arXiv subjects

Neerja Mhaskar

Publications and source records attributed to Neerja Mhaskar.

6 recordsLinked to original sources

arXiv2025

Efficient Algorithms to Compute Closed Substrings

A closed string $u$ is either of length one or contains a border that occurs only as a prefix and as a suffix in $u$ and nowhere else within $u$. In this paper, we present fast $\mathcal{O}(n\log n)$ time algorithms to compute all $\mathcal{O}(n^2)$ closed substrings by introducing a compact representation for all closed substrings of a string $ w[1..n]$, using only $\mathcal{O}(n \log n)$ space. These simple and space-efficient algorithms also compute maximal closed strings. Furthermore, we compare the performance of these algorithms and identify classes of strings where each performs best. Finally, we show that the exact number of MCSs ($M(f_n)$) in a Fibonacci word $ f_n $, for $n \geq 5$, is $\approx \left(1 + \frac{1}{\phi^2}\right) F_n \approx 1.382 F_n$, where $ \phi $ is the golden ratio.

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arXiv2025

On the Complexity of Finding Approximate LCS of Multiple Strings

Finding an Approximate Longest Common Substring (ALCS) within a given set $S=\{s_1,s_2,\ldots,s_m\}$ of $m \ge 2$ strings is a key problem in computational biology, such as identifying related mutations across multiple genetic sequences. We study several variants of ALCS problems that, given integers $k$ and $t \le m$, seek the longest string $u$ -- or the longest substring $u$ of any string in $S$ -- that lies within distance $k$ of at least one substring in $t$ distinct strings from $S$. While the general problems are NP-hard, we present efficient algorithms for restricted cases under Hamming and edit distances using the $LCP_k$ and $k$-errata tree data structures. Our methods achieve run times of $\mathcal{O}(N^2)$, $\mathcal{O}(k\ell N^2)$, and $\mathcal{O}(mN\log^k \ell)$, where $\ell$ is the length of the longest string and $N$ is the sum of the lengths of all the strings in $S$. We also establish conditional lower bounds under the Strong Exponential Time Hypothesis and extend our study to indeterminate strings.

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arXiv2024

V-Words, Lyndon Words and Galois Words

We say that a family $\mathcal{W}$ of strings over $\Sigma^+$ forms a Unique Maximal Factorization Family (UMFF) if and only if every $w \in \mathcal{W}$ has a unique maximal factorization. Further, an UMFF $\mathcal{W}$ is called a circ-UMFF whenever it contains exactly one rotation of every primitive string $x \in \Sigma^+$. $V$-order is a non-lexicographical total ordering on strings that determines a circ-UMFF. In this paper we propose a generalization of circ-UMFF called the substring circ-UMFF and extend combinatorial research on $V$-order by investigating connections to Lyndon words. Then we extend these concepts to any total order. Applications of this research arise in efficient text indexing, compression, and search problems.

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arXiv2022

String Covering: A Survey

The study of strings is an important combinatorial field that precedes the digital computer. Strings can be very long, trillions of letters, so it is important to find compact representations. Here we first survey various forms of one potential compaction methodology, the cover of a given string x, initially proposed in a simple form in 1990, but increasingly of interest as more sophisticated variants have been discovered. We then consider covering by a seed; that is, a cover of a superstring of x. We conclude with many proposals for research directions that could make significant contributions to string processing in future.

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arXiv2022

Practical KMP/BM Style Pattern-Matching on Indeterminate Strings

In this paper we describe two simple, fast, space-efficient algorithms for finding all matches of an indeterminate pattern $p = p[1..m]$ in an indeterminate string $x = x[1..n]$, where both $p$ and $x$ are defined on a "small" ordered alphabet $\Sigma$ $-$ say, $\sigma = |\Sigma| \le 9$. Both algorithms depend on a preprocessing phase that replaces $\Sigma$ by an integer alphabet $\Sigma_I$ of size $\sigma_I = \sigma$ which (reversibly, in time linear in string length) maps both $x$ and $p$ into equivalent regular strings $y$ and $q$, respectively, on $\Sigma_I$, whose maximum (indeterminate) letter can be expressed in a 32-bit word (for $\sigma \le 4$, thus for DNA sequences, an 8-bit representation suffices). We first describe an efficient version KMP Indet of the venerable Knuth-Morris-Pratt algorithm to find all occurrences of $q$ in $y$ (that is, of $p$ in $x$), but, whenever necessary, using the prefix array, rather than the border array, to control shifts of the transformed pattern $q$ along the transformed string $y$. We go on to describe a similar efficient version BM Indet of the Boyer- Moore algorithm that turns out to execute significantly faster than KMP Indet over a wide range of test cases. A noteworthy feature is that both algorithms require very little additional space: $\Theta(m)$ words. We conjecture that a similar approach may yield practical and efficient indeterminate equivalents to other well-known pattern-matching algorithms, in particular the several variants of Boyer-Moore.

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arXiv2020

A New Approach to Regular & Indeterminate Strings

In this paper we propose a new, more appropriate definition of regular and indeterminate strings. A regular string is one that is "isomorphic" to a string whose entries all consist of a single letter, but which nevertheless may itself include entries containing multiple letters. A string that is not regular is said to be indeterminate. We begin by proposing a new model for the representation of strings, regular or indeterminate, then go on to describe a linear time algorithm to determine whether or not a string $x = x[1..n]$ is regular and, if so, to replace it by a lexicographically least (lex-least) string $y$ whose entries are all single letters. Furthermore, we connect the regularity of a string to the transitive closure problem on a graph, which in our special case can be efficiently solved. We then introduce the idea of a feasible palindrome array MP of a string, and prove that every feasible MP corresponds to some (regular or indeterminate) string. We describe an algorithm that constructs a string $x$ corresponding to given feasible MP, while ensuring that whenever possible $x$ is regular and if so, then lex-least. A final section outlines new research directions suggested by this changed perspective on regular and indeterminate strings.

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