghtng functon from Seden [2002] and the prevou analy framework, we fnd that the compettve rato at leat 3.04. (run our programmng 2DHSH.c lke./2dhsh+.exe old > yourfle ) Even f we ue the new weghtng functon, by the prevou analy framework, the compettve rato tll at leat 3.04, by runnng our programmng 2DHSH.c lke./2dhsh+.exe new1 > yourfle. We alo fnd that f we ue the old weghtng functon from Seden [2002] wth the new analy framework, the compettve rato at leat 2.79. (run our programmng 2DHSH.c lke./2dhsh+.exe old2 > yourfle ) The reaon that Lemma 4.3 doe not work very well wth the old weght functon, that, the reultng value F G away from the maxmum weght of tem n a ngle bn. 5. CONCLUDING REMARKS When the parameter n Super Harmonc uch a α, β, γ and φ()andϕ() are gven, we can calculate the weghtng functon of Super Harmonc W j B ( ). Then, the weghtng functon W, j (x, y) for algorthm H SH+ can be calculated a well a f, j (y) and g, j (x). For each (, j), we call the API of GLPK to olve P( f, j )andp(g, j ). To ue our program under lnux ytem: Intall GLPK, Comple: gcc -o 2DHSH+.exe 2DHSH+.c -lglpk Run:./2DHSH+.exe new2 > yourfle If there an error meage lke Could not load *.o when you comple the ource, then try to et "LD LIBRARY PATH" a follow: LD LIBRARY PATH= $LD LIBRARY PATH:/ur/local/lb, then export LD LIBRARY PATH. When we ue the tool for olvng the mxed nteger program, there are two fle whch are neceary: one the model fle for the lnear or nteger program telf (refer to Appendx), and the other the data fle where the data tored. We wrte a program to generate the data and then call the tool GLPK. (Actually, we call the API (Applcaton Program Interface) of GLPK. To download the ource fle, go to: http://te.google.com/te/xnhan2009/home/fle/2dhsh.c). Our framework can be appled to 3D onlne bn packng to reult n an algorthm H H SH+ wth t compettve rato 2.5545 1.69103( 4.3198). APPENDIXES A. VALUES OF F I,J AND G I,J (, j) = (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (1, 7) λ, j 0.500000 0.500000 0.540000 0.550000 0.565000 0.565000 0.600000 P( f, j ) 1.598272 1.598272 1.605095 1.606845 1.609490 1.609490 1.615665 P(g, j ) 1.598272 1.597872 1.574422 1.581742 1.585430 1.587508 1.575580 P( f, j )P(g, j ) 2.554474 2.553834 2.527096 2.541614 2.551734 2.555079 2.545610 (, j) = (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (2, 7) λ, j 0.500000 0.500000 0.530000 0.550000 0.565000 0.565000 0.600000 P( f, j ) 1.597328 1.597328 1.597148 1.597028 1.596938 1.596938 1.596729 P(g, j ) 1.609235 1.598326 1.586301 1.595016 1.602278 1.604268 1.589545 P( f, j )P(g, j ) 2.570476 2.553051 2.533557 2.547285 2.558739 2.561917 2.538073

50:16 X. Han et al. (, j) = (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (3, 7) λ, j 0.500000 0.500000 0.530000 0.550000 0.565000 0.565000 0.600000 P( f, j ) 1.573676 1.573676 1.572837 1.572777 1.572732 1.572732 1.572627 P(g, j ) 1.609235 1.598326 1.586301 1.595016 1.602278 1.604268 1.589545 P( f, j )P(g, j ) 2.532414 2.515247 2.494992 2.508604 2.519954 2.523084 2.499762 (, j) = (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (4, 7) λ, j 0.500000 0.500000 0.535000 0.550000 0.565000 0.565000 0.600000 P( f, j ) 1.581245 1.581245 1.577140 1.575380 1.573621 1.573621 1.569515 P(g, j ) 1.609235 1.598326 1.586855 1.595016 1.602278 1.604268 1.589545 P( f, j )P(g, j ) 2.544594 2.527344 2.502692 2.512755 2.521378 2.524510 2.494814 (, j) = (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (5, 7) λ, j 0.500000 0.500000 0.535000 0.550000 0.565000 0.565000 0.600000 P( f, j ) 1.585370 1.585370 1.580113 1.577860 1.575607 1.575607 1.570350 P(g, j ) 1.609542 1.598326 1.587240 1.595374 1.602747 1.604737 1.589740 P( f, j )P(g, j ) 2.551720 2.533939 2.508019 2.517277 2.525300 2.528436 2.496449 (, j) = (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) (6, 7) λ, j 0.500000 0.500000 0.530000 0.550000 0.565000 0.565000 0.600000 P( f, j ) 1.586853 1.586853 1.582237 1.579160 1.576853 1.576853 1.571468 P(g, j ) 1.609785 1.598326 1.586682 1.595657 1.603117 1.605107 1.589894 P( f, j )P(g, j ) 2.554493 2.536309 2.510507 2.519798 2.527881 2.531019 2.498468 (, j) = (7, 1) (7, 2) (7, 3) (7, 4) (7, 5) (7, 6) (7, 7) λ, j 0.500000 0.515000 0.535000 0.555000 0.565000 0.570000 0.600000 P( f, j ) 1.568686 1.560602 1.549821 1.539044 1.533655 1.530958 1.517143 P(g, j ) 1.621572 1.609605 1.602462 1.612258 1.622822 1.638219 1.624359 P( f, j )P(g, j ) 2.543738 2.511952 2.483529 2.481335 2.488849 2.508043 2.464386 B. MODEL FILE FOR GLPK AND USAGE OF OUR PROGRAM 2DHSH+.C param I:=50; param c{ n 1..I}>=0; param w{ n 1..I}; var x{ n 1..I}, nteger, >=0; maxmze f: um{ n 1..I} w[]*x[] + (1-um{ n 1..I} c[]*x[]) * 38/37;.t. x0: um{ n 1..I} c[]*x[] <= 1; x1: um{ n 1..7} x[] <= 1; x7: um{ n 8..13} x[] <= 2; x14: x[14] <= 3; x15: x[15] <= 3; x16: x[16] <= 4; x17: x[17] <= 5; x18: x[18] + x[19] <= 6; y715: 2*x[7] + x[15] <= 3.9; y71317: 3*x[7] + 2*x[13] + x[17] <= 5.9;

