Online Algorithms for -Space Bounded Multi Dimensional Bin Pacing and Hypercube Pacing Yong Zhang Francis Y.L. Chin Hing-Fung Ting Xin Han Abstract In this paper, we study -space bounded multi-dimensional bin pacing and hypercube pacing. A sequence of items arrive over time, each item is a d-dimensional hyperbox (in bin pacing) or hypercube (in hypercube pacing), and the length of each side is no more than. These items must be paced without overlapping into d-dimensional hypercubes with unit length on each side. In d-dimensional space, any two dimensions i and j define a space P ij. When an item arrives, we must pac it into an active bin immediately without any nowledge of the future items, and 90 -rotation on any plane P ij is allowed. The objective is to minimize the total number of bins used for pacing all these items in the sequence. In the -space bounded variant, there is only one active bin for pacing the current item. If the active bin does not have enough space to pac the item, it must be closed and a new active bin is opened. For d-dimensional bin pacing, an online algorithm with competitive ratio d is given. Moreover, we consider d-dimensional hypercube pacing, and give a d+ -competitive algorithm. These two results are the first study on -space bounded multi dimensional bin pacing and hypercube pacing. Introduction Bin pacing is a very fundamental problem in computer science, and has been well studied for more than thirty years. Given a sequence of items, we pac them into unit-size bins without overlapping. The objective is to minimize the number of bins for all items in the sequence. Department of Computer Science, The University of Hong Kong, Hong Kong. yzhang@cs.hu.h, Research supported by NSFC (7086) Department of Computer Science, The University of Hong Kong, Hong Kong. chin@cs.hu.h, Research supported by HK RGC grant HKU-77/09E Department of Computer Science, The University of Hong Kong, Hong Kong. hfting@cs.hu.h, Research supported by HK RGC grant HKU-77/08E School of Software, Dalian University of Technology, China. hanxin.mail@gmail.com, Research supported by NSFC(0065)
We focus on the online version of bin pacing, where the items arrive over time, when pacing the current item, we have no information of the future items. The positions of the paced items in the bin are fixed and cannot be repaced. To measure the performance of online bin pacing, we study a general used method called asymptotic competitive ratio. Consider an online algorithm A and an optimal offline algorithm OP T. For any sequence S of items, let A(S) be the cost (number of bins used) incurred by algorithm A and OP T (S) be the corresponding optimal cost incurred by algorithm OP T. The asymptotic competitive ratio for algorithm A is: R A = lim sup S { A(S) OP T (S) = }. OP T (S) In the online bin pacing, there are two models: bounded space model and unbounded space model. If we do not impose a limit on the number of bins available for pacing the items (called active bins), we call it unbounded space. Otherwise, if the number of active bins is bounded by a constant, and each item can only be paced into one of the active bins, we call it bounded space, which is more realistic in many applications. If none of the active bins has enough space to pac the arrival item, one of the active bins must be closed and a new active bin will be opened to pac that item. In this paper, we consider -space bounded multi-dimensional bin pacing and hypercube pacing. In the -space bounded variant, the number of active bins is only one. If an item cannot be paced into the active bin, we have to close it and open a new bin to pac this item. In the -space bounded d-dimensional bin pacing problem (d ), each item is a d-dimensional hyperbox such that the length on each side is no more than, while in the d-dimensional hypercube pacing, each item is a d-dimensional hypercube with side length no more than. The items must be paced into d-dimensional hypercubes with side length. Any two dimensions i and j define a plane P ij. 90 -rotation of the item in any plane P ij is allowed in -space bounded bin pacing, otherwise, the competitive ratio is unbounded []. To understand this problem clearly, we give an example for the -space bounded - dimensional bin pacing. In Figure (a), there are four items to be paced into unit square bins, and the arrival order is A, B, C and D. After the pacing position of A is fixed, we have two choices to pac