Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing

Transcription

1 SIAM REVIEW Vol. 44, No. 1, pp c 2002 Society for Idustrial ad Applied Mathematics Perfect Packig Theorems ad the Average-Case Behavior of Optimal ad Olie Bi Packig E. G. Coffma, Jr. C. Courcoubetis M. R. Garey D. S. Johso P. W. Shor R. R. Weber M. Yaakakis Abstract. We cosider the oe-dimesioal bi packig problem uder the discrete uiform distributios U{j, k}, 1 j k 1, i which the bi capacity is k ad item sizes are chose uiformly from the set {1, 2,..., j}. Note that for 0 < u = j/k 1 this is a discrete versio of the previously studied cotiuous uiform distributio U(0, u], where the bi capacity is 1 ad item sizes are chose uiformly from the iterval (0, u]. We show that the average-case performace of heuristics ca differ substatially betwee the two types of distributios. I particular, there is a olie algorithm that has costat expected wasted space uder U{j, k} for ay j, k with 1 j < k 1, whereas o olie algorithm ca have o( 1/2 ) expected waste uder U(0, u] for ay 0 < u 1. Our U{j, k} result is a applicatio of a geeral theorem of Courcoubetis ad Weber that covers all discrete distributios. Uder each such distributio, the optimal expected waste for a radom list of items must be either Θ(), Θ( 1/2 ), or O(1), depedig o whether certai perfect packigs exist. The perfect packig theorem eeded for the U{j, k} distributios is a itriguig result of idepedet combiatorial iterest, ad its proof is a corerstoe of the paper. We also survey other recet results comparig the behavior of heuristics uder discrete ad cotiuous uiform distributios. Key words. bi packig, olie, average-case aalysis, approximatio algorithms AMS subject classificatios. 68W25, 68W40 PII. S Itroductio. Suppose oe is give items of sizes 1, 2, 3,..., j, oe of each size, ad is asked to pack them ito bis of capacity k with as little wasted space as possible, i.e., oe is asked to fid a least cardiality partitio (packig) of the set of items such that the sizes of the items i each block (bi) sum to at most k. For what Published electroically February 1, This paper origially appeared i SIAM Joural o Discrete Mathematics, Volume 13, Number 3, 2000, pages Columbia Uiversity, New York, NY (egc@ee.columbia.edu). Departmet of Computer Sciece, Athes Uiversity of Ecoomics ad Busiess, Athes, Greece (courcou@csi.forth.gr). Bell Labs, Murray Hill, NJ AT&T Labs Research, Florham Park, NJ (dsj@research.att.com, shor@research.att.com). Statistical Laboratory, Uiversity of Cambridge, Cambridge CB3 0WB, UK (R.R.Weber@ statslab.cam.ac.uk). Avaya Labs Research, Baskig Ridge, NJ (mihalis@research.avayalabs.com). 95

2 96 COFFMAN ET AL. values of j ad k ca the set be packed perfectly (i.e., so that the sizes of the items i each block sum to exactly k)? Clearly the sum of the item sizes must be divisible by k, but what other coditios must be satisfied? Surprisigly, the divisibility costrait is ot oly ecessary but sufficiet. Readers might wat to try their had at provig this. Relatively short proofs exist, as illustrated i the ext sectio, but a certai igeuity is required to fid oe. The exercise serves as a warm-up for the followig more geeral ad more difficult theorem, also proved i the ext sectio, i which there are r copies of each size for some r 1. Theorem 1 (perfect packig theorem). For positive itegers k, j, ad r, with k j, oe ca perfectly pack a list L cosistig of rj items, r each of sizes 1 through j, ito bis of size k if ad oly if the sum of the rj item sizes is a multiple of k. I set-theoretic terms, the questio aswered by Theorem 1 is a itriguig puzzle i pure combiatorics. But our motivatio to work o it came from its relevace to certai fudametal questios about the average-case aalysis of algorithms. I particular, cosider the stadard bi packig problem, i which oe is give a bi capacity b ad a list of items L = (a 1, a 2,..., a ), where each a i has a positive size s i b, ad is asked to fid a packig of these items ito a miimum umber of bis. I most real-world applicatios of bi packig, as i Theorem 1, the item sizes are draw from some fiite set. However, the usual average-case aalysis of bi packig heuristics has assumed that item sizes are chose accordig to cotiuous probability distributios, which by their ature allow a ucoutable umber of possible item sizes (see [3, 10], for example). The assumptio of a cotiuous distributio has the advatage of sometimes simplifyig the aalysis ad has bee justified o the grouds that cotiuous distributios should serve as reasoable approximatios for discrete oes. But there are reasos to ask whether this is actually true. For example, cosider the cotiuous uiform distributios U(0, u], 0 < u 1, where the bi capacity is 1 ad item sizes are chose uiformly from the iterval (0, u], ad the discrete uiform distributios U{j, k}, 1 j k 1, where the bi capacity is k ad item sizes are chose uiformly from the set {1, 2,..., j}. The limit of the distributios U{mj, mk}, as m, is equivalet to U(0, j/k] (after scalig by dividig the item sizes ad bi capacities by mk). However, i the limit, combiatorial questios such as those addressed by Theorem 1 evaporate. This suggests that somethig importat (ad iterestig) may i fact be lost i the trasitio from discrete to cotiuous models. The results i this paper show that this is ideed the case. To describe the results, we eed the followig otatio. If A is a bi packig algorithm ad L is a list of items, the A(L) is the umber of bis used whe A is applied to L, ad s(l) is the sum of the item sizes i L divided by the bi capacity. Note that s(l) is a lower boud o the umber of bis eeded to pack L. The waste i the packig of L by A is deoted by W A (L) = A(L) s(l). Let OPT(L) deote a algorithm that always produces a optimal packig. I what follows, L (F ) will deote a list of items whose sizes are idepedet samples from a give distributio F. The expected waste rate EW A (F ) for a algorithm A ad distributio F is defied to be the expected value of W A (L (F )) as a fuctio of. I what follows we typically abbreviate this as simply the expected waste. We say a distributio F is a bouded waste distributio if EW OP T (F ) = O(1). As a cosequece of Theorem 1 ad a classificatio theorem of Courcoubetis ad Weber [11], we ca prove the followig. Theorem 2. For ay j, k with 1 < j < k 1, EW OP T (U{j, k}) = O(1). This i itself does ot represet a departure from the cotiuous model, sice U(0, u] is also a bouded waste distributio for all u, 0 < u < 1 [3, 16]. The distictio comes whe we cosider olie algorithms. I a olie algorithm, items are

