Explicit Combinatoric Structures of Palindromes and Chromatic Number of Restriction Graphs

TL;DR

This paper establishes tight combinatorial bounds for reconstructing strings from their palindromic fingerprints, including the minimal alphabet size needed, and introduces specific strings demonstrating these bounds.

Contribution

It provides the first tight bounds on the minimal alphabet size for palindrome fingerprint reconstruction and constructs shortest strings that require a large alphabet for reconstruction.

Findings

Tight bounds for the minimal alphabet size in palindrome fingerprint reconstruction.

Construction of shortest strings requiring at least k characters for reconstruction.

Resolution of an open problem from prior research.

Abstract

The palindromic fingerprint of a string $S[1\ldots n]$ is the set $PF(S) = \{(i,j)~|~ S[i\ldots j] \textit{ is a maximal }\\ \textit{palindrome substring of } S\}$. In this work, we consider the problem of string reconstruction from a palindromic fingerprint. That is, given an input set of pairs $PF \subseteq [1\ldots n] \times [1\ldots n]$ for an integer $n$, we wish to determine if $PF$ is a valid palindromic fingerprint for a string $S$, and if it is, output a string $S$ such that $PF= PF(S)$. I et al. [SPIRE2010] showed a linear reconstruction algorithm from a palindromic fingerprint that outputs the lexicographically smallest string over a minimum alphabet. They also presented an upper bound of $\mathcal{O}(\log(n))$ for the maximal number of characters in the minimal alphabet. In this paper, we show tight combinatorial bounds for the palindromic fingerprint reconstruction problem. We present the string $S_k$, which is the shortest string whose fingerprint $PF(S_k)$ cannot be reconstructed using less than $k$ characters. The results additionally solve an open problem presented by I et al.