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## Вычисление длиннейшей общей подстроки с одной ошибкой

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

We consider a compact text index based on evenly spaced sparse suffix trees of a text \cite{KU-96}. Such a tree is defined by partitioning the text into blocks of equal size and constructing the suffix tree only for those suffixes that start at block boundaries. We propose a new pattern matching algorithm on this structure. The algorithm is based on a notion of suffix links different from that of~\cite{KU-96} and on the packing of several letters into one computer word.

We study a new variant of the string matching problem called {\em cross-document string matching}, which is the problem of indexing a collection of documents to support an efficient search for a pattern in a selected document, where the pattern itself is a substring of another document. Several variants of this problem are considered, and efficient linear-space solutions are proposed with query time bounds that either do not depend at all on the pattern size or depend on it in a very limited way (doubly logarithmic). As a side result, we propose an improved solution to the {\em weighted level ancestor} problem.

Given a set of $N$ strings $A = \set{\alpha_1, \ldots, \alpha_N}$ of total length $n$ over alphabet~$\Sigma$ one may ask to find, for a fixed integer $K$, $2 \le K \le N$, the longest substring $\beta$ that appears in at least $K$ strings in $A$. It is known that this problem can be solved in $O(n)$ time with the help of suffix trees. However, the resulting algorithm is rather complicated. Also, its running time and memory consumption may depend on~$\abs{\Sigma}$. This paper presents an alternative, remarkably simple approach to the above problem, which relies on the notion of suffix arrays. Once the suffix array of some auxiliary $O(n)$-length string is computed, one needs a simple $O(n)$-time postprocessing to find the requested longest substring. Since a number of efficient and simple linear-time algorithms for constructing suffix arrays has been recently developed (with constant not depending on $|\Sigma|$), our approach seems to be quite practical.

This book constitutes the refereed proceedings of the 23rd Annual Symposium on Combinatorial Pattern Matching, CPM 2012, held in Helsinki, Finalnd, in July 2012. The 33 revised full papers presented together with 2 invited talks were carefully reviewed and selected from 60 submissions. The papers address issues of searching and matching strings and more complicated patterns such as trees, regular expressions, graphs, point sets, and arrays. The goal is to derive non-trivial combinatorial properties of such structures and to exploit these properties in order to either achieve superior performance for the corresponding computational problems or pinpoint conditions under which searches cannot be performed efficiently. The meeting also deals with problems in computational biology, data compression and data mining, coding, information retrieval, natural language processing, and pattern recognition.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.