Efficient Data Structures for Range Shortest Unique Substring Queries

Abstract

Let T[1,n] be a string of length n and T[i,j] be the substring of T starting at position i and ending at position j. A substring T[i,j] of T is a repeat if it occurs more than once in T; otherwise, it is a unique substring of T. Repeats and unique substrings are of great interest in computational biology and information retrieval. Given string T as input, the Shortest Unique Substring problem is to find a shortest substring of T that does not occur elsewhere in T. In this paper, we introduce the range variant of this problem, which we call the Range Shortest Unique Substring problem. The task is to construct a data structure over T answering the following type of online queries efficiently. Given a range [α,β], return a shortest substring T[i,j] of T with exactly one occurrence in [α,β]. We present an O(nlogn)-word data structure with O(logwn) query time, where w=Ω(logn) is the word size. Our construction is based on a non-trivial reduction allowing for us to apply a recently introduced optimal geometric data structure [Chan et al., ICALP 2018]. Additionally, we present an O(n)-word data structure with O(nlogϵn) query time, where ϵ>0 is an arbitrarily small constant. The latter data structure relies heavily on another geometric data structure [Nekrich and Navarro, SWAT 2012].

Internal shortest absent word queries in constant time and linear space

“…. 14] is 3; a shortest absent word is 7,9], FTR [2][9] = 0 < 7 so the length of a shortest absent word is 2; a shortest absent word is [9] = 0 tells us that the minimum in this range came from index k = 9.…”

“…We consider T , SP 2 and EP 2 from Example 5. Let [a, b] = [2,7]. The length of a shortest absent word of T [2 .…”

“…, σ} and 1 Supported by the Israel Science Foundation grant 592/17. 2 Supported by the Ministry of Science and Higher Education of the Russian Federation (Ural Mathematical Center project No. 075-02-2021-1387)…”

“…In the related range shortest unique substring problem, defined by Abedin et al [2], the task is to construct a data structure over T to be able to answer the following type of online queries efficiently. Given a range [i, j], return a shortest string with exactly one occurrence (starting position) in [i, j].…”

“…Abedin et al presented a data structure of size O(n log n) supporting O(log w n)-time queries, where w = Θ(log n) is the word size. Additionally, Abedin et al [2] presented a data structure of size O(n) supporting O( √ n log n)-time queries, where is an arbitrarily small positive constant.…”