E. G. Coffman, Jr. and G. S. Lueker. Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley-Interscience Publication, New York, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ACM 1 58113 349 9 01 0007 . 5.00. moves one position left or right, depending on the flip of a fair coin. Such random walks may be generalized to more complicated lattices and to finite or infinite graphs, and have had several interesting applications in computer science (see, for instance, [3, 8, 20, 22], as well as the discussion below) We refer the reader to Kemeny and Snell [21] for basic facts regarding random walks. In this paper we consider quantum variations of random walks on one dimensional lattices we refer to such processes as quantum walks. In direct analogy to classical random ....

E. Coffman and G. Lueker. Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley, 1991.

....of the optimal packing that ensures that for any item size distribution the tails of the distribution of # # # decline rapidly enough with # [24] so that as # ## , # # # # # # # # # # # # # # # # # # # # # # # # # # # # 0 7803 7476 2 02 17.00 (c) 2002 IEEE. converge to the same limit [6]. Therefore the asymptotic expected performance ratio is given by # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # (9) To find the asymptotic expected ....

E. G. Coffman, Jr. and G. S. Lueker. Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley-Interscience Publication, New York, 1991.

....processor utilization and stability condition in the worst case as compared with FCFS. These results also show that FPMPFS improves performance more than FPFS. 16 Average Case Analysis. The performance of NF in the average case where s(p i ) is uniformly distributed on (0; 1] is given as follows [24]. Here, h ac (A) denotes the number of bins to pack n items by algorithm A in the average case and h(opt) denotes the number of bins to pack n items in the optimal case. h ac (NF ) h(opt) 4 3 ; n 1 (14) Next, the performance of FF in the average case is given by h ac (FF ) h(opt) ....

....denotes the number of bins to pack n items by algorithm A in the average case and h(opt) denotes the number of bins to pack n items in the optimal case. h ac (NF ) h(opt) 4 3 ; n 1 (14) Next, the performance of FF in the average case is given by h ac (FF ) h(opt) n 2 3 ) 15) [24, 25]. Since s(p i ) is uniformly distributed on (0,1] h(opt) n=2. Then, h ac (FF ) h(opt) 1 (n 2 3 ) n=2 = 1 (n 0 1 3 ) 16) is derived from (15) and then, h ac (FF ) h(opt) 1; n 1: 17) In the same way, the performance of FFD in the average case is given by h ac (FFD) ....

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E. G. Coffman and G. S. Lueker. Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley, 1991.

....to zero, then the resulting mapping is optimum with respect to transition count, but could be area expensive, as memory space would be wasted. The algorithm we use for mapping the logical memories into physical memories is a variant of the best fit decreasing heuristic for the bin packing problem [20]. The general strategy is to consider the logical memories one at a time, largest first, and assign it to a physical memory module based on its own size and the sizes of the available memory modules. We start with the largest logical memory ( since this is a candidate that possibly accounts for ....

E. G. Coffman, Jr. and G. S. Lueker, Probabilistic Analysis of Packing and Partitioning Algorithms. New York: Wiley, 1991.

....Our choice of methods covered here is aimed at closing a gap between analytical and probabilistic methods. There are excellent books on analytical methods (cf. Knuth s three volumes [64, 65, 66] Sedgewick and Flajolet [84] and probabilistic methods (cf. Alon and Spencer [5] Coffman and Lueker [17], and Motwani and Raghavan [75] however, remarkably very few books have been dedicated to both analytical and probabilistic analysis of algorithms (with possible exceptions of Hofri [46] and Mahmoud [73] Finally, before we launch our journey through probabilistic and analytical methods, we ....

....Q n) 1 (pr. In the example below, we will actually prove that H n = 2 log Q n) 1 (pr. by establishing a lower bound. 2 Let us look now at the second moment method. Setting in the Chebyshev inequality = E[X] we easily prove that PrfX = 0g Var[X] E[X] 2 : But, one can do better (cf. [5, 17]) Using Schwarz s inequality for a random variable X we obtain the following chain of inequalities E[X] 2 = E[I(X 6= 0)X] 2 E[I(X 6= 0) E[X 2 ] PrfI(X 6= 0)gE[X 2 ] which finally implies the second moment inequality PrfX 0g E[X] 2 E[X 2 ] 8) 17 Actually, another ....

