Approximation of Two-Dimensional Rectangle Packing

Transcription

1 pproximtion of Two-imensionl Rectngle Pcking Pinhong hen, Yn hen, Mudit Goel, Freddy Mng S70 Project Report, Spring My 18, Introduction 1-d in pcking nd -d in pcking re clssic NP-complete prolems tht re motivted y industril pplictions of stock cutting, dt storge, etc. In this pper, we look t slightly more generl prolem tht is primrily motivted y VLSI design. In VLSI design, one wnts to pck given numer of rectngles (which might e RM chips, trnsistors, etc.) into wfer of minimum re to reduce the size of the chip. This is the -d rectngulr pcking prolem. The optimiztion prolem is Given set of N rectngles, minimize the re of ounding ox tht contins ll the N rectngles, without ny rottions nd such tht no two rectngles overlp. The corresponding decision prolem is Given set of N rectngles nd ox of length L nd height H, cn ll the N rectngles e pcked in the ox? y reduction from the -d in pcking, it cn e esily shown tht the prolem is NP-hrd. lso it cn e esily seen tht the prolem is in NP, nd hence it is NP-complete. Though lot of reserch hs gone into 1-d in pcking nd -d in pcking [GJ84],[GJ79],[LR90], the rectngle pcking prolem is reltively less reserched. offmn, et.l. [GJ84], in ddition to discussing the in-pcking prolems nd its vritions, present rief survey of vrition closer to the rectngle pcking prolem. In this vrition of the prolem, clled strip-pcking, they try to minimize the re fter fixing the width of the ounding ox. This prolem rises in vriety of industril settings where the rw mteril comes in the form of rolls nd from this we my wish to cut rectngulr ptterns for vrious purposes. ue to the economic importnce of efficient stock cutting, this vrint hs een extensively studied nd vrious heuristics nd pproximtion lgorithms exist for tht. References to ppers studying this prolem cn e found in [GJ84]. In ech of these lgorithms, the rectngles re first sorted ccording to sorting rule, like decresing width (W), incresing height (IH), etc. nd then pcked ccording to some pcking rule, like the OTTOM-LEFT (L) rule where the sorted rectngles re pcked in turn, ech item eing plced s close to the ottom of the strip s it will fit nd then s fr to the left s it cn e plced t tht ottom-most level. For this simple pproch, the LW lgorithm, where we use the OTTOM-LEFT lgorithm fter ordering the locks y decresing width gives us the est pproximtion rtio of 3. nother similr set of lgorithms re sed on different type of pcking rule clled level lgorithms. The rectngles re ll sorted y decresing heights. Two pcking rules clled NEXT FIT nd FIRST FIT give good results with the symptotic worst cse rtios for the two eing nd 1.7 respectively. These lgorithms require guillotine cuts, i.e. edge-to-edge cuts of the strip prllel to the ottom of the strip. Without this constrint, the est lgorithm sed on similr ides is n lgorithm y ker, et. l. [K81] which gives n symptotic worst cse rtio of 1.5. Klietmn nd Krieger [KK75] ddress the rectngle pcking prolem for the specil cse when ll rectngles re squres. They consider collection of squres whose totl re dd up to unity. They show tht ounding ox of = p 3y p is unique rectngle tht will suffice. Hochum nd Mss [HM85] propose shifting strtegy to ddress the specil cse of squre pcking. They consider nother vrition where they try to pck mximl numer of k k squres (for nturl numer k)into n re tht is given y n squres of unit size on rectiliner grid. For ny nturl numer l 1, the pproximtion rtio is (1+1=l) nd the run-time is O(k l n l ). They lso propose wys to otin pproximtions to pcking prolems in higher dimensions, with ritrry orienttions nd with ojects other thn squres. Hwng, Ko nd Horng [HKH94] present genetic lgorithm to solve the rectngle pcking prolem where 90 o rottion of locks is llowed nd squre pcking is preferred. They lso ddress the in pcking nd strip 1

