Average-case analysis of the Modified Harmonic algorithm

The study on one-dimensional bin packing problem may bring about many important applications such as multiprocessor scheduling, resource allocating, real-world planning and packing. Harmonic algorithms (including H K, RH, etc.) for bin packing have been famous for their linear-time and on-line properties for a long time. This paper profoundly analyzes the average-case performance of harmonic algorithms for pieces of i.i.d. sizes, provides the average-case performance ratio of H K under (0,d] (d ≤ 1) uniform distribution and the average-case performance ratio of RH under (0,1] uniform distribution. We also finished the discussion of the worst-case performance ratio of RH. Moreover, we propose a new improved version of RH that obtains better worst- and average-case performance ratios.

This paper presents a framework for an error correction decoding algorithm which can perform soft-input/soft-output decoding on arbitrary linear block codes over arbitrary finite fields. The algorithm produces estimates of individual symbols by detecting the symbol as a scalar multiple of a column of the generator matrix for the code, with other columns forming an interfering "noise" which is probabilistically modeled. Only the generator matrix is used for the decoder; the parity check matrix is not employed. The posterior probability so obtained is iteratively fed back as a prior input for turbo-like operation. Methods of incorporating statistical dependence among symbols are presented. The decoder can be used for channels with arbitrary noise distributions. Computation over larger fields is achieved through the use of an extended Hadamard transform, which is used to compute the distribution of arbitrary linear combinations of random variables distributed over the field. Results are presented for some codes of various lengths and design distances.

The use of the repeated Cauchy principal value affords greater facility in the application of inversion formulae involving characteristic functions. Formula (2) below is especially useful in obtaining the inversion formula (1) for the distribution of the ratio of linear combinations of random variables which may be correlated. Formulae (1), (10), (12) generalize the special cases considered by Cramer [2], Curtiss [4], Geary [6], and are free of some restrictions they impose. The results are further generalized in section 6, where inversion formulae are given for the joint distribution of several ratios. In section 7, the joint distribution of several ratios of quadratic forms in random variables $X_1, X_2,\cdots,X_n$ having a multivariate normal distribution is considered.

Before learning anything about statistical inference in beginning service courses in biostatistics, students learn how to calculate the mean and variance of linear combinations of random variables. Practical precalculus examples of the importance of these exercises can be helpful for instructors, the target audience of this paper. We shall present applications to the "1-sample" and "2-sample" methods for randomized short-term 2-treatment crossover studies, where patients experience both treatments in random order with a "washout" between the active treatment periods. First, we show that the 2-sample method is preferred as it eliminates "conditional bias" when sample sizes by order differ and produces a smaller variance. We also demonstrate that it is usually advisable to use the differences in posttests (ignoring baseline and post washout values) rather than the differences between the changes in treatment from the start of the period to the end of the period ("delta of delta"). Although the intent is not to provide a definitive discussion of crossover designs, we provide a section and references to excellent alternative methods, where instructors can provide motivation to students to explore the topic in greater detail in future readings or courses.

The extended Gini is a family of measures of variability which is mainly used in the areas of finance and income distribution. Each index in the family is defined by specifying one parameter, which reflects the social evaluation of the marginal utility of income. The higher the parameter, the more weight is attached to the lower portion of the cumulative distribution, reflecting higher concern for poverty. In this paper we list and investigate the properties of the equivalents of the correlation coefficient that are associated with the extended Gini family. In addition, we show that the extended Gini of a linear combination of random variables can be decomposed, in a way which is equivalent to the decomposition of the variance, with, in addition, terms that reflect additional properties of the random variables. The implication of these properties is that any decomposition that is performed with the coefficient of variation can be replicated by an infinite number of indices that are based on the Extended Gini coefficient.

