A probabilistic analysis of the Next-Fit bin packing algorithm

An infinite supply of pieces, with i.i.d. sizes, is packed under the Next-Fit packing procedure. The process A_n, the number of bins required to pack n pieces, is investigated, and the first two moments are computed when the piece sizes are uniformly distributed. For this special case expressions for the distribution of A_n are also presented.     

有向Kirkman填充设计的谱

§ 1　IntroductionLet X be a set of v points.A packing(directed packing) of X is a collection of subsets(ordered subsets) of X(called blocks) such that any pair(ordered pair) of distinct pointsfrom X occur together in atmostone block in the collection.A packing(directed packing)is called resolvable ifitsblock setadmitsa partition into parallel classes,each parallel classbeing a partition of the pointset X.A Kirkman triple system KTS(v) is a collection Tof3 -subsets of X(triples) suchthat …     

Efficient Approximation Algorithms for Tiling and Packing Problems with Rectangles

We provide improved approximation algorithms for several rectangle tiling and packing problems (RTILE, DRTILE, and d-RPACK) studied in the literature. Most of our algorithms are highly efficient since their running times are near-linear in the sparse input size rather than in the domain size. In addition, we improve the best known approximation ratios.     

Finite and Uniform Stability of Sphere Packings

The main purpose of this paper is to discuss how firm or steady certain known ball packing are, thinking of them as structures. This is closely related to the property of being locally maximally dense. Among other things we show that many of the usual best-known candidates, for the most dense packings with congruent spherical balls, have the property of being uniformly stable, i.e., for a sufficiently small ε > 0 every finite rearrangement of the balls of this packing, where no ball is moved more than ε , is the identity rearrangement. For example, the lattice packings D _d and A _d for d ≥ 3 in E ^d are all uniformly stable. The methods developed here can work for many other packings as well. We also give a construction to show that the densest cubic lattice ball packing in E ^d for d ≥ 2 is not uniformly stable. A packing of balls is called finitely stable if any finite subfamily of the packing is fixed by its neighbors. If a packing is uniformly stable, then it is finitely stable. On the other hand, the cubic lattice packings mentioned above, which are not uniformly stable, are nevertheless finitely stable. Received April 22, 1996, and in revised form October 11, 1996.     

Dimensions of random recursive sets

We prove a theorem that generalizes the equality among the packing, Hausdorff, and upper and lower Minkowski dimensions for a general class of random recursive constructions, and apply it to constructions with finite memory. Then we prove an upper bound on the packing dimension of certain random distribution functions on [0, 1]. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 20–26.     

Kirkman Packing Designs KPD ({3, 5*},v)

A Kirkman packing design KPD ({3, 5^*},v) is a resolvable packing with maximum possible number of parallel classes, each class containing one block of size 5 and all other blocks of size three. Such designs can be used to construct certain threshold schemes in cryptography. In this paper, direct and recursive constructions are discussed for such designs. The existence of a KPD ({3, 5^*},v) for is established with a few possible exceptions.     

Production-inventory systems in stochastic environment and stochastic lead times

We consider a production-inventory system where the production and demand rates are modulated by a finite state Continuous Time Markov Chain (CTMC). When the inventory position (inventory on hand – backorders+inventory on order) falls to a reorder point r, we place an order of size q from an external supplier. We consider the case of stochastic leadtimes, where the leadtimes are i.i.d. exponential(μ) random variables, and orders may or may not be allowed to cross. We derive the distribution of the inventory level, and analyze the long run holding, backlogging, and ordering cost rate per unit time. We use simulation to study the sensitivity of the system to the distribution of the lead times.     

Insensitive bounds for the moments of the sojourn time distribution in the M/G/1 processor-sharing queue

This paper studies the M/G/1 processor-sharing (PS) queue, in particular the sojourn time distribution conditioned on the initial job size. Although several expressions for the Laplace-Stieltjes transform (LST) are known, these expressions are not suitable for computational purposes. This paper derives readily applicable insensitive bounds for all moments of the conditional sojourn time distribution. The instantaneous sojourn time, i.e., the sojourn time of an infinitesimally small job, leads to insensitive upper bounds requiring only knowledge of the traffic intensity and the initial job size. Interestingly, the upper bounds involve polynomials with so-called Eulerian numbers as coefficients. In addition, stochastic ordering and moment ordering results for the sojourn time distribution are obtained. AMS Subject Classification: 60K25, 60E15 This work has been partially funded by the Dutch Ministry of Economic Affairs under the program ‘Technologische Samenwerking ICT-doorbraakprojecten’, project TSIT1025 BEYOND 3G.     

