21,642 research outputs found
Interleaving schemes for multidimensional cluster errors
We present two-dimensional and three-dimensional interleaving techniques for correcting two- and three-dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. Correction of multidimensional error clusters is required in holographic storage, an emerging application of considerable importance. Our main contribution is the construction of efficient two-dimensional and three-dimensional interleaving schemes. The proposed schemes are based on t-interleaved arrays of integers, defined by the property that every connected component of area or volume t consists of distinct integers. In the two-dimensional case, our constructions are optimal: they have the lowest possible interleaving degree. That is, the resulting t-interleaved arrays contain the smallest possible number of distinct integers, hence minimizing the number of codewords required in an interleaving scheme. In general, we observe that the interleaving problem can be interpreted as a graph-coloring problem, and introduce the useful special class of lattice interleavers. We employ a result of Minkowski, dating back to 1904, to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions. For the case t≡0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For t≠0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice
Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets
Consider the following problem: given a set system (U,I) and an edge-weighted
graph G = (U, E) on the same universe U, find the set A in I such that the
Steiner tree cost with terminals A is as large as possible: "which set in I is
the most difficult to connect up?" This is an example of a max-min problem:
find the set A in I such that the value of some minimization (covering) problem
is as large as possible.
In this paper, we show that for certain covering problems which admit good
deterministic online algorithms, we can give good algorithms for max-min
optimization when the set system I is given by a p-system or q-knapsacks or
both. This result is similar to results for constrained maximization of
submodular functions. Although many natural covering problems are not even
approximately submodular, we show that one can use properties of the online
algorithm as a surrogate for submodularity.
Moreover, we give stronger connections between max-min optimization and
two-stage robust optimization, and hence give improved algorithms for robust
versions of various covering problems, for cases where the uncertainty sets are
given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and
http://arxiv.org/abs/0912.1045 appeared in ICALP 201
An Integer Programming Formulation of the Minimum Common String Partition problem
We consider the problem of finding a minimum common partition of two strings
(MCSP). The problem has its application in genome comparison. MCSP problem is
proved to be NP-hard. In this paper, we develop an Integer Programming (IP)
formulation for the problem and implement it. The experimental results are
compared with the previous state-of-the-art algorithms and are found to be
promising.Comment: arXiv admin note: text overlap with arXiv:1401.453
Computational Performance Evaluation of Two Integer Linear Programming Models for the Minimum Common String Partition Problem
In the minimum common string partition (MCSP) problem two related input
strings are given. "Related" refers to the property that both strings consist
of the same set of letters appearing the same number of times in each of the
two strings. The MCSP seeks a minimum cardinality partitioning of one string
into non-overlapping substrings that is also a valid partitioning for the
second string. This problem has applications in bioinformatics e.g. in
analyzing related DNA or protein sequences. For strings with lengths less than
about 1000 letters, a previously published integer linear programming (ILP)
formulation yields, when solved with a state-of-the-art solver such as CPLEX,
satisfactory results. In this work, we propose a new, alternative ILP model
that is compared to the former one. While a polyhedral study shows the linear
programming relaxations of the two models to be equally strong, a comprehensive
experimental comparison using real-world as well as artificially created
benchmark instances indicates substantial computational advantages of the new
formulation.Comment: arXiv admin note: text overlap with arXiv:1405.5646 This paper
version replaces the one submitted on January 10, 2015, due to detected error
in the calculation of the variables involved in the ILP model
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