8,272 research outputs found

    Improved space for bounded-space, on-line bin-packing

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    The author presents a sequence of linear-time, bounded-space, on-line, bin-packing algorithms that are based on the "HARMONIC" algorithms {\text{H}}_k introduced by Lee and Lee [J. Assoc. Comput. Mach., 32 (1985), pp. 562–572]. The algorithms in this paper guarantee the worst case performance of {\text{H}}_k, whereas they only use $O ( \log \log k ) instead of k active bins. For k\geqq 6, the algorithms in this paper outperform all known heuristics using k active bins. For example, the author gives an algorithm that has worst case ratio less than 17/10 and uses only six active bin

    A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game

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    We prove a tight lower bound on the asymptotic performance ratio ρ\rho of the bounded space online dd-hypercube bin packing problem, solving an open question raised in 2005. In the classic dd-hypercube bin packing problem, we are given a sequence of dd-dimensional hypercubes and we have an unlimited number of bins, each of which is a dd-dimensional unit hypercube. The goal is to pack (orthogonally) the given hypercubes into the minimum possible number of bins, in such a way that no two hypercubes in the same bin overlap. The bounded space online dd-hypercube bin packing problem is a variant of the dd-hypercube bin packing problem, in which the hypercubes arrive online and each one must be packed in an open bin without the knowledge of the next hypercubes. Moreover, at each moment, only a constant number of open bins are allowed (whenever a new bin is used, it is considered open, and it remains so until it is considered closed, in which case, it is not allowed to accept new hypercubes). Epstein and van Stee [SIAM J. Comput. 35 (2005), no. 2, 431-448] showed that ρ\rho is Ω(logd)\Omega(\log d) and O(d/logd)O(d/\log d), and conjectured that it is Θ(logd)\Theta(\log d). We show that ρ\rho is in fact Θ(d/logd)\Theta(d/\log d). To obtain this result, we elaborate on some ideas presented by those authors, and go one step further showing how to obtain better (offline) packings of certain special instances for which one knows how many bins any bounded space algorithm has to use. Our main contribution establishes the existence of such packings, for large enough dd, using probabilistic arguments. Such packings also lead to lower bounds for the prices of anarchy of the selfish dd-hypercube bin packing game. We present a lower bound of Ω(d/logd)\Omega(d/\log d) for the pure price of anarchy of this game, and we also give a lower bound of Ω(logd)\Omega(\log d) for its strong price of anarchy

    Locality-preserving allocations Problems and coloured Bin Packing

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    We study the following problem, introduced by Chung et al. in 2006. We are given, online or offline, a set of coloured items of different sizes, and wish to pack them into bins of equal size so that we use few bins in total (at most α\alpha times optimal), and that the items of each colour span few bins (at most β\beta times optimal). We call such allocations (α,β)(\alpha, \beta)-approximate. As usual in bin packing problems, we allow additive constants and consider (α,β)(\alpha,\beta) as the asymptotic performance ratios. We prove that for \eps>0, if we desire small α\alpha, no scheme can beat (1+\eps, \Omega(1/\eps))-approximate allocations and similarly as we desire small β\beta, no scheme can beat (1.69103, 1+\eps)-approximate allocations. We give offline schemes that come very close to achieving these lower bounds. For the online case, we prove that no scheme can even achieve (O(1),O(1))(O(1),O(1))-approximate allocations. However, a small restriction on item sizes permits a simple online scheme that computes (2+\eps, 1.7)-approximate allocations
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