On the Greedy Algorithm for the Shortest Common Superstring Problem with Reversals

We study a variation of the classical Shortest Common Superstring (SCS) problem in which a shortest superstring of a finite set of strings $S$ is sought containing as a factor every string of $S$ or its reversal. We call this problem Shortest Common Superstring with Reversals (SCS-R). This problem has been introduced by Jiang et al., who designed a greedy-like algorithm with length approximation ratio $4$. In this paper, we show that a natural adaptation of the classical greedy algorithm for SCS has (optimal) compression ratio $\frac12$, i.e., the sum of the overlaps in the output string is at least half the sum of the overlaps in an optimal solution. We also provide a linear-time implementation of our algorithm.Comment: Published in Information Processing Letter

Nondeterministic State Complexity for Suffix-Free Regular Languages

We investigate the nondeterministic state complexity of basic operations for suffix-free regular languages. The nondeterministic state complexity of an operation is the number of states that are necessary and sufficient in the worst-case for a minimal nondeterministic finite-state automaton that accepts the language obtained from the operation. We consider basic operations (catenation, union, intersection, Kleene star, reversal and complementation) and establish matching upper and lower bounds for each operation. In the case of complementation the upper and lower bounds differ by an additive constant of two.Comment: In Proceedings DCFS 2010, arXiv:1008.127

Finding approximate palindromes in strings

We introduce a novel definition of approximate palindromes in strings, and provide an algorithm to find all maximal approximate palindromes in a string with up to $k$ errors. Our definition is based on the usual edit operations of approximate pattern matching, and the algorithm we give, for a string of size $n$ on a fixed alphabet, runs in $O(k^2 n)$ time. We also discuss two implementation-related improvements to the algorithm, and demonstrate their efficacy in practice by means of both experiments and an average-case analysis

Detecting One-variable Patterns

Given a pattern $p = s_1x_1s_2x_2\cdots s_{r-1}x_{r-1}s_r$ such that $x_1,x_2,\ldots,x_{r-1}\in\{x,\overset{{}_{\leftarrow}}{x}\}$, where $x$ is a variable and $\overset{{}_{\leftarrow}}{x}$ its reversal, and $s_1,s_2,\ldots,s_r$ are strings that contain no variables, we describe an algorithm that constructs in $O(rn)$ time a compact representation of all $P$ instances of $p$ in an input string of length $n$ over a polynomially bounded integer alphabet, so that one can report those instances in $O(P)$ time.Comment: 16 pages (+13 pages of Appendix), 4 figures, accepted to SPIRE 201