Palindromic Trees for a Sliding Window and Its Applications

Abstract

The palindromic tree (a.k.a. eertree) for a string S of length n is a tree-like data structure that represents the set of all distinct palindromic substrings of S, using O(n) space [Rubinchik and Shur, 2018]. It is known that, when S is over an alphabet of size σ and is given in an online manner, then the palindromic tree of S can be constructed in O(n log σ) time with O(n) space. In this paper, we consider the sliding window version of the problem: For a fixed window length d, we propose two algorithms to maintain the palindromic tree of size O(d) for every sliding window S[i..i+d−1] over S, one running in O(n log σ ′ ) time with O(d) space where σ ′ ≤ d is the maximum number of distinct characters in the windows, and the other running in O(n+dσ) time with dσ +O(d) space. We also present applications of our algorithms for computing minimal unique palindromic substrings (MUPS) and for computing minimal absent palindromic words (MAPW) for a sliding window.which enumerate all distinct palindromes in a given sting of length n over an integer alphabet of size σ = n O(1) . For the same problem in the online model, Kosolobov et al. [13] proposed an O(n log σ)-time and O(n)-space algorithm for a general ordered alphabet. Kosolobov et al.'s algorithm is a combination of Manacher's algorithm and Ukkonen's online suffix tree construction algorithm [21]. Rubinchik and Shur [19] proposed a new data structure called eertree, which permits efficient access to distinct palindromes in a string without storing the string itself. Eertrees can be utilized for solving problems related to palindromic structures, such as the palindrome counting problem and the palindromic factorization problem [19]. The size of the eertree of S is linear in the number p S of distinct palindromes in S [19]. It is known that p S is at most |S|+ 1, and that it can be much smaller than the length |S| of the string, e.g., for S = abc n/3 , p S = 4 since all distinct palindromes in S are a, b, c, and the empty string. Thus, the size of the eertree of S can be much smaller than that of the suffix tree of S which is Θ(n). Therefore, it is of significance if one can build eertrees without suffix trees. Rubinchik and Shur [19] indeed proposed an online eertree construction algorithm running in O(n log σ) time without suffix trees.Recently, a concept of palindromic structures called minimal unique palindromic substrings (MUPS ) is introduced by Inoue et al. [12]. A palindromic substring w = S[i..j] of a string S is called a MUPS of S if w occurs in S exactly once, and S[i+1..j −1] occurs at least twice in S. MUPSs are utilized for solving the shortest unique palindromic substring (SUPS ) problem [12], which is motivated by an application in molecular biology. Watanabe et al. [22] proposed an algorithm to solve the SUPS problem based on the run-length encoding (RLE ) version of eertrees, named e 2 rtre 2 .