CHAPTER III - NUMERICAL CHARACTERISTICS OF PROBABILITY DISTRIBUTIONS

Supplementary Appendix to: A Coskewness Shrinkage Approach for Estimating the Skewness of Linear Combinations of Random Variables

This work investigates the use of the tolerance limits on the treatment couch position to detect mistakes in patient positioning and warn users of possible treatment errors. Computer controlled radiotherapy systems use the position of the treatment couch as a surrogate for patient position, and a tolerance limit is applied against a planned position. When the couch is out of tolerance, a warning is sent to a user to indicate a possible mistake in setup. A tight tolerance may catch all positioning mistakes while at the same time sending too many warnings; a loose tolerance will not catch all mistakes. We developed a statistical model of the absolute position for the three translational axes of the couch. The couch position for any fraction is considered a random variable xi. The ideal planned couch position xp is unknown before a patient starts treatment and must be estimated from the daily positions of xi. As such, xp is also a random variable. The tolerance, tol, is applied to the difference between the daily and planned position, di=xi−xp. The di is a linear combination of random variables and therefore the density of di is the convolution of distributions of xi and xp. Tolerance limits are based on the standard deviation of di such that couch positions that are more than two standard deviations away are considered out of tolerance. Using this framework, we investigated two methods of setting xp and tolerance limits. The first, called first day acquire (FDA), is to take couch position on the first day as the planned position. The second is to use the cumulative average (CumA) over previous fractions as the planned position. The standard deviation of di shrinks as more samples are used to determine xp and, as a result, the tolerance limit shrinks as a function of fraction number when a CumA technique is used. The metrics of sensitivity and specificity were used to characterize the performance of the two methods to correctly identify a couch position as in‐ or out‐of‐tolerance. These two methods were tested using simulated and real patient data. Five clinical sites with different indexed immobilization were tested. These were whole brain, head and neck, breast, thorax, and prostate. Analysis of the head and neck data shows that it is reasonable to model the daily couch position as a random variable in this treatment site. Using an average couch position for xp increased the sensitivity of the couch interlock and reduced the chances of acquiring a couch position that was a statistical outlier. Analysis of variation in couch position for different sites allowed the tolerance limit to be set specifically for a site and immobilization device. The CumA technique was able to increase the sensitivity of detecting out‐of‐tolerance positions while shrinking tolerance limits for a treatment course. Making better use of the software interlock on the couch positions could have a positive impact on patient safety and reduce mistakes in treatment delivery.PACS number: 87.55.Ne, 87.55.Qr, 87.55.tg, 87.55.tm

Multi-capacity bin-packing is a generalization of the classical one-dimensional bin-packing problem in which the bin capacity and the item sizes are represented by d-dimensional vectors. Previous work in d-capacity bin-packing algorithms analyzed variants of single capacity bin-packing, extended to deal with the d capacities. These extended algorithms are oblivious to the additional capacity information, however, and do not scale well with increasing d. We provide new packing algorithms which use the additional capacity information to provide better packing and show how these algorithms might lead to better multi-resource allocation and scheduling solutions.

This paper studies a variation of online bin packing where there is a capacitated buffer to temporarily store items during packing, and item size is bounded within $$(\alpha , 1/2]$$(?,1/2] for some $$0\le \alpha < 1/2$$0≤?<1/2. The problem is motivated by surgery scheduling such that we regard the planned uniform available time interval in each day as a unit size bin and surgeries as items to be packed. Our focus is on the asymptotic performance of NF (Next Fit) and NF-based online algorithms. We show that the classical NF algorithm without use of the buffer has an asymptotic competitive ratio of $$2/(1+\alpha )$$2/(1+?). An NF-based algorithm which makes use of the buffer is further proposed, and proved to be asymptotic 13/9-competitive for any given buffer size not less than 1. We also present a lower bound of 4/3.

We study online algorithms for bin packing and bin covering in two different probabilistic settings in which the item sizes are drawn randomly or the items are adversarial but arrive in random order. We prove several results on the expected performance of well-known online algorithms in these settings. In particular, we prove that the simple greedy algorithm Dual Next-Fit for bin covering performs in the random-order setting strictly better than in the worst case, proving a conjecture by Christ et al. (Theoret Comput Sci 556:71–84, 2014).

