1,051 research outputs found
Improved Lower Bounds for Online Hypercube Packing
Packing a given sequence of items into as few bins as possible in an online fashion is a widely studied problem. We improve lower bounds for packing hypercubes into bins in two or more dimensions, once for general algorithms (in two dimensions) and once for an important subclass, so-called Harmonic-type algorithms (in two or more dimensions). Lastly, we show that two adaptions of the ideas from the best known one-dimensional packing algorithm to square packing also do not help to break the barrier of 2
A tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game
We prove a tight lower bound on the asymptotic performance ratio of
the bounded space online -hypercube bin packing problem, solving an open
question raised in 2005. In the classic -hypercube bin packing problem, we
are given a sequence of -dimensional hypercubes and we have an unlimited
number of bins, each of which is a -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 -hypercube bin packing problem is a variant of the
-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 is and , and conjectured that
it is . We show that is in fact . 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 , using probabilistic arguments. Such packings also lead to
lower bounds for the prices of anarchy of the selfish -hypercube bin packing
game. We present a lower bound of for the pure price of
anarchy of this game, and we also give a lower bound of for
its strong price of anarchy
Coloring translates and homothets of a convex body
We obtain improved upper bounds and new lower bounds on the chromatic number
as a linear function of the clique number, for the intersection graphs (and
their complements) of finite families of translates and homothets of a convex
body in \RR^n.Comment: 11 pages, 2 figure
Rotated sphere packing designs
We propose a new class of space-filling designs called rotated sphere packing
designs for computer experiments. The approach starts from the asymptotically
optimal positioning of identical balls that covers the unit cube. Properly
scaled, rotated, translated and extracted, such designs are excellent in
maximin distance criterion, low in discrepancy, good in projective uniformity
and thus useful in both prediction and numerical integration purposes. We
provide a fast algorithm to construct such designs for any numbers of
dimensions and points with R codes available online. Theoretical and numerical
results are also provided
Online algorithms for 1-space bounded multi dimensional bin packing and hypercube packing
In this paper, we study 1-space bounded multi-dimensional bin packing and hypercube packing. A sequence of items arrive over time, each item is a d-dimensional hyperbox (in bin packing) or hypercube (in hypercube packing), and the length of each side is no more than 1. These items must be packed without overlapping into d-dimensional hypercubes with unit length on each side. In d-dimensional space, any two dimensions i and j define a space P ij. When an item arrives, we must pack it into an active bin immediately without any knowledge of the future items, and 90 {ring operator}-rotation on any plane P ij is allowed. The objective is to minimize the total number of bins used for packing all these items in the sequence. In the 1-space bounded variant, there is only one active bin for packing the current item. If the active bin does not have enough space to pack the item, it must be closed and a new active bin is opened. For d-dimensional bin packing, an online algorithm with competitive ratio 4 d is given. Moreover, we consider d-dimensional hypercube packing, and give a 2 d+1-competitive algorithm. These two results are the first study on 1-space bounded multi dimensional bin packing and hypercube packing. © 2012 The Author(s).published_or_final_versionSpringer Open Choice, 28 May 201
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