3 PERFECT PACKING THEOREMS 97 assiged to bis i the order i which they occur i the iput list L. Each assigmet must be made without kowledge of the sizes or umber of items later i the list, ad oce a item is packed it caot subsequetly be moved. This mirrors may practical situatios but clearly is a substatial restrictio o the power of a algorithm. I particular, we ca prove the followig. Theorem 3. If A is a olie algorithm ad u (0, 1], the it caot be the case that E[W A (U(0, u])] = o( 1/2 ). I cotrast, i the discrete uiform case there is a sigle olie algorithm that has bouded expected waste for all the distributios U{j, k}, 1 j < k 1. This is the recetly discovered sum-of-squares algorithm (SS) of [12], defied as follows. Suppose we are packig iteger-sized items ito bis of capacity b. Whe a olie algorithm packs a item x from such a list, the oly thig relevat about the curret packig is the umber N h of bis whose curret cotets total h, 1 h b 1. SS chooses the bi ito which x is to be placed (either a ew, previously empty bi or oe that is already partially filled but has eough room for x) so as to miimize the resultig sum b 1 h=1 N h 2. Note that SS ca be implemeted to take O(b) time per item [12], ad so rus i liear time for ay fixed bi size. As show i [12], algorithm SS performs well o average i a surprisigly geeral sese. Let us say that a discrete distributio F is ay triple (b, S, p), where b is a itegral bi size, S = {s 1,..., s d } is a fiite set of itegral item sizes i the rage from 1 to b 1, ad p = (p 1,..., p d ) is a ratioal probability vector, where p i > 0 is the probability of item size s i ad d i=1 p i = 1. (We igore the possibility of items of size b sice such items always must start a ew bi ad completely fill it, leavig the rest of the packig uaffected.) The followig specialized versio of the result of [12] suffices for our eeds. Theorem (see [12]). For ay discrete distributio F = (b, S, p) with 1 S, EW SS (F ) = Θ ( EW OP T (F ) ). Hece by Theorem 2, EW SS (U{j, k}) = O(1) for all 1 < j < k 1. The curret paper is orgaized as follows. The proof of the perfect packig theorem (Theorem 1) appears i sectio 2. Sectio 3 the presets the proof of Theorem 2. We begi by describig the classificatio result of Courcoubetis ad Weber [11] upo which the proof depeds. This result says that for ay discrete distributio F, EW OP T (F ) must be oe of Θ(), Θ( 1/2 ), or O(1). Which case applies depeds o the existece of certai perfect packigs ad is i geeral NPhard. However, Theorem 1 allows us avoid this complexity i the case of the discrete uiform distributios U{j, k}. Theorem 3, this paper s cotributio to the theory of cotiuous distributios, is proved i sectio 4. We coclude i sectio 5 with a survey of the results that have bee proved about the average-case behavior of bi packig algorithms uder discrete ad cotiuous distributios. As we shall see, there are other sigificat differeces betwee the discrete ad cotiuous cases. 2. The Perfect Packig Theorem. We begi our proof of Theorem 1 with three lemmas that list a umber of special istaces that lead to perfect packig. The first lemma takes care of the special case, r = 1. Lemma 4. Suppose m, j, ad k are positive itegers such that j k ad mk = j(j + 1)/2. The the set of j items, oe each of sizes 1,..., j, perfectly packs ito m bis of size k. Proof. The proof is by iductio. Pick j ad k ad assume the theorem is true for all pairs that are smaller i lexicographic order tha (k, j). The theorem is clearly true for k 2 or j 2, so assume k, j > 2.

4 98 COFFMAN ET AL. If j > k/2, the we ca start by perfectly packig bis with pairs of items (j i, k j + i), 0 i < j k/2, after which the remaiig items are those of sizes 1,..., k j 1, plus the item of size k/2 if k is eve. Sice the sum of the sizes of the items that have bee packed at this poit is a multiple of k, the sum of the sizes of the remaiig items is also a multiple of k. If k is odd, the upacked items are a istace of (k, k j 1), with k j 1 < j ad the iductio hypothesis applies. If k is eve, the k/2 divides j(j + 1)/2 ad all remaiig items are o larger tha k/2. Thus the items 1,..., k j 1 form a istace of (k/2, k j 1) ad by the iductio hypothesis ca be perfectly packed ito bis of size k/2. These half-bis ad the item of size k/2 ca the be combied ito bis of size k. Now suppose j k/2. If k is eve, the we have a istace of (k/2, j) ad the iductio hypothesis applies. If k is odd, first ote that k/2 j ad j > 2 imply k > j + 1, which together with mk = j(j + 1)/2 implies j > 2m. Thus we ca costruct m pairs of items each of total size k = 2j 2m + 1 by combiig j i with k j + i, 0 i m 1. If we place oe pair i each of our m bis, we ow have m bis with gaps of size k k = k 2j + 2m 1 ad items of sizes 1,..., j 2m. Because mk = j(j + 1)/2, the sum of these item sizes must be m(k k ), ad so a applicatio of the iductio hypothesis to the istace (k k, j 2m) completes the proof. Lemma 5. Cosider r > 1 sets, the ith of which cosists of j items of cosecutive sizes, l i + 1,..., l i + j, for some l i 0. Suppose either (a) r is eve or (b) j is odd. The these rj items perfectly pack ito j bis of size equal to the sum of the average item sizes i the r groups, i.e., r(j + 1)/2 + r i=1 l i. Proof. The lemma will follow if we ca show that for l i = 0, 1 i r, ad bis of size r(j + 1)/2 it is possible to pack perfectly the items ito the j bis i such a maer that each bi cotais exactly oe item from each of the r sets. If r is eve, the we simply take two of the sets ad pack the ith largest item i oe set with the ith smallest item i the other set, i.e., as the pair (i, j i + 1), i = 1,..., j. This fills j bis to level j + 1. By repeatig this r/2 times we fill j bis of size r(j + 1)/2. If r ad j are both odd, the a extra step is required. The idea is first to pack items i triples, oe item from each of three sets, such that the sum of each triple is the same. It is easiest to appreciate the costructio by cosiderig a example, say j = 9. The triples, which each sum to 15, are give i the colums below I geeral, the triples are (i, i + (j + 1)/2, j + 1 2i), i = 1,..., (j 1)/2, ad (i, i (j 1)/2, 2j 2i + 1), i = (j + 1)/2,..., j. The result of packig these oe per bi is to fill all j bis to level 3(j + 1)/2. The umber of remaiig sets is eve ad the remaiig spaces i the j bis are equal. Thus, the procedure for case (a) ca be applied to complete the packig i each bi. The followig lemma provides part of the iductio step used i the proof of Theorem 1. Lemma 6. Cosider a quadruple (k, j, r, m) of positive itegers such that k j ad mk = rj(j + 1)/2. The there exists a perfect packig of r copies of 1,..., j ito m bis of size k if there exists a perfect packig for each lexicographically smaller quadruple of this form, ad if ay oe of the followig holds:

5 PERFECT PACKING THEOREMS 99 (a) j k/2. (b) r does ot divide k. (c) k or r is eve. (d) j (r 1)k/2r. Proof. First, usig the argumets of Lemma 4, we demostrate how to reduce the problem to a smaller istace if (a) holds. If j k/2 ad k is odd, the we ca pack bis with pairs (j i, k j + i), i = 0,..., j (k + 1)/2. The remaiig items, which are of sizes 1,..., k j 1, defie the smaller istace (k, k j 1, r, m ), where m = m r(2j + 1 k)/2. If j > k/2 ad k is eve, the we ca pack bis i the same way, i = 0,..., j 1 k/2. The remaiig items, which are of sizes 1,..., k j 1 ad k/2, ca be packed ito bis of size k/2 by the iductio hypothesis that there exists a perfect packig for (k/2, k j 1, r, m ), where m = 2m r(2j k). If (b) holds, the r ad m must have a commo factor p > 1 ad the problem reduces to the istace (k, j, r/p, m/p). Now suppose either (a) or (b) holds but (c) does. If k is eve, the k/2 divides rj(j + 1)/2. Sice (a) does t hold, j < k/2. Thus the problem reduces to a smaller istace i which the bi size is k/2. If r is eve, the sice (b) does ot hold, k is divisible by r ad so must be eve too. Thus the same argumet applies. Fially, for case (d), assume that (a), (b), ad (c) do ot hold, i.e., j < k/2, r divides k, ad k ad r are both odd. Let r 1 = (r+1)/2, k 1 = kr 1 /r ad r 2 = (r 1)/2, k 2 = kr 2 /r. Note that r 1 +r 2 = r ad k 1 +k 2 = k. The fact that r is odd implies that r 1 ad r 2 are itegers. The fact that r divides k implies that k 1 ad k 2 are itegers, with mk 1 = r 1 j(j + 1)/2 ad mk 2 = r 2 j(j + 1)/2. Sice by assumptio j < k/2, we have k 1 j, ad hece by hypothesis for the istace (k 1, j, r 1, m), we ca pack r 1 copies of 1,..., j ito m bis of size k 1. Similarly, if also j k 2, the we ca pack r 2 copies of 1,..., j ito m bis of size k 2. Sice k 1 + k 2 = k we ca combie pairs of bis of sizes k 1 ad k 2 ito bis of size k. Thus there is a reductio to smaller istaces if j k 2 = (r 1)k/2r, i.e., if (d) holds. Proof of the perfect packig theorem. Istaces for which the theorem is to be proved are described by the quadruples of Lemma 6. Notice that it would be eough to specify the triple (k, j, r); however, it is helpful to metio m explicitly. The proof of the theorem is by iductio o (k, j, r) uder lexicographical orderig. By Lemma 4 it is true for r = 1. Assume all quadruples that are smaller tha (k, j, r, m) ca be perfectly packed ad r > 1. We show there exists a perfect packig of r copies of 1,..., j ito m bis of size k. By Lemma 6, we eed oly cosider the case whe k ad r are odd, r divides k, ad (r 1)k/2r < j < k/2. Note that i this case (r 1)k/2r is a iteger ad k/2r is 0.5 more tha a iteger. We show below that we ca perfectly pack all the items of sizes from j + 1 (r 1)k/2r through j ito bis of size k. (Note that the lower boud o this rage is greater tha 1 because of the above lower boud o j.) The theorem the follows because the remaiig items form a smaller quadruple, so by the iductio hypothesis they ca be perfectly packed ito bis of size k. To follow the costructio below, the reader may fid it helpful to cosider a specific example. Cosider the quadruple (k, j, r, m) = (165, 77, 5, 91). Note that k ad r are odd, r divides k, ad j lies betwee (r 1)k/2r = 66 ad k/2 = We show below how to perfectly pack all items of sizes 12,..., 77. The remaiig items form the smaller quadruple (165, 11, 5, 2). To pack all items of sizes from j +1 (r 1)k/2r through j, we divide the rage of item sizes ito itervals, i.e., sets of cosecutive itegers. Each iterval is symmetric about a multiple of k/2r ad has oe of two legths depedig o whether the iterval