B: rotation and without rotation. If we pac B without rotation in the same bin with A as shown in Figure (b), when item C arrives, we have to open a new bin since the current active bin does not have enough space for pacing C. In the optimal solution, these four items can be paced into one bin (Figure (c)), since item B, C and D can be rotated and the free space in the bin can accommodate all of them in their order of arrival. Related wors: Both the offline and online version of the bin pacing problem have been well studied. The offline bin pacing is NP-Hard [5]. For one-dimensional bin pacing, Simchi- Levi gave a.5-approximation algorithm [9]. Johnson and Gary [0] gave an asymptotic
A B C D (a) four items arrive in order A, B, C, and D D C A B C D A B (b) non-optimal pacing into two bins (c) optimal pacing into one bin Figure : Example of optimal pacing and non-optimal pacing 7/60-approximation algorithm. An AFPTAS was given by Karmarar and Karp [6]. For two-dimensional bin pacing. Chung et. al. [5] presented an approximation algorithm with an asymptotic performance ratio of.5. Caprara [] improved the upper bound to.6903. Bansal et al. [] devised a randomized algorithm with an asymptotic performance ratio of at most.55. As for the offline lower bound of the approximation ratio, Bansal et al. [] showed that the two-dimensional bin pacing problem does not admit any asymptotic polynomial time approximation scheme. The online bin pacing has been studied for more than thirty years. For one-dimensional online bin pacing, Johnson et al. [] showed that the First Fit algorithm (FF) has an asymptotic competitive ratio of.7. Yao [3] improved the algorithm to obtain a better upper bound of 5/3. Lee et al. [] introduced the class of Harmonic algorithms, and showed that an asymptotic competitive ratio of.63597 is achievable. Ramanan et al. [7] further improved the upper bound to.67. The best nown upper bound is.58889, which was given by Seiden [8]. As for the lower bound of the competitive ratio of one dimensional bin pacing, Yao [3] showed that no online algorithm can have an asymptotic competitive ratio less than.5. The best nown lower bound is.50 [3]. For two-dimensional online bin pacing, Coppersmith and Raghan [6] gave the first online algorithm with asymptotic competitive ratio 3.5. Csiri et al. [7] improved the upper bound to 3.065. Based on the techniques of the Improved Harmonic, Han et.al [6] improved the upper bound to.783. Seiden and van Stee [30] showed an upper bound of.6603 by implementing the Super Harmonic Algorithm. The best nown upper bound of the competitive ratio for two dimensional bin pacing is.555, which was given by Han et. al [7]. The best nown lower bound is.907 [3]. For bounded space online bin pacing, Harmonic algorithm by Lee et al. [] can be applied for one dimensional case, the competitive ratio is.6903 when the number of active bins goes to infinity. Csiri and Johnson [8] presented an.7-competitive algorithm (K-Bounded Best Fit algorithms (BBF K )) for one dimensional bin pacing using K active bins, where K. For multi-dimensional case, Epstein et al. [3] gave a.6903 d - competitive algorithm using (M ) d active bins, where M 0 is an integer such that 3
M /( ( ε) /(d+) ), ε > 0 and d is the dimension of the bin pacing problem. For the -space bounded variant, Fujita [] first gave an O((log log m) )-competitive algorithm, where m is the width of the square bin and the size of each item is a b (a, b are integers and a, b m). Chin et al. proposed an 8.8-competitive pacing strategy [9], then they further improved the upper bound to 5.55 [33], they also gave the lower bound 3 for -space bounded two dimensional bin pacing. For a special case where the items are squares (or hypercubes), there are also many results [0, 8, 9, 3 5]. For bounded space online square pacing, Epstein and van Stee [] gave a.369-competitive algorithm, they also proved that the lower bound of the competitive ratio is at least.3633. For bounded space d-dimensional hypercube pacing, an O(d/ log d)-competitive algorithm was given [], however, to achieve this bound, the number of active bins is very large. Moreover, they proved that the asymptotic competitive ratio of bounded space hypercube pacing is lower bounded by Ω(log d). Januszewsi and Lassa [9] proved that any sequence of square items with a total area of at most 5/6 can be paced into a unit bin. Han et al. [8] studied a variant in which any paced item