scholarly journals Online Bandwidth packing with symmetric distribution

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Marc Lelarge
Keyword(s):  
Bin Packing ◽  
Queueing System ◽  
Packing Problem ◽  
Symmetric Distribution ◽  
Heavy Load ◽  
Average Case ◽  
Matching Problems ◽  
Model Average ◽  
Monotonicity Properties ◽  
Best Fit

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.

scholarly journals Analysis of an Approximation Algorithm for Scheduling Independent Parallel Tasks

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Keqin Li
Keyword(s):  
Approximation Algorithm ◽  
Bin Packing ◽  
Random Variables ◽  
Packing Problem ◽  
Performance Ratio ◽  
Worst Case ◽  
Average Case ◽  
Parallel Tasks ◽  
International Audience ◽  
Worst Case Performance Ratio

International audience In this paper, we consider the problem of scheduling independent parallel tasks in parallel systems with identical processors. The problem is NP-hard, since it includes the bin packing problem as a special case when all tasks have unit execution time. We propose and analyze a simple approximation algorithm called H_m, where m is a positive integer. Algorithm H_m has a moderate asymptotic worst-case performance ratio in the range [4/3 ... 31/18] for all m≥ 6; but the algorithm has a small asymptotic worst-case performance ratio in the range [1+1/(r+1)..1+1/r], when task sizes do not exceed 1/r of the total available processors, where r>1 is an integer. Furthermore, we show that if the task sizes are independent, identically distributed (i.i.d.) uniform random variables, and task execution times are i.i.d. random variables with finite mean and variance, then the average-case performance ratio of algorithm H_m is no larger than 1.2898680..., and for an exponential distribution of task sizes, it does not exceed 1.2898305.... As demonstrated by our analytical as well as numerical results, the average-case performance ratio improves significantly when tasks request for smaller numbers of processors.

scholarly journals Best Fit Bin Packing with Random Order Revisited

Algorithmica ◽  
2021 ◽  
Author(s):  
Susanne Albers ◽  
Arindam Khan ◽  
Leon Ladewig
Keyword(s):  
Online Algorithms ◽  
Bin Packing ◽  
Packing Problem ◽  
Random Order ◽  
Monotonicity Property ◽  
Performance Measure ◽  
Upper And Lower Bounds ◽  
Analysis Model ◽  
Matching Problem ◽  
Best Fit

AbstractBest Fit is a well known online algorithm for the bin packing problem, where a collection of one-dimensional items has to be packed into a minimum number of unit-sized bins. In a seminal work, Kenyon [SODA 1996] introduced the (asymptotic) random order ratio as an alternative performance measure for online algorithms. Here, an adversary specifies the items, but the order of arrival is drawn uniformly at random. Kenyon’s result establishes lower and upper bounds of 1.08 and 1.5, respectively, for the random order ratio of Best Fit. Although this type of analysis model became increasingly popular in the field of online algorithms, no progress has been made for the Best Fit algorithm after the result of Kenyon. We study the random order ratio of Best Fit and tighten the long-standing gap by establishing an improved lower bound of 1.10. For the case where all items are larger than 1/3, we show that the random order ratio converges quickly to 1.25. It is the existence of such large items that crucially determines the performance of Best Fit in the general case. Moreover, this case is closely related to the classical maximum-cardinality matching problem in the fully online model. As a side product, we show that Best Fit satisfies a monotonicity property on such instances, unlike in the general case. In addition, we initiate the study of the absolute random order ratio for this problem. In contrast to asymptotic ratios, absolute ratios must hold even for instances that can be packed into a small number of bins. We show that the absolute random order ratio of Best Fit is at least 1.3. For the case where all items are larger than 1/3, we derive upper and lower bounds of 21/16 and 1.2, respectively.

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