6 100 COFFMAN ET AL. is symmetric about a odd or eve multiple of k/2r. To form the itervals, we first take the largest iterval that is symmetric about (r 1)k/2r; this is the iterval [(r 1)k/r j, j]. Note that this iterval does ot iclude (r 2)k/2r sice j < k/2 = rk/2r. Next we take the largest iterval that ca be formed from the remaiig items that is symmetric about (r 2)k/2r, obtaiig the iterval [j k/r+1, (r 1)k/r j 1]. Cotiuig i this fashio ad takig itervals symmetric about further multiples of k/2r, we ed up with itervals of two kids. First, there are (r 1)/2 itervals cetered o eve multiples of k/2r, with the iterval cetered o (r 1 2i)k/2r beig [(r 1 i)k/r j, j ik/r], where i rages from 0 to (r 3)/2. Secod, there is a equal umber of itervals cetered o odd multiples of k/2r, with the iterval cetered o (r 2i)k/2r beig [j ik/r + 1, (r i)k/r j 1], where i rages from 1 to (r 1)/2. Note that the smallest edpoit is j ik/r + 1 for i = (r 1)/2, which equals j + 1 (r 1)k/2r as claimed above. For the umerical example above, there are two itervals of each type. Itervals of the first type are [22, 44] ad [55, 77]; they are of legth 23 ad symmetric about 33 ad 66. Itervals of the secod type are [12, 21] ad [45, 54]; they are of legth 10 ad symmetric about 16.5 ad I geeral, itervals of the first type have odd legth 2j (r 1)k/r + 1 ad are symmetric about a eve multiple of k/2r. Itervals of the secod type have eve legth k 2j 1 ad are symmetric about a odd multiple of k/2r. Our pla is to use Lemma 5 to perfectly pack ito bis of size k those items whose sizes lie i itervals of the same type. We begi by cosiderig all those itervals of the first type. These have odd legths ad they are symmetric about poits ik/r, i = 1,..., (r 1)/2. There are r items of each size i each of these itervals. Our strategy is to partitio these itervals ito groups that satisfy the hypotheses of Lemma 5(b). That is, we arrage for the midpoits of the itervals withi each group to sum to k. Sice the midpoits correspod to the average item sizes for the correspodig itervals, ad the umber of items i the itervals is odd, Lemma 5(b) implies that we ca perfectly pack the items i the itervals of each group. Costructig these groups is a bi packig problem i which the midpoits of the itervals take o the role of item sizes. I what follows we write items i quotes whe speakig of the midpoits of itervals, possibly ormalized, ad viewig them as items to be perfectly packed i bis of some required size. I cosiderig itervals of the first type, it is as though we had r items of each of the sizes ik/r, i = 1,..., (r 1)/2, ad wished to pack them ito bis of size k. After a ormalizatio that multiplies each item size by r/k, this is equivalet to the problem of packig r items of each of the sizes 1,..., (r 1)/2 ito bis of size r. That is, we have a smaller versio (k, j, r, m ) of our packig problem, with j = (r 1)/2, k = r = r, ad m = r j (j + 1)/2k. But by the iductio hypothesis this meas that the desired packig ca be achieved. I the example, it is as though we had 5 items of sizes 33 ad 66 that are to be packed i bis of size 165. Normalizig by a factor of 1/33, this is equivalet to the problem istace (5, 2, 5, 3). We must ow pack items whose sizes lie i itervals of the secod type. These itervals are of eve legth, symmetric about the poits ik/2r, for i odd ad i = 1,..., r 2. Agai, there are r items of each size i these itervals. As above, we exhibit a reductio to a smaller perfect packig problem. After we multiply item sizes by 2r/k the problem is equivalet to perfectly packig r copies of items of sizes 1, 3, 5,..., r 2 ito bis of size 2r. For the example, this is 5 copies of items of sizes 1 ad 3, to be perfectly packed ito 2 bis of size 10. Ufortuately, if the sum of the item sizes is a odd multiple of r the items caot be perfectly packed ito bis of size 2r. For this reaso, ad also because it is coveiet to do so eve

7 PERFECT PACKING THEOREMS 101 whe the sum of the item sizes is a multiple of 2r, we cosider perfect packigs ito bis of sizes r ad 2r. Assume for the momet that r copies of items of sizes 1, 3, 5,..., r 2 ca be perfectly packed ito bis of sizes r ad 2r. If they are packed etirely ito bis of size 2r, the the umber of items i each bi must be eve (as all item sizes are odd), ad so Lemma 5 applies ad implies that the origial items ca be perfectly packed ito bis of size k. O the other had, suppose a bi of size r is required. The set of items that are packed ito a bi of size r correspods to a set of itervals whose midpoits sum to k/2. Recall that the itervals are of eve legth. We divide each such iterval ito its first half ad its secod half, obtaiig twice as may itervals, whose midpoits ow sum to k. Now we ca agai use Lemma 5 to costruct the perfect packig. The fial step i the proof is to show that we ca ideed perfectly pack r copies of each of the item sizes 1, 3, 5,..., r 2 ito bis of sizes r ad 2r. We shall use a differet packig depedig upo whether r = 4α + 1 or r = 4α + 3. For the case r = 4α + 1, the items are perfectly packed by the followig simple procedure. We begi by packig oe bi with (1, 1, r 2), oe bi with (2i + 1, 2i + 1, r 2i 2, r 2i) for each i = 1,..., α 1, ad oe bi with (2i 1, 2i+1, r 2i, r 2i) for each i = 1,..., α (otig that i the fial case, whe i = α, we get three items of size 2i + 1 = r 2i). This packs four items of each size larger tha 1, ad three items of size 1. We ca apply this packig α times, leavig us with oe item of each size larger tha 1 ad α + 1 items of size 1. The we pack oe bi with (1, 2i 1, r 2i) for each i = 1,..., α. This uses up all the remaiig items (where α + 1 items of size 1 are used because there are two items of size 1 whe i = 1). For the umerical example, i which α = 1, this costructio says that we should pack 5 copies of 1 ad 3 ito bis of size 5 ad 10 by first packig oe bi with (1, 1, 3) ad the oe bi with (1, 3, 3, 3). This leaves oe item of size 3 ad two of size 1. These perfectly pack ito a bi of size 5. Whe r = 4α + 3 the procedure is very similar to that above. We begi by packig oe bi with (1, 1, r 2), oe bi with (2i + 1, 2i + 1, r 2i 2, r 2i) for each i = 1,..., α, ad oe bi with (2i 1, 2i + 1, r 2i, r 2i) for each i = 1,..., α. As before, this packs four items of each size larger tha 1, ad three items of size 1. We apply this packig α times, leavig us with three items of each size larger tha 1 ad α + 3 items of size 1. The we pack oe bi with (2i 1, 2i + 1, r 2i, r 2i) for each i = 1,..., α. This leaves us with α + 2 items of size 1, two items of size 2α + 1, ad oe item of each other size. Fially, as before, we pack oe bi with (1, 2i 1, r 2i) for each i = 1,..., α + 1, which uses up all remaiig items. 3. Proof of Theorem 2. Recall the theorem statemet: For ay distributio U{j, k}, with j < k 1, EW OP T (U{j, k}) = Θ(1). We rely o a geeral result of Courcoubetis ad Weber [11]. Suppose F = (b, S, p) is a discrete distributio as defied i sectio 1. Note that a packig of items with sizes from S = {s 1,..., s d } ito a bi of size b ca be viewed as a oegative iteger vector c = (c 1,..., c d ), where d i=1 c is i b. Of particular iterest are those vectors that give rise to a sum of exactly b, which we shall call perfect packig cofiguratios. For istace, if S = {1, 2, 3} ad b = 7, oe such cofiguratio would be (1, 0, 2). Let P S,b deote the set of all perfect packig cofiguratios for a give S ad b. Let Λ S,b be the covex coe i R d spaed by all oegative liear combiatios of cofiguratios i P S,b. Theorem (Courcoubetis ad Weber [11]). For ay discrete distributio F = (b, S, p), the followig hold.