We study stochastic variants of Packing Integer Programs (PIP) --- the problems of finding a maximum-value 0/1 vector x satisfying Ax ≤ b, with A and b nonnegative. Many combinatorial problems belong to this broad class, including the knapsack problem, maximum clique, stable set, matching, hypergraph matching (a.k.a. set packing), b-matching, and others. PIP can also be seen as a multidimensional knapsack problem where we wish to pack a maximum-value collection of items with vector-valued sizes. In our stochastic setting, the vector-valued sizes of each item is known to us apriori only as a probability distribution, and the size of an item is instantiated once we commit to including the item in our solution.Following the framework of [3], we consider both adaptive and non-adaptive policies for solving such problems, adaptive policies having the flexibility of being able to make decisions based on the instantiated sizes of items already included in the solution. We investigate the adaptivity gap for these problems: the maximum ratio between the expected values achieved by optimal adaptive and non-adaptive policies. We show tight bounds on the adaptivity gap for set packing and b-matching, and we also show how to find efficiently non-adaptive policies approximating the adaptive optimum. For instance, we can approximate the adaptive optimum for stochastic set packing to within O(d1/2), which is not only optimal with respect to the adaptivity gap, but it is also the best known approximation factor in the deterministic case. It is known that there is no polynomial-time d1/2-e approximation for set packing, unless NP = ZPP. Similarly, for b-matching, we obtain algorithmically a tight bound on the adaptivity gap of O(λ) where λ satisfies Σ λbj+1 = 1.For general Stochastic Packing, we prove that a simple greedy algorithm provides an O(d)-approximation to the adaptive optimum. For A ∈ [0, 1]dxn, we provide an O(λ) approximation where Σ 1/λbj = 1. (For b = (B, B,..., B), we get λ = d1/B.) We also improve the hardness results for deterministic PIP: in the general case, we prove that a polynomial-time d1-e-approximation algorithm would imply NP = ZPP. In the special case when A ∈ [0,1]dxn and b = (B,B,...,B), we show that a d1/B-∈-approximation would imply NP = ZPP. Finally, we prove that it is PSPACE-hard to find the optimal adaptive policy for Stochastic Packing in any fixed dimension d ≥ 2.

Best-fit is the best known algorithm for on-line binpacking, in the sense that no algorithm is known to behave better both in the worst case (when Best-fit has performance ratio 1.7) and in the average uniform case, with items drawn uniformly in the interval [0, l] (then Best-fit has expected wasted space O(n’/2(log n)“/“)). In practical applications, Best-fit appears to perform within a few percent of optimal. In this paper, in the spirit of previous work in computational geometry, we study the expected performance ratio, taking the worst-case multiset of items L, and assuming that the elements of L are inserted in random order, with all permutations equally likely. We show a lower bound of 1.08 . . . and an upper bound of 1.5 on the random order performance ratio of Best-fit. The upper bound contrasts with the result that in the worst case, any (deterministic or randomized) on-line bin-packing algorithm has performance ratio at least 1.54. . . .

Average-case performance analysis of scheduling random parallel tasks with precedence constraints on mesh connected multicomputer systems

The two-dimensional strip packing problem is a generalization of the classic one-dimensional bin packing problem. It has many important applications such as costume clipping, material cutting, real-world planning, packing, scheduling etc. Average-case performance analysis of approximation algorithms attracts a lot of attention because it plays a crucial role in selecting an appropriate algorithm for a given application. While approximation algorithms for two-dimensional packing are frequently presented, the results of their average-case performance analyses have seldom been reported due to intractability. In this paper, we analyze the average-case performance of Next Fit Decreasing Height (NFDH) algorithm, one of the first strip packing algorithms proposed by Coffman, Jr. in 1980. We prove that the expected height of packing with NFDH algorithm, when the heights and widths of the rectangle items are independent and both obey (0, 1] uniform distribution, is about n/3, where n is the number of rectangle items. We also validate the theoretical result with experiments.