can be removed so as to guarantee a good competitive ratio and presented a pacing algorithm that is 3-competitive. Note that in the above two studies, there is only one bin to pac the square items. The remaining part of this paper is organized as follows. In Section, we show the d -competitive algorithm for -space bounded d-dimensional bin pacing. In Section 3, a d+ -competitive algorithm is given for -space bounded d-dimensional hypercube pacing. -Space Bounded d-dimensional Bin Pacing Let d be the highest dimension of the item and hypercube, each item a is associated with a vector (a, a,..., a d ), where a i ( i d) is the length in the i-th dimension of item a. In d-dimensional space, any two dimensions i and j define a space P ij. For -space bounded multi-dimensional bin pacing problem, rotation 90 in any plane P ij is allowed. Otherwise, the performance ratio is unbounded. Consider an example of a sequence with n items: {A, B, A, B,...}, where A = (/n,,,..., ) and B = (, /n,,,..., ). If rotation is not allowed, any two adjacent items cannot be paced into the same bin by any online algorithm, thus, the number of used bins is n. In the optimal pacing, all A items can be paced into one bin, all B items can be paced into another bin, only two bins is enough to pac all these items. In this way, the performance ratio is n. If rotation is allowed, the first half part of items in the sequence can be paced into one bin by rotate B items 90 in the plane P. Similarly, the second half of items can be paced into another bin. Since rotation 90 in any plane P ij is allowed, we may assume that the lengths in dimensions of any item a is non-increasing, i.e., a i a j (i < j) for each item. Denote the size of an item a = (a, a,..., a d ) to be d i= a i. We say a ( + )-
dimensional hyperbox b = (b, b,..., b + ) is a ( +, h)-hyperbox if b =... = b =, b = / and b + = h. Let o i and r i ( i < d) be the average occupancy ratio in the worst case and competitive ratio for pacing i-dimensional items by our algorithm, respectively. Our target is to design an algorithm with competitive ratio as smaller as possible. Since any algorithm cannot pac items with total sizes more than into one bin, we set r i = /o i. The target can be done by designing algorithm with the average occupancy ratio as larger as possible. According to the algorithm, o i (r i ) until o d (r d ) can be recursively computed. We say an item is large w.r.t. its (i +)-th dimension if i j= a j o i, and small w.r.t. its (i+)-th dimension otherwise. For a small item a w.r.t. its ( +)-th dimension, we have a + a o / = r /. From Table and Lemma in the later part of this paper, we have r / >, thus, a + a < /. A small items a w.r.t. its ( + )-th dimension can be paced into a ( +, h)-hyperbox such that h/ < a + h and h = j o / (for some j = 0,,,...).. Pacing strategy Roughly speaing, items are recursively paced by the strategy from higher dimension to lower dimension. For an incoming item a, if it is large w.r.t. the d-th dimension, we pac it in a top-down order along the d-th dimension. Otherwise, it is small w.r.t. the d-th dimension, we first pac it into a (d, h)-hyperbox, then pac this (d, h)-hyperbox into the bin in a bottom-up manner. Since the length of (d )-th dimension of the (d, h)-hyperbox is /, the lower part along dimension d is partitioned into the left side and the right side: the left side is the area such that the (d )-th dimension is in the range [0, /], while the right side is in the range [/, ]. When pacing the (d, h)-hyperbox into the bin, we try to balance the heights of the left and right sides. When pacing small item into (d, h)-hyperbox, we have no need to consider the length of the d-th dimension. In another word, pacing small item into (d, h)-hyperbox can be regarded as pacing (d )-dimensional item into a bin, therefore, the dimension of the pacing problem is decreased by one. By implementing this idea, we can recursively pac items from higher dimension into lower dimension. Note that in our algorithm, small items can be only paced into a ( +, h)-hyperbox with h < a +. Let o + be the average occupancy ratio for pacing small items into ( +, h)-hyper-box, we have the following fact. Fact. o + = o /. Proof. Since the length of the -th dimension is no more than /, any small item a can be paced into the corresponding ( +, h)-hyperbox such that, h/ < a + h. Pacing small items can be regarded as pacing general items into -dimensional bin by doubling the length in the -th dimension of the small item. Thus, the average occupancy ratio 5