8 102 COFFMAN ET AL. (a) If p lies i the iterior of Λ S,b, the EW OP T (F ) = O(1). (b) If p lies o the boudary of Λ S,b, the EW OP T (F ) = Θ( 1/2 ). (c) If p lies outside of Λ S,b, the EW OP T (F ) = Θ(). I geeral it is NP-hard to determie which of the three cases applies to a give distributio (as ca be proved by a straightforward trasformatio from the PARTI- TION problem [13]). However, for the distributios U{j, k}, j < k 1, we ca use the followig lemma, which we shall prove usig the perfect packig theorem, to show that (a) applies. Lemma 7. For each i, j, k with 1 i j < k 1, there exist positive itegers r i, s i, m i < 2k 2 such that the set of r i j + s i items cosistig of r i + s i items of size i together with r i items of each of the other j 1 sizes ca be packed perfectly ito m i bis of size k. Note that this lemma implies that the j-dimesioal vector ē = (1/j, 1/j,..., 1/j) is strictly iside the appropriate coe whe S = {1, 2,..., j}, j < k 1. This is because ē is i the iterior of the coe spaed by vectors of the form (r i,..., r i, r i + s i, r i,..., r i ), i = 1,..., j, ad those vectors are sums of perfect packig cofiguratios by Lemma 7. The proof of Theorem 2 thus follows from case (a) of the above theorem. Proof of Lemma 7. We make use of the perfect packig theorem. There are two cases. If k i + j, we simply set r i = k i ad s i = m i = j(j + 1)/2. Note that the total size of r i items each of the sizes 1,..., j equals r i j(j + 1)/2, so by the perfect packig theorem, we ca perfectly pack them ito j(j + 1)/2 = m i bis of size r i = k i. The remaiig s i = m i items of size i ca the go oe per bi to fill these bis up to size precisely k. O the other had, suppose k < i + j. Now thigs are a bit more complicated. We have r i = 2(k i), s i = (k j)(k j 1), ad m i = s i + r i (2j k + 1)/2. By the perfect packig theorem r i items each of the sizes 1,..., k j 1 perfectly pack i s i bis of size k i. (Such items exist because by assumptio j < k 1.) We the add the additioal s i items of size i to these bis, oe per bi, to brig each bi up to size k. There remai r i items each of sizes k j through j, for a total of r i (j (k j 1)) = 2(m i s i ) items. These ca be used to completely fill the remaiig m i s i bis with pairs of items of sizes (j, k j), (j 1, k j + 1),..., ( k/2, k/2 ). Note that if k is eve, the last bi type cotais two items of size k/2, but we have a eve umber of such items by our choice of r i = 2(k i), so this presets o difficulty. It is easy to verify that i both cases r i, s i, ad m i are all less tha 2k Proof of Theorem 3. Recall the theorem statemet: If L has item sizes geerated accordig to U(0, u] for 0 < u 1, ad A is ay olie algorithm, the there exists a costat c > 0 such that E[W A (L )] > c 1/2 for ifiitely may. Proof. Let w(t) deote the amout of empty space i partially filled bis after t items have bee packed. We show that for ay > 0 the expected value of the average of w(1),..., w() is Ω( 1/2 u 3 ). This implies that E[w()] must be Ω( 1/2 u 3 ), i.e., ot o( 1/2 ). Cosider packig item a t+1. Let υ(t) deote the umber of oempty bis that have a gap of at least u 2 /8 after the first t items have bee packed. There are at most υ(t) bis ito which oe ca put a item larger tha u 2 /8. Therefore, if a t+1 is to leave a gap of less tha δ i its bi, either it must have size less tha u 2 /8 or its size must be withi δ of the empty space i oe of these υ(t) bis with gaps larger tha u 2 /8. The probability of this is at most [u 2 /8 + δυ(t)]/u. By choosig δ = u 2 1/2 /8, coditioig o whether υ(t) is greater or less tha 1/2, ad otig that the size of a t+1 is distributed as U(0, u] idepedet of υ(t), we have

9 PERFECT PACKING THEOREMS 103 Now E[w(t)] P (a t+1 leaves gap < δ) P (υ(t) 1/2 ) + u/4. t 1 δ P (a s+1 is last i a bi ad leaves gap δ) t 1 = δ [P (a s+1 is last i a bi) P (a s+1 is last i a bi ad leaves gap < δ)] t 1 δ [P (a s+1 is last i a bi) P (a s+1 leaves gap < δ)] t 1 t 1 t 1 δ P (a s+1 is last i a bi) P (υ(s) 1/2 ) u/4. Let S t be the sum of the first t item sizes, ad ote that S t is a lower boud o the umber of bis ad hece o the umber of items that are the last item i a bi. We thus have [ t 1 ] E P (a s+1 is last i a bi) E[S t ] = tu/2. Usig the fact that δ = u 2 1/2 /8, we the have [ ] t 1 E[w(t)] (u 2 1/2 /8) tu/4 P (υ(s) 1/2 ). If 1 P (υ(s) 1/2 ) u/24, we have for all t /2, This implies E[w(t]) (u 2 1/2 /8)[u/8 u/24] = u 3 1/2 /96. E [ 1 ] w(t) u 3 1/2 /192. t=1 O the other had, if 1 P (υ(s) 1/2 ) u/24, the [ ] 1 E w(t) 1 P (υ(t) 1/2 ) 1/2 (u 2 /8) t=1 t=1 1/2 (u 2 /8)(u/24) = u 3 1/2 /192. These imply that E[w()] is Ω( 1/2 ). It should be oted that the above proof relies heavily o the fact that the distributio is cotiuous, sice this is the reaso why the uio of 1/2 itervals of size δ caot cover the full probability space. Our discrete distributios U{j, k} do ot have this failig, ad for this reaso we ca obtai sigificatly better average-case behavior for them.

