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 in 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(n log n)-word data structure with O(log w n) query time, where w = Ω(log n) is the word size. Our construction is based on a non-trivial reduction allowing us to apply a recently introduced optimal geometric data structure [Chan et al. ICALP 2018].

Computing Minimal Unique Substrings for a Sliding Window

“…Although not mentioned explicitly in [16], the size of their data structure (except for the input string) and query time can be respectively written as O(m) space and O( √ log m∕ log log m + occ) time with respect to the number m of MUSs of the input string T. Note that all the above algorithms for the SUS problems compute all MUSs of the given string (or some data structure which is essentially equivalent to MUSs) in the preprocessing. We also refer to [1,3,7,17] for related results on the SUS problems.…”

Efficient Data Structures for Range Shortest Unique Substring Queries

“…We conclude this paper in Section 4 with some future proposals. An early version of this paper appeared as [33]. When compared to that early version ( [33]), Theorem 2 is new.…”