On the Density of 2-Saturated Lattice Packings of Discs

 In a recent paper of G. Fejes Tóth, G. Kuperberg and W. Kuperberg [1] a conjecture has been published concerning the greatest lower bound of the density of a 2-saturated packing of unit discs in the plane. (A packing of unit discs is said to be 2-saturated if none of the discs could be replaced by two other ones of the same size to generate a new packing. A packing of the unit disc is a lattice packing if the centers form a point lattice.) In the present note we study this problem for lattice packings, however, in a more general form in which the removed unit disc is replaced by two discs of radius r. A corollary of our results supports the above conjecture proving that a lattice packing cannot be 2-saturated except if its density is larger than the conjectured bound. (Received 6 December 2000; in revised form March 29, 2001)     

A robust APTAS for the classical bin packing problem

Bin packing is a well studied problem which has many applications. In this paper we design a robust APTAS for the problem. The robust APTAS receives a single input item to be added to the packing at each step. It maintains an approximate solution throughout this process, by slightly adjusting the solution for each new item. At each step, the total size of items which may migrate between bins must be bounded by a constant factor times the size of the new item. We show that such a property cannot be maintained with respect to optimal solutions. A preliminary version of this paper appeared in Proceedings of the 33rd International Colloquium on Automata, Languages and Programming (ICALP2006), part I, pp. 214–225.     

Packing items to feed assembly lines

We treat a practical application of packing problems in feeding assembly lines. This study was motivated by a real situation encountered in the shop floor of a major automobile industry plant in Brazil. The assembly line feed problem (LFP) consists in how pack the items in the available containers to meet the line work centers’ requirements with a minimum total cost over the planning horizon. LFP is a variable-sized bin packing problem that has two special features: (i) a cardinality constraint on each bin’s size; and, (ii) a cost structure such that each bin’s cost varies according to the items that are packed in it. We propose an integer programming model and a GRASP heuristic for LFP. Numerical results on real-life test instances are reported.     

Fisher information in<Emphasis Type="Italic">k</Emphasis>-records

We derive some general results on the Fisher information (FI) contained in the upper (or lower)k-record values and associatedk-record times generated from an i.i.d. sample of fixed size from a continuous distribution. We apply the results to obtain the FI in both upper and lowerk-record data from an exponential distribution. We propose two estimators of the exponential mean, based on the upper and lowerk-record data, and discuss their small sample properties. We also considerk-record data from an inverse sampling plan, and present general formulas for the FI contained in it. Supported in part by Fonde Nacional de Desarrollo Cientifico y Tecnologico (FONDECYT) grant # 1020479 of Chile.     

The average-case analysis of some on-line algorithms for bin packing

In this paper we give tighter bounds than were previously known for the performance of the bin packing algorithms Best Fit and First Fit when the inputs are uniformly distributed on [0, 1]. We also give a general lower bound for the performance of any on-line bin packing algorithm on this distribution. We prove these results by analyzing optimal matchings on points randomly distributed in a unit square. We give a new lower bound for the up-right matching problem. A preliminary version of this paper appeared inProc. 25th IEEE Symposium on Foundations of Computer Science, 193–200. This research was done while the author was at the Massachusetts Institute of Technology Dept. of Mathematics and was supported by a NSF Graduate Fellowship and by Air Force grant OSR-82-0326.     

Nonparametric density estimation for linear processes with infinite variance

We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of the kernel density estimators with the order of bandwidths fixed as h =  cn ^−1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations. In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties.     