A New Upper Bound 2.5545 on 2D Onlne Bn Packng 50:17 end; y131524: 4*x[13] + 3*x[15] + x[24] <= 11.9; y71118: 5*x[7] + 3.53*x[11] + 1.47 *x[18] <= 9; y7132036: 12*x[7]+8*x[13] + 3*x[20] + x[36] <=23; y7132130: 9*x[7] + 6*x[13] + 2*x[21] + x[30] <=17; other{ n 20..50}: x[] <= -13; ACKNOWLEDGMENTS The author wh to thank the referee for ther ueful comment on the earler draft of the artcle. Ther uggeton have helped mprove the preentaton of the paper. REFERENCES BANSAL, N., CAPRARA, A., AND SVIRIDENKO, M. 2009. A new approxmaton method for et coverng problem, wth applcaton to multdmenonal bn packng. SIAM J. Comput. 39, 4, 1256 1278. BANSAL, N., CORREA, J. R., KENYON, C., AND SVIRIDENKO, M. 2006. Bn packng n multple dmenon: Inapproxmablty reult and approxmaton cheme. Math. Oper. Re. 31, 1, 31 49. BLITZ,D.,VAN VLIET,A.,AND WOEGINGER, G. J. 1996. Lower bound on the aymptotc wort-cae rato of onlne bn packng alorthm. Unpublhed manucrpt. BROWN, D. 1979. A lower bound for on-lne one-dmenonal bn packng algorthm. Tech. rep. R864, Coordnated Scence Lab., Urbana, Illno. CAPRARA, A. 2002. Packng 2-dmenonal bn n harmony. In Proceedng of the Sympoum on Foundaton of Computer Scence (FOCS). 490 499. CHUNG, F., GAREY, M., AND JOHNSON, D. 1982. On packng two-dmenonal bn. SIAM J. Alg. Dc. Meth. 3, 1, 66 76. COFFMAN, E., GAREY, M., AND JOHNSON, D. 1987. Approxmaton Algorthm for Bn Packng: A Survey, chapter 2. PWS, Boton, MA. COPPERSMITH, D. AND PAGHAVAN, P. 1989. Multdmenonal on-lne bn packng: Algorthm and wort cae analy. Oper. Re. Lett 8, 17 20. CSIRIK, J., FRENK, J. B. G., AND LABBÉ, M. 1993. Two-dmenonal rectangle packng: On-lne method and reult. Dc. Appl. Math. 45, 3, 197 204. CSIRIK, J. AND VAN VLIET, A. 1993. An on-lne algorthm for multdmenonal bn packng. Oper. Re. Lett 13, 149 158. EPSTEIN, L. AND VAN STEE, R. 2004. Optmal onlne bounded pace multdmenonal packng. In Proceedng of the ACM-SIAM Sympoum on Dcrete Algorthm (SODA 04). 214 223. EPSTEIN, L. AND VAN STEE, R. 2005a. Onlne quare and cube packng. Acta Inf. 41, 9. EPSTEIN, L. AND VAN STEE, R. 2005b. Optmal onlne algorthm for multdmenonal packng problem. SIAM J. Comput. 35, 2, 431 448. GALAMBOS, G. 1991. A 1.6 lower-bound for the two-dmenonal on-lne rectange bn-packng. Acta Cybern. 10, 1-2, 21 24. GALAMBOS,G. AND VAN VLIET, A. 1994. Lower bound for 1-, 2- and 3-dmenonal on-lne bn packng algorthm. Computng 52, 3, 281 297. GLP. http://www.gnu.org/oftware/glpk. HAN,X.,FUJITA,S.,AND GUO, H. 2001. A two-dmenonal harmonc algorthm wth performance rato 2.7834. IPSJ SIG Note 93, 43 50. HAN, X., YE, D., AND ZHOU, Y. 2006. Improved onlne hypercube packng. In Proceedng of the 4th Workhop on Approxmaton and Onlne Algorthm (WAOA). 226 239. JOHNSON, D., DEMERS, A., ULLMAN, J., GAREY, M., AND GRAHAM, R. 1974. Wort-cae performance bound for mple one-dmenonal packng algorthm. SIAM J. Comput. 3, 4, 299 325. LEE, C. AND LEE, D. 1985. A mple on-lne packng algorthm. J. ACM 32, 562 572. LIANG, F. 1980. A lower bound for onlne bn packng. Inf. Proc. Lett. 10, 76 79. MIYAZAWA, F. AND WAKABAYASHI, Y. 2003. Cube packng. Theoret. Comput. Sc. 297, 1 3, 355 366. RAMANAN, P., BROWN, D., LEE, C., AND LEE, D. 1989. On-lne bn packng n lnear tme. J. Algor. 10, 305 326.

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