2 () () (c) (d) Figure 1: Visuliztion of encoding of topologicl constrints sed on sequence pirs. () corresponds to the sequence pir h;i; (), (c) nd (d) re h;i; h;indh;i respectively. pcking prolems nd present heuristics for them in the form of genetic lgorithm. They use the slicing tree structure to represent solution. lthough this cnnot express ll the possile pckings, they clim tht it represents most good nd ner optiml pckings. The verge pcking density of their experiments using this lgorithm is 88%. With resonle increse in the running time of their heuristic, they were le to chieve n verge pcking density of 96:5%. Sequence-Pir sed lgorithms.1 Sequence pir Sequence pir is wy of encoding topologicl constrints of ojects (of ny shpe) on plne. It consists of pir of sequences of identicl lphets, one for ech of the oject to e plced. The motivtion is tht given two ojects nd, t lest one of the four reltion holds: ove, elow, to the left of, or to the right of. These topologicl reltions cn e esily encoded using two sequences. We use the following scheme: the reltion tht lies ove is encoded y h;i; tht lies elow y h;i; tht lies to the left y h;i;ndtht lies elow y h;i. In fct, this encoding scheme cn e esily visulized on the tilted grid. Figure 1 shows the four reltions nd their corresponding encoding. more sophiscted exmple is shown in Figure nd Figure. It is not hrd to see tht, for every plcement of ojects on plne, there exists sequence pir which represents ll the topologicl reltions mong the ojects. However, this encoding is not unique. Given lyout of the ojects, it is possile to hve more thn one sequence pir to represent the topologicl reltions. onversely, for every sequence pir, there exists plcement of the ojects which would stisfy the topologicl constrints imposed y

3 () () T S T (c) S (d) Figure : Visuliztion of sequence pirs on ojects ;; nd. () Topologicl constrints sed on the sequences h;i. () The corresponding lyout of the ojects. (c) Longest pth from s to t gives the width of the ounding ox. (d) Longest pth from s to t gives the height of the ounding ox. the pir. gin, this plcement is not unique: one cn dd ritrry mount of spce mong ll the ojects while respecting their reltive positions imposed y the sequence pir.. Properties of sequence pir We re concerned with finding the minimum re of the ounding ox where the given rectngles cn e pcked in. It is therefore importnt to find out from the sequence pir the minimum height nd width of the ounding ox efficiently. It turns out tht they cn e computed firly efficiently y finding the longest pth directed cyclic grph. To simplify discussion, we only consider the computtion of the width of the ounding ox. onsider Figure. For ech rectngle in Figure, there is corresponding vertex in the directed cyclic grph Figure c. Moreover, there re two extr vertices s;t which re the source nd sink. There is n edge from u to v if rectngle v lies on the right of u, s constrined y the sequence pir. The corresponding edge weight is the width of rectngle u. Moreover, there is n edge from the source s to every vertex, with edge weight 0. Finlly, there is n edge from every vertex to the sink t, with weight equl to the corresponding width of the rectngle. It is not hrd to see tht width of the ounding ox is given y the length of the longest pth from to source s to sink t. Using FS, for exmple, one cn compute the longest pth in time liner to the numer of edges, or in the worst cse, qudrtic in the numer of vertices in the grph. The height of the ounding ox cn e computed similrly, s shown in Figure d. The re of the ounding ox is simply the product of the computed width nd height...1 Greedy lgorithm Our first heuristics is greedy pproch. We mintin prtil sequence pir. t ech itertion, we consider new rectngle nd try to insert it into the sequence pir such tht the resulting ounding ox hs the minimum re. For instnce, suppose there re four rectngles ;; nd. First we fix n order of the rectngles to 3