Improved lower bounds on the randomized complexity of graph properties

We prove a lower bound of Ω(n^4/3 log ^1/3n) on the randomized decision tree complexity of any nontrivial monotone n‐vertex graph property, and of any nontrivial monotone bipartite graph property with bipartitions of size n. This improves the previous best bound of Ω(n^4/3) due to Hajnal (Combinatorica 11 (1991) 131–143). Our proof works by improving a graph packing lemma used in earlier work, and this improvement in turn stems from a novel probabilistic analysis. Graph packing being a well‐studied subject in its own right, our improved packing lemma and the probabilistic technique used to prove it may be of independent interest. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Random packings by cubes

Y. Itoh’s problem on random integral packings of the d-dimensional (4 × 4)-cube by (2 × 2)-cubes is formulated as follows: (2 × 2)-cubes come to the cube K ₄ sequentially and randomly until it is possible in the following way: no (2 × 2)-cubes overlap, and all their centers are integer points in K ₄. Further, all admissible positions at every step are equiprobable. This process continues until the packing becomes saturated. Find the mean number M of (2 × 2)-cubes in a random saturated packing of the (4 × 4)-cube. This paper provides the proof of the first nontrivial exponential bound of the mean number of cubes in a saturated packing in Itoh’s problem: M ≥ (3/2)^d. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 187–196, 2005.     

Outlier Eigenvalues for Deformed I.I.D. Random Matrices

We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has i.i.d. entries with variance 1/N. Under mild assumptions, as N grows the empirical distribution of the eigenvalues of A + Y converges weakly to a limit probability measure β on the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e., eigenvalues in the complement of the support of β. Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of Y or the Jordan decomposition of A. We also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function. © 2016 Wiley Periodicals, Inc.     

Packing random intervals

Summary Letn random intervalsI ₁, ...,I _n be chosen by selecting endpoints independently from the uniform distribution on [0.1]. Apacking is a pairwise disjoint subset of the intervals; itswasted space is the Lebesgue measure of the points of [0,1] not covered by the packing. In any set of intervals the packing with least wasted space is computationally easy to find; but its expected wasted space in the random case is not obvious. We show that with high probability for largen, this best packing has wasted space . It turns out that if the endpoints 0 and 1 are identified, so that the problem is now one of packing random arcs in a unit-circumference circle, then optimal wasted space is reduced toO(1/n). Interestingly, there is a striking difference between thesizes of the best packings: about logn intervals in the unit interval case, but usually only one or two arcs in the circle case.     

Curved Hexagonal Packings of Equal Disks in a Circle

For each k ≥ 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we introduce packings of h(k) equal disks inside a circle which we call the curved hexagonal packings. The curved hexagonal packing of 7 disks (k = 1, m(1)=1) is well known and one of the 19 disks (k = 2, m(2)=1) has been previously conjectured to be optimal. New curved hexagonal packings of 37, 61, and 91 disks (k = 3, 4, and 5, m(3)=1, m(4)=3, and m(5)=12) were the densest we obtained on a computer using a so-called ``billiards' simulation algorithm. A curved hexagonal packing pattern is invariant under a rotation. For , the density (covering fraction) of curved hexagonal packings tends to . The limit is smaller than the density of the known optimum disk packing in the infinite plane. We found disk configurations that are denser than curved hexagonal packings for 127, 169, and 217 disks (k = 6, 7, and 8). In addition to new packings for h(k) disks, we present the new packings we found for h(k)+1 and h(k)-1 disks for k up to 5, i.e., for 36, 38, 60, 62, 90, and 92 disks. The additional packings show the ``tightness' of the curved hexagonal pattern for k ≤ 5: deleting a disk does not change the optimum packing and its quality significantly, but adding a disk causes a substantial rearrangement in the optimum packing and substantially decreases the quality. Received May 15, 1995, and in revised form March 5, 1996.     

Kirkman packing and covering designs with spanned holes of size 2

A Kirkman holey packing (resp. covering) design, denoted by KHPD(g^u) (resp. KHCD(g^u)), is a resolvable (gu, 3, 1) packing (resp. covering) design of pairs with u disjoint holes of size g, which has the maximum (resp. minimum) possible number of parallel classes. Each parallel class contains one block of size δ, while other blocks have size 3. Here δ is equal to 2, 3, and 4 when gu ≡ 2, 3, and 4 (mod 3) in turn. In this paper, the existence problem of a KHPD(2^u) and a KHCD(2^u) is solved with one possible exception of a KHPD(2⁸). © 2004 Wiley Periodicals, Inc.