10 104 COFFMAN ET AL. Table 1 Expected waste i the symmetric case. U(0, 1] Ref U{j, k}, j = k 1, k Ref OP T, F F D, BF D Θ( 1/2 ) [20, 21] Θ( 1/2 ) [10] SS Θ( 1/2 k 1/2 ) [12] F F Θ( 2/3 ) [9] Θ( 1/2 k 1/2 ), k = O( 1/3 ) [9] Θ( 2/3 ), k = Ω( 1/3 ) [9] BF Θ( 1/2 log 3/4 ) [26] Θ( 1/2 log 3/4 k) [6] Best olie Θ( 1/2 log 1/2 ) [26, 27] Θ( 1/2 log 1/2 k) The upper boud is proved i the referece; the lower boud is cojectured based o experimets. The upper ad lower bouds here appear to follow from the correspodig results for the cotiuous case, but the details of the upper boud i particular still eed to be worked out. 5. Cocludig Remarks. The results i this paper were amog the first to be obtaied about the average-case behavior of bi packig algorithms uder discrete distributios. Sice they were aouced i [4], may additioal results have bee proved, illustratig further cotrasts with (ad similarities to) the case of cotiuous distributios. I this cocludig sectio we survey the literature ad poit out some of the remaiig ope problems. Let us begi by cosiderig symmetric uiform distributios, as represeted by U(0, 1] ad U{k 1, k}, k 1. (I geeral, a symmetric distributio is oe that satisfies p(s ) = p(s b ) for all, 0 b.) Table 1 summarizes what is kow about average-case behavior uder these distributios. A horizotal lie separates the offlie algorithms from the olie oes. Except where oted, all results i this table are theorems. Four famous classical algorithms have bee extesively studied. First fit (F F ) is a olie algorithm i which each item is placed i the first bi that has room for it, where bis are sequeced accordig to the order i which they received their first item. If o bi has room, a ew bi is started. Best fit (BF ) is similar, except ow the item is placed i the bi with the smallest gap large eough to cotai it (ties broke i favor of the earlier bi). First fit decreasig (F F D) ad best fit decreasig (BF D) are the correspodig offlie algorithms i which the list is first sorted so that the items are i oicreasig order by size, ad the F F (BF ) is applied. From a worst-case poit of view, F F ad BF are equivalet: i a asymptotic sese each ca produce packigs that use 70% more bis tha optimal, but either ca do ay worse [15]. The correspodig offlie versios F F D ad BF D each ca use 2/9 = % more bis tha optimal but ca do o worse [14, 15]. The results i Table 1 show that these algorithms perform much better o average tha i the worst case, sice they ow have subliear expected waste, a surprise whe it was first observed empirically i [2]. The offlie versios cotiue to have a advatage over their olie couterparts, but it is of reduced practical sigificace. Ad ow there is a distictio i the behavior of F F ad BF, with BF beig the better of the two. The above remarks apply equally well to the discrete ad cotiuous cases. As to the compariso betwee these cases, we oce agai have a sigificat differece for olie algorithms. For ay fixed value of k, the olie algorithms i the table all have Θ( 1/2 ) expected waste, i cotrast to the expected wastes i the cotiuous case of Θ( 2/3 ) for F F, Θ( 1/2 log 3/4 ) for BF ad Θ( 1/2 log 1/2 ) for the best possible olie algorithm. (Here the otatio Θ(f()) meas that the lower boud is take i the Hardy ad Littlewood sese of ot o(f()), i.e., greater tha cf() for some

11 PERFECT PACKING THEOREMS 105 Table 2 Possibilities for expected waste i the osymmetric case. U(0, u], u < 1 Ref U{j, k}, 1 j k 2 Ref OP T Θ u(1) [3] Θ k (1) [ ] F F D, BF D Θ u(1), u 1/2 [3, 16] Θ k (1), Θ k ( 1/2 ), Θ k () [5] Θ u( 1/3 ), u > 1/2 [3, 16] F F, BF Θ u() [4] Θ k (1), Θ k (),...? [1, 7, 8, 19] Best olie Ω u ( 1/2 ), O u( 1/2 log 3/4 ) [ ] [25] Θ k (1) [ ] + [12] Results proved i this paper. Θ k ( 1/2 ) is ot ruled out by theorems, but o occurreces are kow either. For BFD ad FFD, it does ot occur for ay k 10,000 [5]. This is cojectured to hold for all u (0, 1), based o experimetal studies. To date it has bee proved oly for u [0.66, 2/3) ad BF [18]. c > 0 ad ifiitely may, rather tha i the stadard Kuthia sese of greater tha cf() for some c > 0 ad all sufficietly large. ) I a sese, however, the olie results for the discrete case are cosistet with those for the cotiuous oe. Although techically a olie algorithm is ot allowed to kow the magitude of, if oe formally sets k = Θ() i the formulas for expected waste for the discrete case, oe gets EW A (U{k 1, k}) = EW A (U(0, 1]) for F F, BF, ad the best possible olie algorithm. SS is ot applicable to cotiuous distributios, but ote that i this asymptotic discrete sese it appears to be worse tha F F ad BF. Ideed, experimets suggest that EW SS (U{ 1, }) = Θ(). Let us ow tur to the osymmetric distributios U(0, u], u < 1, ad U{j, k}, j < k 1. The kow results for these distributios are summarized i Table 2. Here for the first time we see differeces betwee the cotiuous ad discrete cases for offlie algorithms. I particular, for A {F F D, BF D}, EW A (U(0, u]) = O(1) for all u 1/2 [3, 16], but for may of the distributios U{j, k} with j k/2 (the correspodig discrete uiform distributios), we have EW A (U{j, k}) = Θ() [4, 5]. Moreover, for u (1/2, 1), EW A (U(0, u]) = Θ( 1/3 ), ad this growth rate ever occurs for U{j, k}. This follows from a theorem i [5] that says that for all discrete (F ) must be either O(1), Θ( 1/2 ), or Θ(). The theorem also provides algorithms that determie the aswers for a give distributio (b, S, p), fid the costats of proportioality whe the expected waste is liear, ad ru i time polyomial i b ad S. Ufortuately, although the aswers for the distributios U{j, k} with k 10,000 have all bee computed [5], these do ot suggest ay simple rule as to how the choice amog O(1), Θ( 1/2 ), ad Θ() might deped o j ad k. (As a example of the type of behavior that ca occur, for the U{j, 151} distributios the choice betwee liear ad bouded expected waste switches back ad forth 10 times as j icreases from 1 to 149.) The results for k 10,000 do, however, exhibit several suggestive patters. First (ad this ca be proved to hold for arbitrarily large k), the expected waste is O(1) wheever j < k or j > k k. Secod, expected waste Θ( 1/2 ) does ot occur for ay U{j, k} with j < k 1 < 10,000, suggestig that it may ever occur. Third, for each U{j, k} with distributios F, EW F F D k 10,000, EW F F D (F ) ad EW BF D (U{j, k}) are either both liear or both bouded. If liear, the costat of proportioality for BF D is ever larger tha that for F F D (but is occasioally smaller). Here agai there is a sese i which the discrete case is asymptotically cosistet with the cotiuous case, eve though the expected waste for F F D ad BF D is always subliear i the latter. As k icreases, the maximum costats of proportioality for the liear expected waste uder U{j, k} appear to decrease. Ideed, it ca be (U{j, k}) ad EW BF D

