Average-case analyses of first fit and random fit bin packing

Abstract

We prove that the First Fit bin packing algorithm is stable under the input distribution U”k − 2; k• for all k ≥ 3, settling an open question from the recent survey by Coffman, Garey, and Johnson [“Approximation algorithms for bin backing: A survey,” Approximation algorithms for NP-hard problems, D. Hochbaum (Editor), PWS, Boston, 1996]. Our proof generalizes the multidimensional Markov chain analysis used by Kenyon, Sinclair, and Rabani to prove that Best Fit is also stable under these distributions [Proc Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 1995, pp. 351–358]. 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”k− 2; k•, as well as present worst case bounds and some results on distributions U”k− 1; k• and U”k; k• for Random Fit. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 240–259, 2000 Correspondence to: Michael Mitzenmacher. *Most of this work was done while at the Max-Planch-Institut fur Informatik, Saarbrucken, Germany. † A substantial portion of this research was done while at the Computer Science Department, UC Berkeley and Digital Equipment Corporation Systems Research Center. Contract grant sponsor: National Science Foundation. Contract grant number: CCR-9505448. © 2000 John Wiley & Sons, Inc.