1,645 research outputs found
Variants of Constrained Longest Common Subsequence
In this work, we consider a variant of the classical Longest Common
Subsequence problem called Doubly-Constrained Longest Common Subsequence
(DC-LCS). Given two strings s1 and s2 over an alphabet A, a set C_s of strings,
and a function Co from A to N, the DC-LCS problem consists in finding the
longest subsequence s of s1 and s2 such that s is a supersequence of all the
strings in Cs and such that the number of occurrences in s of each symbol a in
A is upper bounded by Co(a). The DC-LCS problem provides a clear mathematical
formulation of a sequence comparison problem in Computational Biology and
generalizes two other constrained variants of the LCS problem: the Constrained
LCS and the Repetition-Free LCS. We present two results for the DC-LCS problem.
First, we illustrate a fixed-parameter algorithm where the parameter is the
length of the solution. Secondly, we prove a parameterized hardness result for
the Constrained LCS problem when the parameter is the number of the constraint
strings and the size of the alphabet A. This hardness result also implies the
parameterized hardness of the DC-LCS problem (with the same parameters) and its
NP-hardness when the size of the alphabet is constant
Algebraic aspects of increasing subsequences
We present a number of results relating partial Cauchy-Littlewood sums,
integrals over the compact classical groups, and increasing subsequences of
permutations. These include: integral formulae for the distribution of the
longest increasing subsequence of a random involution with constrained number
of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as
new proofs of old formulae; relations of these expressions to orthogonal
polynomials on the unit circle; and explicit bases for invariant spaces of the
classical groups, together with appropriate generalizations of the
straightening algorithm.Comment: LaTeX+amsmath+eepic; 52 pages. Expanded introduction, new references,
other minor change
Orthogonal polynomial ensembles in probability theory
We survey a number of models from physics, statistical mechanics, probability
theory and combinatorics, which are each described in terms of an orthogonal
polynomial ensemble. The most prominent example is apparently the Hermite
ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE),
and other well-known ensembles known in random matrix theory like the Laguerre
ensemble for the spectrum of Wishart matrices. In recent years, a number of
further interesting models were found to lead to orthogonal polynomial
ensembles, among which the corner growth model, directed last passage
percolation, the PNG droplet, non-colliding random processes, the length of the
longest increasing subsequence of a random permutation, and others. Much
attention has been paid to universal classes of asymptotic behaviors of these
models in the limit of large particle numbers, in particular the spacings
between the particles and the fluctuation behavior of the largest particle.
Computer simulations suggest that the connections go even farther and also
comprise the zeros of the Riemann zeta function. The existing proofs require a
substantial technical machinery and heavy tools from various parts of
mathematics, in particular complex analysis, combinatorics and variational
analysis. Particularly in the last decade, a number of fine results have been
achieved, but it is obvious that a comprehensive and thorough understanding of
the matter is still lacking. Hence, it seems an appropriate time to provide a
surveying text on this research area.Comment: Published at http://dx.doi.org/10.1214/154957805100000177 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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