Reconstructing words from a fixed palindromic length sequence

Journal of Combinatorial Theory, Series A, 2015

We regard a finite word u = u 1 u 2 · · · u n up to word isomorphism as an equivalence relation on {1, 2, . . . , n} where i is equivalent to j if and only if u i = u j . Some finite words (in particular all binary words) are generated by palindromic relations of the form k ∼ j +i−k for some choice of 1 ≤ i ≤ j ≤ n and k ∈ {i, i + 1, . . . , j}. That is to say, some finite words u are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. In this paper we study the function µ(u) defined as the least number of palindromic relations required to generate u. We show that if x is an infinite word such that µ(u) ≤ 2 for each factor u of x, then x is ultimately periodic. On the other hand we establish the existence of non-ultimately periodic words for which µ(u) ≤ 3 for each factor u of x, and obtain a complete classification of such words on a binary alphabet (which includes the well known class of Sturmian words). In contrast, for the Thue-Morse word, we show that the function µ is unbounded.

Theoretical Computer Science, 2019

We investigate the least number of distinct palindromic sub-arrays in two-dimensional words over a finite alphabet = {a 1 , a 2 • • • , a q } for a given alphabet size q. We discuss the case for both periodic as well as aperiodic words.

Cornell University - arXiv, 2021

We investigate the scattered palindromic subwords in a finite word. We start by characterizing the words with the least number of scattered palindromic subwords. Then, we give an upper bound for the total number of palindromic subwords in a word of length n in terms of Fibonacci number Fn by proving that at most Fn new scattered palindromic subwords can be created on the concatenation of a letter to a word of length n − 1. We propose a conjecture on the maximum number of scattered palindromic subwords in a word of length n with q distinct letters. We support the conjecture by showing its validity for words where q ≥ n 2 .

ArXiv, 2019

A two-dimensional ($2$D) word is a $2$D palindrome if it is equal to its reverse and it is an HV-palindrome if all its columns and rows are $1$D palindromes. We study some combinatorial and structural properties of HV-palindromes and its comparison with $2$D palindromes. We investigate the maximum number number of distinct non-empty HV-palindromic sub-arrays in any finite $2$D word, thus, proving the conjecture given by Anisiua et al. We also find the least number of HV-palindromes in an infinite $2$D word over a finite alphabet size $q$.

ArXiv, 2017

A palindrome is a string that reads the same as its reverse, such as "aibohphobia" (fear of palindromes). Given an integer $d>0$, a $d$-near-palindrome is a string of Hamming distance at most $d$ from its reverse. We study the natural problem of identifying a longest $d$-near-palindrome in data streams. The problem is relevant to the analysis of DNA databases, and to the task of repairing recursive structures in documents such as XML and JSON. We present an algorithm that returns a $d$-near-palindrome whose length is within a multiplicative $(1+\epsilon)$-factor of the longest $d$-near-palindrome. Our algorithm also returns the set of mismatched indices of the $d$-near-palindrome, using $\mathcal{O}\left(\frac{d\log^7 n}{\epsilon\log(1+\epsilon)}\right)$ bits of space, and $\mathcal{O}\left(\frac{d\log^6 n}{\epsilon\log(1+\epsilon)}\right)$ update time per arriving symbol. We show that $\Omega(d\log n)$ space is necessary for estimating the length of longest $d$-near-pa...

International Journal of Open Problems in Computer Science and Mathematics, 2017

A palindrome is a word, or a sequence of characters which reads the same backward as forward. In this article we study palindrome words on the Arnoux Rauzy substitutions sequences.

Advances in Applied Mathematics, 2009

In this paper we prove that for any infinite word w whose set of factors is closed under reversal, the following conditions are equivalent:

Theoretical Computer Science, 2007

We study the relation between the palindromic and factor complexity of infinite words. We show that for uniformly recurrent words one has P(n) + P(n + 1) ≤ ∆C(n) + 2, for all n ∈ N. For a large class of words it is a better estimate of the palindromic complexity in terms of the factor complexity then the one presented in [2]. We provide several examples of infinite words for which our estimate reaches its upper bound. In particular, we derive an explicit prescription for the palindromic complexity of infinite words coding r-interval exchange transformations. If the permutation π connected with the transformation is given by π(k) = r + 1 − k for all k, then there is exactly one palindrome of every even length, and exactly r palindromes of every odd length.

2009

We show that there exists an uniformly recurrent infinite word whose set of factors is closed under reversal and which has only finitely many palindromic factors.

Information Processing Letters

The 2-LCPS problem, first introduced by Chowdhury et al. [Fundam. Inform., 129(4):329-340, 2014], asks one to compute (the length of) a longest palindromic common subsequence between two given strings A and B. We show that the 2-LCPS problem is at least as hard as the well-studied longest common subsequence problem for 4 strings. Then, we present a new algorithm which solves the 2-LCPS problem in O σM 2 + n time, where n denotes the length of A and B, M denotes the number of matching positions between A and B, and σ denotes the number of distinct characters occurring in both A and B. Our new algorithm is faster than Chowdhury et al.'s sparse algorithm when σ = o(log 2 n log log n).