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Average-Case Analyses of First Fit and Random Fit Bin Packing Susanne Albers Michael Mitzenmachery

Summary: Average-Case Analyses of First Fit and Random Fit Bin Packing
Susanne Albers Michael Mitzenmachery
We prove that the First Fit bin packing algorithm is stable under the input distribution
U fk 2; kg for all k 3, settling an open question from the recent survey by Co man, Garey,
and Johnson 3]. Our proof generalizes the multi-dimensional Markov chain analysis used by
Kenyon, Rabani, and Sinclair to prove that Best Fit is also stable under these distributions 10].
Our proof is motivated by an analysis of Random Fit, a new simple packing algorithm related
to First Fit, that is interesting in its own right. We show that Random Fit is stable under
the input distributions U fk 2; kg, as well as present worst-case bounds and some results on
distributions U fk 1; kg and U fk; kg for Random Fit.
1 Introduction
In the one-dimensional bin packing problem, one is given a sequence a1;:::;an 2 (0;1] of items to
pack into bins of unit capacity so as to minimize the number of bins used. A great deal of literature
has focused on this problem, perhaps because, as Co man, Garey, and Johnson 3] observe in their
recent survey on bin packing, \The classical one-dimensional bin packing problem has long served
as a proving ground for new approaches to the analysis of approximation algorithms." For example,
recently the study of Best Fit bin packing under discrete uniform distributions has led to a novel
analysis technique, based on the theory of multi-dimensional Markov chains. In this paper we
extend this approach to analyze First Fit and a new bin packing algorithm, called Random Fit,


Source: Albers, Susanne - Institut für Informatik, Humboldt-Universität zu Berlin


Collections: Computer Technologies and Information Sciences