4 consider. Strting with null sequence pir, we first try to insert lock during the first itertion, resulting in the sequence pir h;i. uring the second itertion, we consider ll the four possile plcement of : h;i, h;i, h;i nd h;i. We pick the one with the minimum re, sy h;i. Similrly during the third itertion, we consider ll the nine possile plce of nd pick out the one with smllest re. In generl, during the k itertion, there re k possile plcements of the kth rectngle to consider. Hence the running time for ech itertion is O(k k )=O(k 4 )=O(n 4 ),wheren is the totl numer rectngles to pck, nd therefore the totl running time would e O(n 4 n)=o(n 5 ). Here we ssume tht the running time for fixing the order of the rectngles is less thn O(n 5 ), which is resonle ssumption. We tested this greedy pproch y considering the rectngles in non-incresing order of re perimeter mximum spect rtio mximum of width nd height The experiment results re presented in the Experiment Result section... Genetic lgorithm Genetic lgorithm is comintoril serch technique mimicking the mechnism of nturl selection. We view the sequence pirs s the genes which encodes the plcement of the rectngles, nd define crossover nd muttion on the genes directly. rossover of two genes re done y exchnging their second sequences. Muttion of gene is done y rndomly swpping two djcent lphets in the sequence. The fitness function of gene is simply given y the re of the ounding ox given corresponding to the sequence pir...3 Simulted nneling Simulted nneling is nother widely used comintoril serch technique in the computer-ided-design industry. We serch through the the possile plcement configurtions y defining move on the sequence pirs. move is similr to muttion in genetic lgorithm: we choose rndom pir of djcent lphets nd swp them, producing new configurtion. 3 Experiments We rn intensive experiments on the lgorithms discussed ove. There re two kinds of input: rndom input generted y our enchmrk file genertor nd the rel-life exmples from the VLSI designs. There re two kinds of distriutions tht we use in generting enchmrk files: uniform distriution nd Gussin distriution. nd there re 5 criteri tht those dt re distriuted on: re, sides, integer sides, squre nd fixed height. For exmple, we hve enchmrk files with uniform distriution on the lock re or Gussin distriution on the lock sides nd so on. The integer sides criteri is used to simulte tht in VLSI design, the side of chips my ll e multiple of some unit length, i.e., they hve corse grnulity or discretized smple vlues. ll squre locks re of the similr concern. To Fixed Height option, it is just like pin-pcking. We know its optiml solution nd cn evlute the performnce esily. The test results for uniform distriution re in Fig. 1 nd those for Gussin distriution in Fig.. For every ctegory, we tested on 5 locks, 5 locks nd 50 locks. We iterted 10 instnces for every cse nd got the verge performnce nd worst performnce. The performnce is mesured y the ounding re divided y the totl re of the locks. Plese note tht the divisor my not e the optiml solution. It is less or equl to the optiml solution, to which is n unknown. So for ech lgorithm, there re two rows, the upper row contins the verge rtio nd the second row hs the worst rtio. We only use the re evlution function for greedy lgorithm. The numers in the columns of rndom re re usully igger thn those of other columns ecuse the enchmrk files my consist of some very thin nd long locks, either or oth in horizontl direction nd verticl direction. So even the optiml solution hs pretty ig rtios. 4

5 Rndom re Rndom Sides Rndom Int Sides Rndom Squre Rndom Fixed Height No. lks GSP vg. se Worst se GSP vg. se Worst se SSP vg. se Worst se Tle 1: Testing results on enchmrk files of uniform distriution. GSP stnds for Greedy lgorithm Sequencing Pir, GSP for Genetic lgorithm Sequencing Pir nd SSP for Simulted nneling Sequencing Pir. Rndom re Rndom Sides Rndom Int Sides Rndom Squre Rndom Fixed Height No. lks GSP vg. se Worst se GSP vg. se Worst se SSP vg. se Worst se Tle : Testing results on enchmrk files of Gussin distriution. GSP stnds for Greedy lgorithm Sequencing Pir, GSP for Genetic lgorithm Sequencing Pir nd SSP for Simulted nneling Sequencing Pir. To view the results more intuitively, we mde some r grphs of the testing results for uniform distriution. The Gussin distriution results re pretty similr. Fig. 3 illumintes the verge performnce of the the greedy lgorithm, genetic lgorithm nd simulted nneling on the five kinds of uniform distriution enchmrk files. In ddition, we lso tested our lgorithms on the rel-life exmples from VLSI design, s shown in Fig. 3. From the results, we cn tell tht ll the three lgorithm perform pretty well on verge nd none of them lwys perform etter thn the rest. From the r grph, it is ovious tht the genetic lgorithm gives the most stle performnce. Usully greedy lgorithm runs much fster thn the genetic lgorithm nd simulted nneling, especilly when the numer of locks is huge. For fixed height locks, greedy lgorithm lwys give the optiml solution, ut the genetic lgorithm nd simulted nneling my not. ircuit Xerox HP mi33 mi49 tpe No. lks GSP GSP SSP Tle 3: Testing results of rel-life exmples. GSP stnds for Greedy lgorithm Sequencing Pir, GSP for Genetic lgorithm Sequencing Pir nd SSP for Simulted nneling Sequencing Pir. We lso oserve tht for the greedy lgorithm optimizing for the perimeter sometimes gives etter minimum re thn optimizing for the re. Fig. 4 is the result optimized for the perimeter. Fig. 5 is the result optimized for the re. lso, optimizing for the re usully results in some extreme spect rtio, while optimizing for the perimeter usully gives some squre-like shpe, which is more cceptle in rel-life pplictions. 4 Liner Progrmming pproch 4.1 Introduction In this section, we use very different pproch, Liner Progrmming(LP), to solve this rectngle pcking prolem. First, we will introduce some Integer Liner Progrmming(ILP) methods, nd compre them with 5