is preserved in the first dimensions. In the ( + )-th dimension, the length is at least h/. Thus, o + = o /. Now we give our algorithm for pacing d-dimensional items with d 3. In case of d =, we use our previous 5.5-competitive algorithm. Note that the occupation ratios o i and competitive ratios r i are all computed by analyzing the algorithm, thus, when pacing an incoming item, these ratios are all nown in advance. Algorithm for Pacing d-dimensional item a: : if a is large w.r.t. the d-th dimension. then : Pac it by a top-down order such that a d alongs the d-th dimension. 3: if overlap happens then : Close this bin then open a new bin for pacing this item. 5: end if 6: else if a is small w.r.t. the d-th dimension. then 7: if there exists (d, h)-hyperbox with enough space for the item then 8: Pac it into the (d, h)-hyperbox. 9: else 0: Open a new (d, h)-hyperbox for this item. : Pac the (d, h)-hyperbox by a bottom-up order in the d-th dimension, : such that the heights of the left part and the right part are balanced. 3: if overlap happens then : Close this bin then open a new bin for pacing this (d, h)-hyperbox. 5: end if 6: end if 7: end if To understand the algorithm clearly, we give an example to show how to pac an incoming item. In the current pacing shown in Figure, there are some large items paced in the upper part of the -th dimension, some small items paced in the (, h)- hyperboxes which located in the lower part of the -th dimension such that the left and right part are balanced. When a large item comes, if it cannot be paced between the upper and lower part without overlapping, we will open a new bin. When a small item comes, we will first try to pac it into some existed (, h)-hyperbox, if no such (, h)-hyperbox, we open a new (, h)-hyperbox then pac it into the right part. The above algorithm recursively pacing items from higher dimension to lower dimension, until dimension 3. We implement the algorithm in [33] for pacing -dimensional items because that the performance ratio r = 5.5 is better than implementing the above algorithm in dimension. 6
-th dimension y y central axle left part right part ( )-th dimension large item (, h)-hyperbox for small items Figure : Pacing -dimensional items into -dimensional hypercube.. Analysis of the strategy When pacing items which are small w.r.t. the -dimension into (, h)-hyperbox, we say the (, h)-hyperboxes with the same height h are of the same type. From Fact, the average occupancy ratio o = o /. Thus, for each ind of (, h)-hyperbox, except the last one, the average occupancy ratio is at least o /. Fact. The total lengths in dimension of the last hyperboxes of each type is at most o /( ). Proof. From previous definition h j = j o /( ) (j = 0,,,...), the length in dimension of each type of (, h j )-hyperbox is fixed. Thus, the total length is at most j h j o /( ). Consider a pacing configuration shown in Figure, suppose the length in -dimension of the upper part is y, the lengths in -dimension of the left and right parts are y and y respectively. W.l.o.g., y y. The current occupancy in this bin is at least y o + (y + y o /( ) ) o The first term is the occupancy of large items, the second term is the occupancy of small items. Since in the lower part, the length in ( )-dimension of each hyperbox is /, we divide in the second term. By Fact, we now the occupancy in this bin is at 7
least y o + (y + y o /( ) ) o If we only count the occupancy in one bin, the performance ratio is unbounded. For example, a bin contains a very small item, the next item is very large and cannot be paced together with the small one. We have to open a new bin for the later item. In this case, the occupancy in the previous bin is very small. This example gives us an heuristic to amortize the occupancies of adjacent two bins: if we have to open a new bin due to an item, this items has contribution to two bins, one is the bin it paced and the other is the previous bin it cannot be paced. To amortize the occupancy of adjacent two bins, item from the upper part compensate half occupancy to the previous bin; item from the lower part compensate the part which is larger than the ratio o / to the previous bin. Now we study two cases of opening a new bin. a large item with length y in dimension cannot be paced into this bin. By amortized analysis, the occupancy in this bin is at least y o = (y + y ) o > ( y ) o = o o/( ) o + (y + y o /( ) ) o o/( ) = o 3 o/( ) + (y + y o /( ) + (y + y o /( ) + (y y ) o o/( ) + y o ) o ) o a small item with length y in dimension cannot be paced into this bin. if (y ) y o, this small item has no contribution to the previous bin. In 8
