The Heaviest Induced Ancestors Problem: Better Data Structures and Applications

Shortest Unique Palindromic Substring Queries in Semi-dynamic Settings

“…They tackled the problem of computing the longest common substring (LCS) for two strings in the after-edit model, and proposed an algorithm running in polylogarithmic time. Afterward, Abedin et al [1] improved the complexities. Also, the problems of computing the longest Lyndon substring [29], the longest palindrome [15], and the set of MUPSs [14] were considered in the after-edit model.…”

“…The operator + denotes the concatenation of strings. For this string and query interval [18,18] (highlighted in blue in the figure), SUPS S ( [18,18]) = { [1,19], [4,22], [16,34], [18,36]} holds. Note that palindromes S[2..18], S [5..21], and S [17..33], which are shorter than 19 and cover the interval [18,18], are not unique since each of them has another occurrence in the artificial gadgets concatenated by + operators.…”

“…Also, we can determine whether there is an expansion of M l , which covers [p, q] by looking at the maximal palindrome centered at c l = center(M l ). Precisely, if the maximal palindrome centered at c l covers [p, q], its shortest contraction covering [p, q] is the only candidate of type (1). Otherwise, there is no SUPS of type (1).…”

“…Precisely, if the maximal palindrome centered at c l covers [p, q], its shortest contraction covering [p, q] is the only candidate of type (1). Otherwise, there is no SUPS of type (1). We emphasize that we can also compute the maximal palindrome centered at c l by querying bidirectional LCE once, without the precomputed maximal palindromes.…”

“…A substring T [i..j] of a string T is said to be a shortest unique palindromic substring (in short, SUPS) for an interval [p, q] if T [i..j] is the shortest substring such that T [i..j] is a palindrome, T [i..j] occurs only once in T , and the occurrence contains [p, q], i.e., [p, q] ⊆ [i, j]. The notion of SUPS was introduced by Inoue et al [19] in 2018 1 , motivated by bioinformatics: for example, in DNA/RNA sequences, the presence of unique palindromic sequences can affect the immunostimulatory activities of oligonucleotides [21,31]. Given a string T of length n, the SUPS problem is to construct a data structure that can compute all SUPSs for any given query interval.…”