Searching of gapped repeats and subrepetitions in a word

A gapped repeat is a factor of the form $uvu$ where $u$ and $v$ are nonempty words. The period of the gapped repeat is defined as $|u|+|v|$. The gapped repeat is maximal if it cannot be extended to the left or to the right by at least one letter with preserving its period. The gapped repeat is called $\alpha$-gapped if its period is not greater than $\alpha |v|$. A $\delta$-subrepetition is a factor which exponent is less than 2 but is not less than $1+\delta$ (the exponent of the factor is the quotient of the length and the minimal period of the factor). The $\delta$-subrepetition is maximal if it cannot be extended to the left or to the right by at least one letter with preserving its minimal period. We reveal a close relation between maximal gapped repeats and maximal subrepetitions. Moreover, we show that in a word of length $n$ the number of maximal $\alpha$-gapped repeats is bounded by $O(\alpha^2n)$ and the number of maximal $\delta$-subrepetitions is bounded by $O(n/\delta^2)$. Using the obtained upper bounds, we propose algorithms for finding all maximal $\alpha$-gapped repeats and all maximal $\delta$-subrepetitions in a word of length $n$. The algorithm for finding all maximal $\alpha$-gapped repeats has $O(\alpha^2n)$ time complexity for the case of constant alphabet size and $O(n\log n + \alpha^2n)$ time complexity for the general case. For finding all maximal $\delta$-subrepetitions we propose two algorithms. The first algorithm has $O(\frac{n\log\log n}{\delta^2})$ time complexity for the case of constant alphabet size and $O(n\log n +\frac{n\log\log n}{\delta^2})$ time complexity for the general case. The second algorithm has $O(n\log n+\frac{n}{\delta^2}\log \frac{1}{\delta})$ expected time complexity

Approximate algorithms for minimization of binary decision diagrams on the basis of linear transformations of variables

Algorithms for an approximate minimization of binary decision diagrams (BDD) on the basis of linear transformations of variables are proposed. The algorithms rely on the transformations of only adjacent variables and have a polynomial complexity relative to the size of the table that lists values of the function involved

Longest Common Extensions in Sublinear Space

The longest common extension problem (LCE problem) is to construct a data structure for an input string $T$ of length $n$ that supports LCE$(i,j)$ queries. Such a query returns the length of the longest common prefix of the suffixes starting at positions $i$ and $j$ in $T$. This classic problem has a well-known solution that uses $O(n)$ space and $O(1)$ query time. In this paper we show that for any trade-off parameter $1 \leq \tau \leq n$, the problem can be solved in $O(\frac{n}{\tau})$ space and $O(\tau)$ query time. This significantly improves the previously best known time-space trade-offs, and almost matches the best known time-space product lower bound.Comment: An extended abstract of this paper has been accepted to CPM 201

Effect of nitrogen ion irradiation parameters on properties of nitrogen-containing carbon coatings prepared by pulsed vacuum arc deposition method

Studies of the effect of nitrogen ion irradiation on the structure and properties of nitrogenated amorphous carbon coatings prepared on polished sitall and silicon substrates by the pulsed vacuum arc deposition method are presented. The techniques used in the investigations were electron energy loss spectroscopy, Raman spectroscopy, and atomic force microscop

Gendarmerie Railway Police of Russian Empire in Early 20th Century: Working with Political Agents

The study examines the historical experience of interaction between the gendarmerie railway police of the Russian Empire and covert informants in political organizations. It explores the establishment of intelligence work on railways, the monitoring of its effectiveness, the verification of the reliability of secret agents, and the identification of provocateurs and blackmailers among them. The materials for analysis are extracted from previously unpublished secret and top-secret case files of the gendarmerie police departments of the railways. The authors provide their own definition of intelligence work. It is emphasized that priority was given to recruiting informants who had a financial interest in collaborating with the police. The study demonstrates that the value of information obtained by secret agents was the main criterion for evaluating the effectiveness of financial resources allocated to intelligence work. The authors highlight the need for caution in establishing relationships between railway gendarmes and informants due to the possibility of assassination attempts against handlers or the provision of misinformation. The study concludes that assigning the duty of recruiting political agents and obtaining information through them was justified by the need to suppress revolutionary movements, but inertia in acquiring informants and the scale of crises in the empire prevented the achievement of the set goal

Counterintelligence Activities of Gendarmerie Railway Police before and during World War I

The article analyzes the role of the gendarmerie railway police in the system of counterintelligence agencies in the Russian Empire before and during World War I. Based on documentary materials, the goals of enemy espionage on railways are revealed. Measures taken by the gendarmerie to restrict photography of railway infrastructure are examined. Through analysis of secret correspondence between gendarmerie leaders and railway department heads, categories of individuals most actively recruited by German and Austro-Hungarian intelligence for espionage are identified: prisoners of war, foreign nationals not involved in combat, and children. The organization of surveillance of foreign officials’ railway transport movements within the Russian Empire is also explored. The conclusion is drawn that the gendarmerie railway police’s ability to carry out counterintelligence tasks was complicated by their simultaneous duties as general and political police, as well as the scale of the infrastructure they were tasked with protecting

Palindromic Decompositions with Gaps and Errors

Identifying palindromes in sequences has been an interesting line of research in combinatorics on words and also in computational biology, after the discovery of the relation of palindromes in the DNA sequence with the HIV virus. Efficient algorithms for the factorization of sequences into palindromes and maximal palindromes have been devised in recent years. We extend these studies by allowing gaps in decompositions and errors in palindromes, and also imposing a lower bound to the length of acceptable palindromes. We first present an algorithm for obtaining a palindromic decomposition of a string of length n with the minimal total gap length in time O(n log n * g) and space O(n g), where g is the number of allowed gaps in the decomposition. We then consider a decomposition of the string in maximal \delta-palindromes (i.e. palindromes with \delta errors under the edit or Hamming distance) and g allowed gaps. We present an algorithm to obtain such a decomposition with the minimal total gap length in time O(n (g + \delta)) and space O(n g).Comment: accepted to CSR 201

A Minimal Periods Algorithm with Applications

Kosaraju in ``Computation of squares in a string'' briefly described a linear-time algorithm for computing the minimal squares starting at each position in a word. Using the same construction of suffix trees, we generalize his result and describe in detail how to compute in O(k|w|)-time the minimal k-th power, with period of length larger than s, starting at each position in a word w for arbitrary exponent $k\geq2$ and integer $s\geq0$. We provide the complete proof of correctness of the algorithm, which is somehow not completely clear in Kosaraju's original paper. The algorithm can be used as a sub-routine to detect certain types of pseudo-patterns in words, which is our original intention to study the generalization.Comment: 14 page

Transition Property For Cube-Free Words

We study cube-free words over arbitrary non-unary finite alphabets and prove the following structural property: for every pair $(u,v)$ of $d$-ary cube-free words, if $u$ can be infinitely extended to the right and $v$ can be infinitely extended to the left respecting the cube-freeness property, then there exists a "transition" word $w$ over the same alphabet such that $uwv$ is cube free. The crucial case is the case of the binary alphabet, analyzed in the central part of the paper. The obtained "transition property", together with the developed technique, allowed us to solve cube-free versions of three old open problems by Restivo and Salemi. Besides, it has some further implications for combinatorics on words; e.g., it implies the existence of infinite cube-free words of very big subword (factor) complexity.Comment: 14 pages, 5 figure