6 () () (c) (d) (e) Figure 3: r grph of the testing results on uniformly distriuted enchmrk files. Ech group consists of the verge performnce on 5 locks, 5 locks nd 50 locks. The left group is the results of greedy lgorithm, the middle is tht of genetic lgorithm nd the right group, simulted nneling. ()Rndom re; () Rndom Sides; (c) Rndom Integer Sides; (d) Rndom Squres; (e)rndom Fixed Height 6

7 Figure 4: Optimizing for the perimeter gives smller re. Normlized re=1.068 Figure 5: Optimizing for the re gives very wide shpe. Normlized re=1.07 rnch-nd-method descried in the previous section. Second, we show rounding ILP pproch s n pproximtion lgorithm. Third, we modify the LP simplex lgorithm s nother pproximtion lgorithm. 4. Formultion in LP Liner progrmming in generl is very powerful method to solve pcking or covering prolem. For rectngle pcking, suppose there re locks,, we cn formulte it generlly y severl sets of constrints: oundry constrints for ech lock : x, w = 0 x + w = W y, h = 0 y + w = H Non-overlpping constrints for ny pir of locks,: or jx, x j w + w jy, y j h + h where (x ;y ) nd (x ;y ) re the center coordintes of lock nd, respectively, h is the height of lock, w is the width of lock, W is the ounding ox width, nd H is the ounding ox height. The ojective function is to minimize W H, ut LP does not llow qudrtic ojective function. Here, we use W + H insted. In generl, this is very good pproximtion s it is shown in the previous greedy lgorithm. It is lso possile to hve severl runs y using dynmic weighting fctors to W nd H nd mke it converge to the glol minimum of W H. 7

8 4.3 Four-it ILP Formultion The non-overlpping constrints incur the discrete nture of this prolem, so we hve to design some integer vriles in ILP to hndle it. The most strightforwrd method is y using integer vriles u ;u ;v ;v, such tht Mu + x, x w + w (1) Mu + x, x w + w () Mv + y, y h + h (3) Mv + y, y h + h (4) u + u + v + v 3 (5) where M is sufficient lrge numer(e.g. sum of height nd width of every lock). ctully, we require t lest one of the following inequlities holds to meet the non-overlpping constrints: x, x w + w x, x w + w y, y h + h y, y h + h (6) (7) (8) (9) 4.4 Two-it ILP Formultion Since tht t lest one of the four Eqs.?? holds stisfies the non-overlpping constrints, we cn thus encode this y 0-1 integer vriles u 0 nd v0, such tht: Mu 0 + Mv 0 + x, x w + w M(1, u 0 )+Mv 0 + x, x w + w Mu 0 + M(1, v 0 )++y, y h + h M(1, u 0 )+M(1, v 0 )+y, y h + h (10) (11) (1) (13) (14) The numer of integer vriles re hlf of the previous method. 4.5 Successive Rounding ILP pproch siclly, we use four-it ILP formultion without integer vrile constrints. t ech run of LP, we select nd round one integer vrile to zero y ssigning proility proportionl to the closeness (to zero) of its LP optiml solution. Tht is, suppose u i is n integer vrile, the proility we select u i s rounded integer vrile is 1, u i (1, u j) t most, O(n ) runs re needed, since it is possile for constrint grph to e complete iprtite grph. However, due to trnsitive reltions, if x x nd x x c,thenx x c is ctully implicitly implied in the LP solution. 8

