Data Structures for Computing Unique Palindromes in Static and Non-Static Strings - Mieno, Takuya; Funakoshi, Mitsuru (Algorithmica, 2024)
A palindromic substring T [ i .. j ] of a string T is said to be a shortest unique palindromic substring (SUPS) in T for an interval [ p , q ] if T [ i .. j ] is a shortest palindromic substring such that T [ i .. j ] occurs only once in T , and [ i , j ] contains [ p , q ]. The SUPS problem is, given a string T of length n , to construct a data structure that can compute all the SUPSs for any given query interval. It is known that any SUPS query can be answered in O ( α ) time after O ( n )-time preprocessing, where α is the number of SUPSs to output (Inoue in J Discrete Algorithms 52-53:122–132, 2018). In this paper, we first show that α is at most 4, and the upper bound is tight. We also show that the total sum of lengths of minimal unique palindromic substrings of string T , which is strongly related to SUPSs, is O ( n ). Then, we present the first O ( n )-bits data structures that can answer any SUPS query in constant time. Also, we present an algorithm to solve the SUPS problem for a sliding window that can answer any query in O ( log log W ) time and update data structures in amortized O ( log σ + log log W ) time, where W is the size of the window, and σ is the alphabet size. Furthermore, we consider the SUPS problem in the after-edit model and present an efficient algorithm. Namely, we present an algorithm that uses O ( n ) time for preprocessing and answers any k SUPS queries in O ( log n log log n + k log log n ) time after single character substitution. Finally, as a by-product, we propose a fully-dynamic data structure for range minimum queries (RmQs) with a constraint where the width of each query range is limited to poly-logarithmic. The constrained RmQ data structure can answer such a query in constant time and support a single-element edit operation in amortized constant time.