Range Shortest Unique Substring queries

Let be a string of length n and be the substring of starting at position i and ending at position j. A substring of is a repeat if it occurs more than once in; otherwise, it is a unique substring of. Repeats and unique substrings are of great interest in computational biology and in information retrieval. Given string as input, the Shortest Unique Substring problem is to find a shortest substring of that does not occur elsewhere in. 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 answering the following type of online queries efficiently. Given a range, return a shortest substring of with exactly one occurrence in. We present an -word data structure with query time, where 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]

More Time-Space Tradeoffs for Finding a Shortest Unique Substring

We extend recent results regarding finding shortest unique substrings (SUSs) to obtain new time-space tradeoffs for this problem and the generalization of finding k-mismatch SUSs. Our new results include the first algorithm for finding a k-mismatch SUS in sublinear space, which we obtain by extending an algorithm by Senanayaka (2019) and combining it with a result on sketching by Gawrychowski and Starikovskaya (2019). We first describe how, given a text T of length n and m words of workspace, with high probability we can find an SUS of length L in O(n(L/m)logL) time using random access to T, or in O(n(L/m)log2(L)loglogσ) time using O((L/m)log2L) sequential passes over T. We then describe how, for constant k, with high probability, we can find a k-mismatch SUS in O(n1+ϵL/m) time using O(nϵL/m) sequential passes over T, again using only m words of workspace. Finally, we also describe a deterministic algorithm that takes O(nτlogσlogn) time to find an SUS using O(n/τ) words of workspace, where τ is a parameter

Space-Time Trade-Offs for the Shortest Unique Substring Problem

Given a string X[1, n] and a position k in [1, n], the Shortest Unique Substring of X covering k, denoted by S_k, is a substring X[i, j] of X which satisfies the following conditions: (i) i leq k leq j, (ii) i is the only position where there is an occurrence of X[i, j], and (iii) j - i is minimized. The best-known algorithm [Hon et al., ISAAC 2015] can find S k for all k in [1, n] in time O(n) using the string X and additional 2n words of working space. Let tau be a given parameter. We present the following new results. For any given k in [1, n], we can compute S_k via a deterministic algorithm in O(n tau^2 log n tau) time using X and additional O(n/tau) words of working space. For every k in [1, n], we can compute S_k via a deterministic algorithm in O(n tau^2 log n/tau) time using X and additional O(n/tau) words and 4n + o(n) bits of working space. For both problems above, we present an O(n tau log^{c+1} n)-time randomized algorithm that uses n/ log c n words in addition to that mentioned above, where c geq 0 is an arbitrary constant. In this case, the reported string is unique and covers k, but with probability at most n^{-O(1)}may not be the shortest. As a consequence of our techniques, we also obtain similar space-and-time tradeoffs for a related problem of finding Maximal Unique Matches of two strings [Delcher et al., Nucleic Acids Res. 1999]

Efficient data structures for range shortest unique substring queries†

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(n log n)-word data structure with O(logw n) query time, where w = Ω(log n) 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(√ n logɛ n) query time, where ɛ > 0 is an arbitrarily small constant. The latter data structure relies heavily on another geometric data structure [Nekrich and Navarro, SWAT 2012]

Substring Complexity in Sublinear Space

Shannon’s entropy is a definitive lower bound for statistical compression. Unfortunately, no such clear measure exists for the compressibility of repetitive strings. Thus, ad hoc measures are employed to estimate the repetitiveness of strings, e.g., the size z of the Lempel–Ziv parse or the number r of equal-letter runs of the Burrows-Wheeler transform. A more recent one is the size γ of a smallest string attractor. Let T be a string of length n. A string attractor of T is a set of positions of T capturing the occurrences of all the substrings of T. Unfortunately, Kempa and Prezza [STOC 2018] showed that computing γ is NP-hard. Kociumaka et al. [LATIN 2020] considered a new measure of compressibility that is based on the function S_T(k) counting the number of distinct substrings of length k of T, also known as the substring complexity of T. This new measure is defined as δ = sup{S_T(k)/k, k ≥ 1} and lower bounds all the relevant ad hoc measures previously considered. In particular, δ ≤ γ always holds and δ can be computed in O(n) time using Θ(n) working space. Kociumaka et al. showed that one can construct an O(δ log n/(δ))-sized representation of T supporting efficient direct access and efficient pattern matching queries on T. Given that for highly compressible strings, δ is significantly smaller than n, it is natural to pose the following question: Can we compute δ efficiently using sublinear working space? It is straightforward to show that in the comparison model, any algorithm computing δ using O(b) space requires Ω(n^{2-o(1)}/b) time through a reduction from the element distinctness problem [Yao, SIAM J. Comput. 1994]. We thus wanted to investigate whether we can indeed match this lower bound. We address this algorithmic challenge by showing the following bounds to compute δ: - O((n3log b)/b2) time using O(b) space, for any b ∈ [1,n], in the comparison model. - Õ(n2/b) time using Õ(b) space, for any b ∈ [√n,n], in the word RAM model. This gives an Õ(n^{1+ε})-time and Õ(n^{1-ε})-space algorithm to compute δ, for any 0 < ε ≤ 1/2. Let us remark that our algorithms compute S_T(k), for all k, within the same complexities