E. Coffman and G. Lueker, Probabilistic Analysis of Packing and Partitioning Algorithms, John Wiley & Sons, New York 1991.

....Let X 1 ; X 2 ; Xm be independent and identically distributed random variables, with P r[X i = 1] p and P r[X i = 0] 1 Gamma p. Let E[Sm ] E[ P X i ] mp 0. Then, for any ffi 0, P r[S m (1 ffi)mp] e ffi (1 ffi) 1 ffi mp i e 1 ffi j (1 ffi)mp . Lemma A. 2 [2] Let X i ; i = 1; 2; n, be independently and identical distributed random variables where P r(X i = 1) P r(X i = Gamma1) 1 2 for all i. Let S i = X 1 X 2 : X i , and S i = S i Gamma E[S i ] Then for y 0 P r max 1in S i y p n 2e Gamma y 2 8 ; and P r ....

....random variables where P r(X i = 1) P r(X i = Gamma1) 1 2 for all i. Let S i = X 1 X 2 : X i , and S i = S i Gamma E[S i ] Then for y 0 P r max 1in S i y p n 2e Gamma y 2 8 ; and P r max 1in S i Gammay p n 2e Gamma y 2 8 : Remark A. 3 [2] We will describe an one dimensional bin packing algorithm Match. The algorithm Match iterates the following until all items are packed. Let S be a list of unpacked items. Choose the largest item in S, say x i . Search for the largest item in S, say x j , that can packed with x i in a bin (that ....

E. G. Coffman Jr. and G. S. Lueker. Probabilistic analysis of packing and partitioning algorithms. Wiley-Interscience, 1991.

....above in Section 1.1. In two dimensional strip packing, a semiinfinite vertical strip of unit width is given, along with a set of rectangular items of width 1. The goal is to pack items on the strip while minimizing the height of the strip. The classical two dimensional bin packing problem [JL91] is a variant on strip packing in which horizontal boundaries are added to the strip at integer heights. Few published analyses exist for the case where the bins are not identical. Israni and Sanders [IS82] present heuristics for packing non identical rectangles. One uses decreasing length, ....

E.G. Coffman Jr. and G. S. Lueker. Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley and Sons, Inc., New York, 1991.

....produced by the algorithm. The wasted space is the total amount of empty space in the bins (formal definitions will be given in the next section) Much work has been done in the worst case analysis of on line algorithms [2] and average case analysis of one dimensional on line bin packing algorithms[4, 6]. It has been shown that for one dimensional openend on line bin packing, the expected wasted space is bounded below by Omega Gamma p n log n) for all algorithms and the Best Fit produces an expected wasted space of order ( p n(log n) 3 4 ) 6] 1 For the off line version of the problem, ....

E. G. Coffman Jr. and G. S. Lueker. Probabilistic analysis of packing and partitioning algorithms. A Wiley-Interscience Publication, 1991.

....to the class of NP hard problems. Therefore, work situated in this field is reported in general for heuristic approaches or and for special scenarios. A review on packing problems is given in [8] For a general overview on work connected with loading and cutting problems see for instance in [6], 10] 31] 33] and [11] Most of the publications are concerned with the problem of finding an optimal loading of identical pieces for only one pallet. This kind of problem has been termed the Manufacturer s Pallet Loading Problem (Hodgson [16] The present paper, by contrast, deals with the ....

E. G. Coffman Jr. and G. S. Luecker. Probabilistic analysis of packing and partitioning algorithms. John Wiley & Sons, New York et al, 1991.

....belongs to the class of NP hard problems. Therefore, work situated in this field is reported in general for heuristic approaches or and for special sceneries. A review on packing problems is given in [6] For a general overview on work connected with packing and cutting problems we refer also to [4], 8] 28] 29] Most of the publications are concerned with the problem of finding an optimal packing of identical pieces for one pallet. This kind of problem has been termed the Manufacturer s Pallet Packing Problem (Hodgson [12] The present paper, by contrast, deals with the Distributer s ....

