Yuto Nakashima

The "runs" theorem

We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of what was known as the “runs” conjecture [R. M. Kolpakov an...

Faster Lyndon factorization algorithms for SLP and LZ78 compressed text

We present two efficient algorithms which, given a compressed representation of a string w of length N, compute the Lyndon factorization of w. Given a straight line program (SLP) S of size n that describes w, the first algorithm runs in O ( n 2 + P ( n , N ) + Q ( n , N ) n log ź n ) time and O ( n 2 + S ( n , N ) ) space, where P ( n , N ) , S ( n , N ) , Q ( n , N ) are respectively the pre-processing time, space, and query time of a data structure for longest common extensions (LCE) on SLPs. Given the Lempel-Ziv 78 encoding of size s for w, the second algorithm runs in O ( s log ź s ) time and space.

The position heap of a trie

The position heap is a text indexing structure for a single text string, recently proposed by Ehrenfeucht et al. [Position heaps: A simple and dynamic text indexing data structure, Journal of Discrete Algorithms, 9(1):100-121, 2011]. In this paper we introduce the position heap for a set of strings, and propose an efficient algorithm to construct the position heap for a set of strings which is given as a trie. For a fixed alphabet our algorithm runs in time linear in the size of the trie. We also show that the position heap can be efficiently updated after addition/removal of a leaf of the input trie.

Constructing LZ78 tries and position heaps in linear time for large alphabets

We propose the first linear time algorithm to compute LZ78 trie over an integer alphabet.We propose a linear time algorithm to construct the position heap of a trie.LZ78 tries and position heaps can be superimposed on the corresponding suffix trees.Both of them can be computed by nearest marked ancestor queries on suffix trees. We present the first worst-case linear-time algorithm to compute the Lempel-Ziv 78 factorization of a given string over an integer alphabet. Our algorithm is based on nearest marked ancestor queries on the suffix tree of the given string. We also show that the same technique can be used to construct the position heap of a set of strings in worst-case linear time, when the set of strings is given as a trie.

Longest Common Abelian Factors and Large Alphabets

Two strings X and Y are considered Abelian equal if the letters of X can be permuted to obtain Y (and vice versa). Recently, Alatabbi et al. (2015) considered the longest common Abelian factor problem in which we are asked to find the length of the longest Abelian-equal factor present in a given pair of strings. They provided an algorithm that uses \(O(\sigma n^2)\) time and \(O(\sigma n)\) space, where n is the length of the pair of strings and \(\sigma \) is the alphabet size. In this paper we describe an algorithm that uses \(O(n^2\log ^2n\log ^*n)\) time and \(O(n\log ^2n)\) space, significantly improving Alatabbi et al.’s result unless the alphabet is small. Our algorithm makes use of techniques for maintaining a dynamic set of strings under split, join, and equality testing (Melhorn et al., Algorithmica 17(2), 1997).

Faster Lyndon Factorization Algorithms for SLP and LZ78 Compressed Text

We present two efficient algorithms which, given a compressed representation of a string w of length N, compute the Lyndon factorization of w. Given a straight line program (SLP)

Inferring strings from Lyndon factorization

\mathcal{S}

Inferring Strings from Lyndon Factorization

of size n and height h that describes w, the first algorithm runs in O(nh(n + logN logn)) time and O(n 2) space. Given the Lempel-Ziv 78 encoding of size s for w, the second algorithm runs in O(s logs) time and space.

On the Size of Lempel-Ziv and Lyndon Factorizations

The Lyndon factorization of a string w is a unique factorization \(\ell_1^{p_1}, \ldots, \ell_m^{p_m}\) of w s.t. l1, …, l m is a sequence of Lyndon words that is monotonically decreasing in lexicographic order. In this paper, we consider the reverse-engineering problem on Lyndon factorization: Given a sequence S = ((s 1, p 1), …, (s m , p m )) of ordered pairs of positive integers, find a string w whose Lyndon factorization corresponds to the input sequence S, i.e., the Lyndon factorization of w is in a form of \(\ell_1^{p_1}, \ldots, \ell_m^{p_m}\) with |l i | = s i for all 1 ≤ i ≤ m. Firstly, we show that there exists a simple O(n)-time algorithm if the size of the alphabet is unbounded, where n is the length of the output string. Secondly, we present an O(n)-time algorithm to compute a string over an alphabet of the smallest size. Thirdly, we show how to compute only the size of the smallest alphabet in O(m) time. Fourthly, we give an O(m)-time algorithm to compute an O(m)-size representation of a string over an alphabet of the smallest size. Finally, we propose an efficient algorithm to enumerate all strings whose Lyndon factorizations correspond to S.

Faster Online Elastic Degenerate String Matching

The Lyndon factorization of a string w is a unique factorization \(\ell_1^{p_1}, \ldots, \ell_m^{p_m}\) of w s.t. l1, …, l m is a sequence of Lyndon words that is monotonically decreasing in lexicographic order. In this paper, we consider the reverse-engineering problem on Lyndon factorization: Given a sequence S = ((s 1, p 1), …, (s m , p m )) of ordered pairs of positive integers, find a string w whose Lyndon factorization corresponds to the input sequence S, i.e., the Lyndon factorization of w is in a form of \(\ell_1^{p_1}, \ldots, \ell_m^{p_m}\) with |l i | = s i for all 1 ≤ i ≤ m. Firstly, we show that there exists a simple O(n)-time algorithm if the size of the alphabet is unbounded, where n is the length of the output string. Secondly, we present an O(n)-time algorithm to compute a string over an alphabet of the smallest size. Thirdly, we show how to compute only the size of the smallest alphabet in O(m) time. Fourthly, we give an O(m)-time algorithm to compute an O(m)-size representation of a string over an alphabet of the smallest size. Finally, we propose an efficient algorithm to enumerate all strings whose Lyndon factorizations correspond to S.