12 106 COFFMAN ET AL. show that the costats for F F D are bouded by a fuctio that declies at least as fast as (log k)/k [5]. (This presumably holds for BF D as well.) The worst case is the distributio U{6, 13}, for which the expected waste for both F F D ad BF D is /624, which is less tha 0.6% of the expected optimal umber of bis. Moreover, this is easily avoided, sice ot oly does SS have bouded expected waste for this distributio, but so do F F ad BF (although this is the oly case we have idetified where the olie F F ad BF algorithms outperform their offlie cousis). Turig ow to the olie algorithms F F ad BF, we observe that their behavior uder discrete uiform distributios appears empirically to be similar to their behavior uder cotiuous oes. Based o extesive experimets, it is cojectured that F F ad BF both have liear expected waste uder U(0, u] for 0 < u < 1, although to date this has bee proved oly for u [0.66, 2/3) ad BF [18]. I the discrete case (U{j, k} with j < k 1) experimets suggest that for sufficietly large k, the expected waste for F F ad BF is bouded whe j = O( k) ad whe j = k 2, but otherwise liear. Some of this has bee proved. I [19] it was show that EW BF (U{k 2, k}) = O(1) for all k > 0, ad this result was exteded to F F i [1]. I [4] it was show that EW F F (U{j, k}) = O(1) if k j(j + 3)/2 ad EW BF (U{j, k}) = O(1) if k j 2. I practice, bouded expected waste is more commo, at least for small k. The growth rates for BF uder U{j, k} with k 11 were completely characterized usig multidimesioal Markov chai argumets i [8], ad liear expected waste oly occurs for U{8, 11}. The oly geeral result provig liear expected waste mirrors the result for the cotiuous case: EW BF (U{j, k}) = Θ() if j/k [.66, 2/3) ad k is sufficietly large [18]. At this poit we do ot kow if expected wastes other tha O(1) ad Θ() are possible for F F or BF uder ay distributios U{j, k} with j < k 1. No geeral classificatio theorem such as those for F F D, BF D, or OP T has bee prove, so the rage of possibilities is ot kow to be limited to O(1), Θ( 1/2 ), ad Θ(), as it was for F F D ad BF D. There is also a gap betwee the lower boud proved i this paper o the best possible olie expected waste for cotiuous distributios U(0, u] ad the best rate kow to be achievable. The former is Ω( 1/2 ) ad the latter is O( 1/2 log 3/4 ), as proved i [25]. The algorithm of [25] works for ay distributio, discrete or cotiuous, but has drawbacks from a pragmatic poit of view: the best curret boud we have o its ruig time is O( 8 log 3 ) [12]. If oe is willig to cosider more specialized algorithms, better ruig times are possible, at least theoretically. For ay fixed distributio F, there is a algorithm A F that rus i time O( log ) ad agai has expected waste of O( 1/2 log 3/4 ) [24]. These algorithms have drawbacks too, however, sice the proof that they exist is ocostructive. The questio of whether practical algorithms exist that attai these bouds, or ideed whether the Ω( 1/2 ) lower boud is achievable, remais ope. Fially, i additio to the ope problems metioed above for the discrete ad cotiuous uiform distributios U(0, u] ad U{j, k}, there is the questio of what happes for arbitrary discrete ad cotiuous distributios. I the discrete case, the above-metioed classificatio theorems apply for BF D, F F D, ad OP T, ad say that the correspodig expected waste must be O(1), Θ( 1/2 ), or Θ(). As also metioed above, the applicable cases for BF D ad F F D ad ay specific distributio F = (b, S, p) ca be determied i time polyomial i b ad S. For OP T there is also a algorithm for determiig which case applies, as oted i [12]. This ivolves solvig up to S + 1 liear programs with S b variables ad S + b costraits. Noe of these algorithms techically rus i polyomial time sice b may be expoetially larger tha its cotributio to istace size (log b). However, all are feasible for b i

13 PERFECT PACKING THEOREMS 107 excess of 1,000, which makes it possible to characterize the behavior of F F D, BF D, ad OP T for may iterestig distributios o a case-by-case basis. The theorem about SS preseted i sectio 1 ca be geeralized to arbitrary discrete distributios if oe replaces SS by a simple variat SS : As i SS, items are packed so as to miimize b 1 h=1 N h 2, but ow the choice must be made subject to the followig additioal costrait: No item may be placed i a partially filled bi if the resultig gap caot be exactly filled with items whose sizes have already bee ecoutered i the list L. The resultig algorithm still rus i time O(b) ad satisfies EW SS (F ) = Θ ( EW OP T (F ) ) for all discrete distributios F [12]. I additio, there is a more complicated radomized variat that rus i time O(b log b), satisfies the above property, ad also has the same costat of proportioality as OP T whe the expected waste is liear [12]. As to the case of arbitrary cotiuous distributios, we as yet have o geeral classificatio theorems, although some partial results have bee proved. Rhee [22] provided a complicated measure-theoretic characterizatio of those F for which (F ) is subliear, but this does ot appear to be computatioally useful. A result of Rhee ad Talagrad [23] implies that if EW OP T (F ) is subliear, it must be EW OP T O( 1/2 ) or better. Rates strictly betwee O(1) ad Θ( 1/2 ) have ot yet bee ruled out, however. Moreover, there is as yet o algorithm with the geeral effectiveess of SS ad its variats. The results of [24, 25] imply that there are olie algorithms whose expected waste is at most O( 1/2 log 3/4 ) worse tha the optimal expected waste. For offlie algorithms, Karmarkar ad Karp have devised a algorithm which i the worst case ever uses more tha the optimal umber of bis plus O ( log 2 (OP T ) ) = O(log 2 ) [17]. This meas that its expected waste is ever more tha the maximum of (F ). Like the algorithms of [24, 25], however, it is impractical, havig a ruig time for which our best curret boud is O( 8 log 2 ). To coclude with a ope problem that hearkes back to the mai result of this paper, ote that our ability to determie the expected waste for F F D, BF D, ad OP T o a case-by-case basis ca oly take us so far, ad more geeral results would be desirable. Results for U(0, 1] ad U{k 1, k} typically cotiue to hold for arbitrary cotiuous ad discrete symmetric distributios, respectively, but the real world is O(log 2 ) ad EW OP T ot domiated by symmetric distributios. It would be ice if we could idetify additioal iterestig classes of osymmetric distributios F for which geeral results about EW OP T (F ) ca be proved, as we did i this paper for the discrete uiform distributios. Are there iterestig classes for which ew perfect packig theorems ca provide us with similar geeral aswers? REFERENCES [1] S. Albers ad M. Mitzemacher, Average-case aalysis of first fit ad radom fit bi packig, i Proceedigs of 9th ACM-SIAM Symposium o Discrete Algorithms, SIAM, Philadelphia, 1998, pp [2] J. L. Betley, D. S. Johso, F. T. Leighto, ad C. C. McGeoch, A experimetal study of bi packig, i Proceedigs of the 21st Aual Allerto Coferece o Commuicatio, Cotrol, ad Computig, Uiversity of Illiois, Urbaa, 1983, pp [3] J. L. Betley, D. S. Johso, F. T. Leighto, C. C. McGeoch, ad L. A. McGeoch, Some uexpected expected behavior results for bi packig, i Proceedigs of the 16th Aual ACM Symposium o Theory of Computig, ACM Press, New York, 1984, pp [4] E. G. Coffma, Jr., C. Courcoubetis, M. R. Garey, D. S. Johso, L. A. McGeoch, P. W. Shor, R. R. Weber, ad M. Yaakakis, Fudametal discrepacies betwee average-case aalyses uder discrete ad cotiuous distributios, i Proceedigs of the 23rd Aual ACM Symposium o Theory of Computig, ACM Press, New York, 1991, pp

