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.

翻译：字符串$S[1\ldots n]$的回文指纹定义为集合$PF(S) = \{(i,j)~|~ S[i\ldots j] \text{ 是 } S \text{ 的最大回文子串}\}$。本文研究了从回文指纹重构字符串的问题：给定整数$n$及输入对集$PF \subseteq [1\ldots n] \times [1\ldots n]$，需判定$PF$是否为某字符串$S$的有效回文指纹，若是则输出满足$PF=PF(S)$的字符串$S$。I等人[SPIRE2010]提出了基于回文指纹的线性重构算法，该算法输出最小字母表上的字典序最小字符串，并证明了最小字母表所需字符数的上界为$\mathcal{O}(\log(n))$。本文给出了回文指纹重构问题的紧组合界，构造了字符串$S_k$——其指纹$PF(S_k)$无法用少于$k$个字符重构的最短字符串。该结果同时解决了I等人提出的一个开放问题。