How to pack better than best fit: tight bounds for average-case online bin packing

Author(s):  
P.W. Shor
2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Marc Lelarge

International audience We consider the following stochastic bin packing process: the items arrive continuously over time to a server and are packed into bins of unit size according to an online algorithm. The unpacked items form a queue. The items have random sizes with symmetric distribution. Our first contribution identifies some monotonicity properties of the queueing system that allow to derive bounds on the queue size for First Fit and Best Fit algorithms. As a direct application, we show how to compute the stability region under very general conditions on the input process. Our second contribution is a study of the queueing system under heavy load. We show how the monotonicity properties allow one to derive bounds for the speed at which the stationary queue length tends to infinity when the load approaches one. In the case of Best Fit, these bounds are tight. Our analysis shows connections between our dynamic model, average-case results on the classical bin packing problem and planar matching problems.


2020 ◽  
Author(s):  
Armand Stefan Rotaru ◽  
Gabriella Vigliocco

A number of recent models of semantics combine linguistic information, derived from text corpora, and visual information, derived from image collections, demonstrating that the resulting multimodal models are better than either of their unimodal counterparts, in accounting for behavioural data. Empirical work on semantic processing has shown that emotion also plays an important role especially for abstract concepts, however, models integrating emotion along with linguistic and visual information are lacking. Here, we first improve on visual and affective representations, derived from state-of-the-art existing models, by choosing models that best fit available human semantic data and extending the number of concepts they cover. Crucially then, we assess whether adding affective representations (obtained from a neural network model designed to predict emojis from co-occurring text) improves the model’s ability to fit semantic similarity/relatedness judgements from a purely linguistic and linguistic-visual model. We find that, given specific weights assigned to the models, adding both visual and affective representations improve performance, with visual representations providing an improvement especially for more concrete words, and affective representations improving especially the fit for more abstract words.


1999 ◽  
Vol 13 (4) ◽  
pp. 387-406 ◽  
Author(s):  
E. G. Coffman ◽  
Anja Feldmann ◽  
Nabil Kahale ◽  
Bjorn Poonen

We study call admission rates in a linear communication network with each call identified by an arrival time, duration, bandwidth requirement, and origin-destination pair. Network links all have the same bandwidth capacity, and a call can be admitted only if there is sufficient bandwidth available on every link along the call's path. Calls not admitted are held in a queue, in contrast to the protocol of loss networks. We determine maximum admission rates (capacities) under greedy call allocation rules such as First Fit and Best Fit for several baseline models and prove that the natural necessary condition for stability is sufficient. We establish the close connections between our new problems and the classic problems of bin packing and interval packing. In view of these connections, it is surprising to find that Best Fit allocation policies are inferior to First Fit policies in the models studied.


2020 ◽  
Vol 12 (01) ◽  
pp. 2050002
Author(s):  
Sergey P. Shary

For the data fitting problem under interval uncertainty, we introduce the concept of strong compatibility between data and parameters. It is shown that the new strengthened formulation of the problem reduces to computing and estimating the so-called tolerable solution set for interval systems of equations constructed from the data being processed. We propose a computational technology for constructing a “best-fit” linear function from interval data, taking into account the strong compatibility requirement. The properties of the new data fitting approach are much better than those of its predecessors: strong compatibility estimates have polynomial computational complexity, the variance of the strong compatibility estimates is almost always finite, and these estimates are rubust. An example considered in the concluding part of the paper illustrates some of these features.


1975 ◽  
Vol 40 (312) ◽  
pp. 405-414 ◽  
Author(s):  
D. K. Bailey ◽  
R. Macdonald

SummaryFluorine, chlorine, zinc, niobium, zirconium, yttrium, and rubidium have been deter-mined on fifteen obsidians from Eburru volcano (Kenya Rift Valley), spanning the range from pantel-leritic trachyte to pantellerite. All pairs of elements show positive correlation coefficients, ranging between 0·769 and 0·998, but with most values better than 0·900. In spite of some very high correlations, only two of the twenty-one best-fit lines pass near the origin of the Cartesian coordinates. Linear distributions are found within two separate groups of elements: F, Zr, Rb; and Cl, Nb, Yt. Zn behaves in general as a member of the second group but seems to be subject to an additional variation. When an element from the fluorine group is plotted against one from the chlorine group the resulting pattern is non-linear. Therefore, although the elements in both groups would generally be considered ‘residual’ (partition coefficients between crystals and liquid approaching zero) there are clearly detectable differences in their variation, and hence their behaviour.Major-element variations in the obsidians are such that a vapour (fluid) phase would be needed to account for any magma evolution. The trace-element patterns are also impossible by closed-system crystal fractionation and suggest that this fluid may have been rich in halogens, with the metallic elements forming preferred ‘complexes’ with either F or Cl. The F-Zr-Rb ‘complex’ also varies quite independently of the important major oxides (e.g. A12O3) in the rocks. In the case of Rb this is but one aspect of a more significant anomaly, in which there is no sign of any influence of alkali feldspar (which partitions Rb) in the variation. This is remarkable because trachytes and rhyolites have normative ab+or > 50 %, and any evolutionary process controlled by crystal ⇋ liquid interactions must be dominated by the melting or crystallization of alkali feldspar. The results on the Eburru obsidians show that if they are an evolutionary series then either, the process was not crystal ⇋ liquid controlled, or that any such process has been overriden (or buffered) by other processes that have superimposed the observed trace-element patterns. In the latter event, the buffering phase may have been a halogen-bearing vapour.The same considerations must apply to other pantellerite provinces where Rb appears to have behaved as a ‘residual’ element.


Author(s):  
Jennifer A Nisbet ◽  
J A Owen ◽  
Gail E Ward

Data obtained from routine analytical radioimmunoassays were processed using five curve-fitting procedures, viz. ‘Amersham’, single binding site, four parameter logistic, a linear logit-log and a polynomial logit-log. The polynomial logit-log procedure gave the best fit, but this was probably due to the inherent flexibility of this curve-fitting process since the analytical precision achieved with it was no better than what was obtained with most of the other procedures. A limited study failed to show that statistical weighting of data before curve fitting had any practical advantage.


Sign in / Sign up

Export Citation Format

Share Document