this case, y (o /) /( ). The amortized occupancy is at least y o y o = (y + y ) o > ( y ) o + (y + y (o ) /( ) ) o + (y (o ) /( ) ) o (o ) /( ) (o ) /( ) = o (o ) /( ) y o o (o ) /( ) (o /) /( ) o if (y ) > y o, this small item has contribution (y ) y o to the previous bin. In this case, y > (o /) /( ). The amortized occupancy is at least y o y o + (y + y (o ) /( ) ) o + (y (o ) /( ) ) o (o ) /( ) + (y ) y o + (y ) y o = (y + y ) o ( y ) o (o ) /( ) + (y ) y o = o (o ) /( ) + (y ) 3 y o o (o ) /( ) (o /) /( ) o + (y ) y o the last inequality holds since 3 and (o /) /( ) < y o /( ) < /. When = 3, the above three formula are equivalent, when > 3, the last two formulas are less. In this paper, the dimension we focused is at least 3, thus, we can say that the amortized occupancy ratio for pacing -dimensional items is at least o = o (o ) /( ) Since r = /o, we have (o /) /( ) o () 9
r = r r /( ) r ( r ) /( ) () In Table, we give the performance ratio r for some lower dimensions ( = to 6). We also compute r /, which will help us to give the upper bound for the performance ratio. 3 5 6 r 5.5 30.86 7.969 58.56 086.38 r /.69 3.3 3.36 3.9 3.57 Table : the performance ratio for = to 6 We can see that the performance ratio is increased very fast, but r / slowly. is increased Lemma. 3.5 < r / < if > 6 and 3.5 < r /( ) <. Proof. Let x = r /, from Equation (), we have x = x x x /( ) x x When x 3.5, the above formula is larger than /. Thus, we can say that x < if x 3.5. x = = < < x x x ( x 3.5 ( x 3.5 x /( ) x /( ) ) x /( ) ) The last inequality holds if x <. Thus, we can say that x > 3.5 if x <. Combine the above two statements and x 6 = 3.57, this lemma can be proved by induction. From Lemma, we conclude that x is in the range (3.5, ) when 6. Therefore, Theorem. The competitive ratio of the algorithm for -space bounded d-dimensional bin pacing is d. (3) 0
3 -Space Bounded d-dimensional Hypercube Pacing In this section, we consider a special case of the multi-dimensional bin pacing, where each item is a hypercube with side length no more than. Since the lengths of each side of a hypercube are same, this ind of items can be paced regularly inside a bin. We will first describe the pacing strategy for hypercubes, then give the performance analysis to show the competitive ratio of this strategy is d+. 3. Pacing strategy Our pacing strategy for hypercubes is based on the following two observations.. In d-dimensional hypercube pacing, an item with side length i < x i can be paced into a hypercube with side length i, and the occupation ratio in this hypercube is at least d.. d-dimensional hypercube can be regularly partitioned: a hypercube with side length i can be partitioned into d smaller hypercubes with side length i. In our pacing strategy, we will find a hypercube with proper size for each incoming item. If a hypercube inside the bin is assigned to pac an item, this hypercube cannot be used for other item. According to the first observation, the occupation ratio in this hypercube is guaranteed. To pac an item into a hypercube with proper size, we may partition the bin regularly according to the method from the second observation. By implementing the partition mentioned above, we can define the pacing configuration of the active bin as follows. Define i -hypercube to be the hypercube with side length i. Let B = (b 0, b, b,...) denote the current pacing configuration of an active bin, where b i denotes the number of empty i -hypercubes. If b j = 0 for all j i, we will ignore b j (j i) in B. Initially, the whole bin is empty, in such configuration, B = (). Suppose an item with side length /3 comes, to pac this item, the bin will be partitioned into d /- hypercubes and use one /-hypercube to pac this item. After that, the configuration will be changed to B = (0, d ). Next, we will describe the pacing strategy based on the current configuration B and the coming item with side length x. W.l.o.g., suppose i < x i. Procedure Hypercube Pacing for an item with side length i < x i If [b i > 0] Select one i -hypercube for pacing this item. Modify B by decrease b i by one. Otherwise, if [there exist b j > 0 for some j < i.]