9 4.6 Modified Simplex pproch From the previous formultion(e.g. -it ILP formultion), it is cler tht we hve to mintin the condition tht one of four inequlity holds; therefore, we cn design the formultion s: U + x, x w + w U + x, x w + w V + y, y h + h V + y, y h + h (15) (16) (17) (18) (19) with n extr constrint tht t lest one of U ;U ;V ;V must e in non-sic fesile solution in simplex lgorithm for LP, tht is, one of them must e zero to ctivte one inequlity. ctully, from our implementtion, it is shown very difficult to control simplex lgorithm fter considering these constrints. First, if we relx these constrints nd use regulr simplex lgorithm, we finlly fils to trnsform it ck to fesile solution suject to these extr constnts. The reson is some of the vriles U s ndv s cn not leve the sic fesile solution with one or two pivotings. This lmost cretes dedlock if we don t use cktrck. Second, s different pproch, we cn mintin these extr constrints ll the wy from the initil solution down to the optiml solution. However, this is clerly very limited locl serch. The qulity of the output is even 1.5 re of tht of the sequence-pir greedy lgorithm. Moreover, the runtime is much more thn tht of the sequence-pir greedy lgorithm. It is possile to introduce some rndomness into selection of pivots; however, this cretes nother prolem tht the initil fesile solution for LP my not e found. In our implementtion, we run this lgorithm 30 times to choose the optimum one nd discrd the run tht cn not find fesile solution. In Fig. 6, it cn e regrded s the serch from fesile solution down to the extr constrints oundry or climing from LP optiml solution up to the fesile solution set. Serch from Fesile Set Extr onstrints LP Fesile Solution onvex Set LP Optiml Solution Trnform Infesi to Fesile Soluti Figure 6: Serch for Optiml Solution in Modified Simplex lgorithm 4.7 Experimentl Results We use pulic domin code for solving mixed integer progrmming prolem here. The ILP sed (ctully mixed integer progrmming) lgorithm cn not hndle ig exmple s expected. ctully, the performnce is fr from the rnch-nd-ound method sed on sequence-pir. The runtime comprison is shown s 9

10 Exmple #Rentngles -it ILP 4-it ILP & t s 0.7s 0.0s t s 6.76s 0.06s t6 6 46s 994s 0.07s t7 7 >1hr >1hr 0.16s t8 8 N/ N/ 3.10s t9 9 N/ N/ 3.15s t10 10 N/ N/ 1.46s Here we compre the output qulity of pproximtion lgorithms: Sequence-Pir Successive Modified Exmple #Rectngles Optiml Greedy lg. Rounding Simplex t t t t t t t The re is in terms of normlized re. The totl of rectngles re is normlized to 1. The column in modified simplex lgorithm is the est solution tken from 30 runs with some rndomness in pivoting section descried previously. The greedy serch sed on sequence pir lwys outperforms the other pproches in these exmples. It my e ecuse the other pproches re somewht similr to rndom serch. 5 onclusion In this project, we study the sequence-pir sed pproch on the two-dimensionl rectngle pcking prolem s well s LP sed pproch. We use extensive exmples from rndom inputs nd rel-life exmples from VLSI designs. The results suggest tht the greedy lgorithm is very efficient nd outputs stisfying pcking res. lso, genetic lgorithm seems le to find sometimes etter result thn the greedy lgorithm. Simulted nneling is not so efficient compred with the genetic lgorithm. LP sed lgorithms need more tricks thn the other pproches. The serch spce (4 n(n,1) ) is much lrger thn the sequence pir pproch((n!) ). We need more sophiscted method to mnipulte the simplex lgorithm to mke it serch through the fesile solution spce. Its performnce is still inferior to the sequence-sed pproches. References [1] T. ORMEN,.LEISERSON nd R. RIVEST, Introduction to lgorithms, MIT Press, [] M.R. GREY nd.s. JOHNSON, omputers nd Intrctility: Guide to the Theory of NP- ompleteness, W. H. Freemn nd o., Sn Frncisco, [3] E.G. OFFMN, JR., M.R. GREY nd.s. JOHNSON, pproximtion lgorithms for in-pcking - n Updted Survey, lgorithm esign for omputer System esign, Springer-Verlg, [4].S. KER,.J.ROWN nd H.P. KTSEFF, 5/4 lgorithm for two-dimensionl pcking, J. of lgorithms,, [5].J. KLEITMN nd M.K. KRIEGER, n optiml ound for two dimensionl in pcking, Proc. 16th nn. Symp. on Foundtions of omputer Science, IEEE omputer Society, Long ech,,

11 [6].S. HOHUM nd W.MSS, pproximtion schemes for covering nd pcking prolems in Imge Processing nd VLSI, J. of M, 3(1), 1985 [7] S. HWNG,.KO nd J. HORNG, On Solving Rectngle in Pcking Prolems Using Genentic lgorithms, Proc. IEEE Interntionl onference on Systems, Mn, nd yernetics,