Coffman, E.G., Luecker, G.S., Probabilistic analysis of packing and partitioning algorithms, New York et al. 1991.

....n Kae(l n (x n ) l n (y n ) Kae( as required. Note that we have established more, namely that f is ae Lipschitz. This completes the proof of the theorem. 3 Application to the stochastic bin packing problem The standard reference on bin packing is the book of Coffman and Lueker [3]. For x 2 [0; 1] n , let nf n (x 1 ; x n ) be the smallest number of unit sized bins required to pack n objects of respective sizes x 1 ; x n . It is well known, and easy to prove, that if X 1 ; X 2 ; is a sequence of independent random variables taking values in [0; 1] ....

E.G. Coffman Jr. and George S. Lueker. Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley, 1991.

....m numbers, and the sums of numbers in each bin being as close as possible. Our grouping problem is known in computer science as the parallel machine scheduling problem which requires the assignment of k tasks to n identical processors so as to minimize the largest task finishing time (makespan) [10]. The constraint that each bin contain the same number of numbers is scheduling with cardinality balance, that is, the number of tasks assigned to each processor is either bk=nc or dk=ne. Several probabilistic analyses of heuristic algorithms have been done for scheduling problems with cardinality ....

E.G. Coffman and G.S. Lueker. Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley, New York, 1991.

.... notation, E[D n ] Theta( p n log n) For up right matching, Shor [Sho] Leighton and Shor [LS] and Rhee and Talagrand [RT] proved that ( E[U n ] Theta( p n log 3=4 n) Subsequent, simpler proofs were given by Coffman and Shor [CS] and Talagrand [Tal] see Chapter 3 in [CLu] for a general treatment. Asymptotic bounds on tail probabilities are also available; indeed, it is often the case that such bounds provide the desired estimates of expected values. The result in ( has also been generalized to the ordered matching problem in d 3 dimensions. If (P Gamma ; P ....

Coffman, E. G., Jr. and Lueker, G. S., Probabilistic Analysis of Packing and Partitioning Algorithms, Wiley-Interscience, New York, 1991.

....The problems studied here all involve the partitioning of a set of positive numbers into a collection of subsets satisfying a sum constraint. The following two problems are among the most fundamental. They have wide ranging applications throughout computer science and operations research [C3] [C4], D1] Bin Packing (BP) Given c 0 and a set S = fX 1 ; X n g with 0 X i c, 1 i n, partition S into a minimum number of subsets such that the sum of the X i s in each subset is no more than c. The X i s are usually called items or pieces and are thought of as being packed into ....

....that tends to 1 as n 1, A LFD (L n ; 2) O(n Gammac log n ) K2] 2 Analytical Techniques We describe and illustrate below a number of the more important techniques that have been successfully applied to the analysis of BP and MS problems. A more extensive discussion appears in [C4]. 2.1 Markov Chains For the simpler BP and MS heuristics, it is sometimes possible to formulate a tractable Markov chain that represents the element by element development of partitions. A state of the Markov chain must represent block sums in a suitable way; given the state space, the transition ....

E. G. Coffman, Jr. and G. S. Lueker. Probabilistic Analysis of Packing and Partitioning Algorithms, Wiley & Sons, New York, 1991.

....known is R 1 A 1:907 (Blitz, van Vliet and Woeginger [6] 4 Final Remarks While worst case analysis of bin packing remains quite active, it must be noted that, in the last 15 years or so, average case analysis has received at least as much attention. The monograph by Coffman and Lueker [17] on the subject describes basic results and analytic techniques. A more recent survey of average case results for one dimensional bin packing can be found in [13] To illustrate typical average case estimates, suppose the n items of L are independent, uniform random draws from [0; 1] and consider ....

E. G. Coffman, Jr. and G. S. Lueker. Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley, New York, 1991.

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E. G. Coffman, Jr. and G. S. Lueker. Probabilistic Analysis of Packing and Partitioning Algorithms. Wiley-Interscience Publication, New York, 1991.

No context found.

E. G. COFFMAN AND G. S. LUEKER, Probabilistic analysis of Packing and Partitioning Algorithms, John Wiley, New York, 1991.

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