14 108 COFFMAN ET AL. [5] E. G. Coffma, Jr., D. S. Johso, L. A. McGeoch, P. W. Shor, ad R. R. Weber, Bi Packig with Discrete Item Sizes, Part III: Average Case Behavior of FFD ad BFD, mauscript. [6] E. G. Coffma, Jr., D. S. Johso, P. W. Shor, ad R. R. Weber, Bi Packig with Discrete Item Sizes, Part V: Tight Bouds o Best Fit, mauscript. [7] E. G. Coffma, Jr., D. S. Johso, P. W. Shor, ad R. R. Weber, Bi Packig with Discrete Item Sizes, Part IV: Markov Chais, Computer Proofs, ad Average-Case Aalysis of Best Fit Bi Packig, mauscript. [8] E. G. Coffma, Jr., D. S. Johso, P. W. Shor, ad R. R. Weber, Markov chais, computer proofs, ad average-case aalysis of Best Fit bi packig, i Proceedigs of the 25th Aual ACM Symposium o Theory of Computig, ACM Press, New York, 1993, pp [9] E. G. Coffma, Jr., D. S. Johso, P. W. Shor, ad R. R. Weber, Bi packig with discrete item sizes, Part II: Tight bouds o First Fit, Radom Structures Algorithms, 10 (1997), pp [10] E. G. Coffma, Jr. ad G. S. Lueker, A Itroductio to the Probabilistic Aalysis of Packig ad Partitioig Algorithms, Wiley, New York, [11] C. Courcoubetis ad R. R. Weber, Stability of o-lie bi packig with radom arrivals ad log-ru average costraits, Probab. Egrg. Iform. Sci., 4 (1990), pp [12] J. Csirik, D. S. Johso, C. Keyo, J. B. Orli, P. Shor, ad R. R. Weber, O the sum-of-squares algorithm for bi packig, i Proceedigs of the 32d ACM Symposium o Theory of Computig, ACM Press, New York, 2000, pp [13] M. R. Garey ad D. S. Johso, Computers ad Itractability: A Guide to the Theory of NP-completeess, W. H. Freema, New York, [14] D. S. Johso, Near-Optimal Bi Packig Algorithms, Ph.D. thesis, Departmet of Mathematics, Massachusetts Istitute of Techology, Cambridge, MA, [15] D. S. Johso, A. Demers, J. D. Ullma, M. R. Garey, ad R. L. Graham, Worst-case performace bouds for simple oe-dimesioal packig algorithms, SIAM J. Comput., 3 (1974), pp [16] D. S. Johso, F. T. Leighto, R. R. Weber, ad P. W. Shor, Average Case Behavior of First Fit Decreasig ad Optimal Packigs for Cotiuous Uiform Distributios U(0, u], mauscript. [17] N. Karmarkar ad R. M. Karp, A efficiet approximatio scheme for the oe-dimesioal bi packig problem, i Proceedigs of the 23rd Aual Symposium o Foudatios of Computer Sciece, IEEE Computer Society Press, Los Alamitos, CA, 1982, pp [18] C. Keyo ad M. Mitzemacher, Liear waste of best fit bi packig o skewed distributios, i Proceedigs of the 41st Aual Symposium o Foudatios of Computer Sciece, IEEE Computer Society Press, Los Alamitos, CA, 2000, pp [19] C. Keyo, Y. Rabai, ad A. Siclair, Biased radom walks, Lyapuov fuctios, ad stochastic aalysis of best fit bi packig, J. Algorithms, 27 (1998), pp [20] W. Ködel, A bi packig algorithm with complexity o(log) i the stochastic limit, i Proceedigs of the 10th Symposium o Mathematical Foudatios of Computer Sciece, J. Gruska ad M. Chytil, eds., Lecture Notes i Comput. Sci. 118, Spriger-Verlag, Berli, 1981, pp [21] G. S. Lueker, A Average-Case Aalysis of Bi Packig with Uiformly Distributed Item Sizes, Tech. Report 181, Departmet of Iformatio ad Computer Sciece, Uiversity of Califoria, Irvie, CA, [22] W. T. Rhee, Optimal bi packig with items of radom sizes, Math. Oper. Res., 13 (1988), pp [23] W. T. Rhee ad M. Talagrad, Optimal bi packig with items of radom sizes. III, SIAM J. Comput., 18 (1989), pp [24] W. T. Rhee ad M. Talagrad, O-lie bi packig with items of radom size, Math. Oper. Res., 18 (1993), pp [25] W. T. Rhee ad M. Talagrad, O-lie bi packig of items of radom sizes. II, SIAM J. Comput., 22 (1993), pp [26] P. W. Shor, The average case aalysis of some o-lie algorithms for bi packig, Combiatorica, 6 (1986), pp [27] P. W. Shor, How to pack better tha Best Fit: Tight bouds for average-case o-lie bi packig, i Proceedigs of the 32d Aual Symposium o Foudatios of Computer Sciece, IEEE Computer Society Press, Los Alamitos, CA, 1991, pp