Let j be the largest integer such that b j > 0 and j < i. Let = j. Repeat Partition one -hypercube into d -hypercubes. = +. Until = i. Using one i -hypercube to pac the coming item, Modify the configuration B by b j = b j, b l = d for j < l i. Otherwise This item cannot be paced into this active bin. Close this bin then open a new bin for pacing this item. Let = 0. Repeat Partition one -hypercube into d -hypercubes. = +. Until = i. Using one i -hypercube to pac the coming item, After pacing, b 0 = 0, b =...b i = d. end Hypercube Pacing 3. Performance analysis Lemma 3. In a configuration B, 0 b i d and all empty i -hypercubes are within a i+ -hypercube. Proof. From the pacing strategy, only when a i -hypercube will be used and b i = 0, we partition a larger hypercube to create d i -hypercubes. After the partition, one i -hypercube will be used, either for pacing an item, or for partitioning to smaller hypercubes. Thus, b i is at most d. When these d i -hypercubes are created, they are within a i+ -hypercube. Only when these d hypercubes are all used up, we will open another d i -hypercubes. Therefore, at any time, all empty i -hypercubes are within a i+ -hypercube. Lemma. In a configuration B, for any i 0, the total size of empty j -hypercubes (j > i) is no more than the size of a i -hypercube. Proof. Suppose in the configuration B, i < j < j <... < j l such that b jx > 0 ( x l). From Lemma 3, we have b jx d. In d-dimensional space, the ratio between the sizes of a j -hypercube and a j -hypercube is d. Thus, the total size of these j x -hypercubes is no more than the size of a i -hypercube. Theorem 5. The competitive ratio of the pacing strategy for hypercubes is d+.
Proof. From the pacing strategy, the occupation ratio of any used i -hypercube is at least d. Thus, For an active bin with configuration B = (b 0, b, b,...), the total occupation in this bin is at least ( b j jd ) d. j 0 Suppose at this time, an item comes and we have to open a new bin according to the pacing strategy. This happens when the item with side length i < x i and b j = 0 for all j i in the configuration B. The size of this item is x d > id d. From Lemma, the total sizes of the empty hypercubes is no more than the size of a i -hypercube. The average occupation in these two bins is at least ( i>i b j jd ) d + id d d Suppose the pacing strategy uses l bins for a sequence of items, and the occupation in the i-th bin is c i. Thus, the total occupation is l i= c i = c + c l l + i= ( c i + c i+ ) > (l ) d The above value is a lower bound of the used bins from the optimal offline algorithm. Thus, the competitive ratio of the pacing strategy is at most l (l ) d In the bin pacing problem, we are interested in the asymptotic competitive ratio, according to the above analysis, this ratio is d+. References [] N. Bansal, J.R. Correa, C. Kenyon and M. Svirideno. Bin Pacing in Multiple Dimensions: In-approximability Results and Approximation Schemes. Mathematics of Operations Research, 3(): 3-9, 006. [] N. Bansal, A. Caprara and M. Svirideno. Improved approximation algorithm for multidimensional bin pacing problems, FOCS 006: 697-708. [3] D. Blitz, A. van Vliet, and G. J. Woeginger. Lower bounds on the asymptotic worst-case ratio of on-line bin pacing algorithms. unpublished manuscript, 996. [] A. Caprara. Pacing -dimensional bins in harmony. FOCS 00: 90-99. 3
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