Abstract
We revisit the exact shortest unique substring (SUS) finding problem, and propose its approximate version where mismatches are allowed, due to its applications in subfields such as computational biology. We design a generic in-place framework that fits to solve both the exact and approximate k-mismatch SUS finding, using the minimum 2n memory words plus n bytes space, where n is the input string size. By using the in-place framework, we can find the exact and approximate k-mismatch SUS for every string position using a total of O(n) and \(O(n^2)\) time, respectively, regardless of the value of k. Our framework does not involve any compressed or succinct data structures and thus is practical and easy to implement.
Authors are listed in alphabetical order. Due to the page limit, a run example for Sect. 5 and all missing proofs and pseudocode can be found in the arXiv version of this paper.
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Notes
- 1.
It is our arbitrary decision to resolve the ties by picking the rightmost choice. Our solution can also be easily modified to find the leftmost choice.
- 2.
- 3.
In actual run, \(t_i^{-1}\) saves the largest number in that list, as we will see more clearly later.
- 4.
For each \(j' < j\), if \(I_{j'}\) covers i, \(I_j\) would also cover i; in such a case, \(B[j] + 1 \ge B[j'] + 1\). For each \(j' \in [{\texttt {pred}}[i], i-1]\), \(I_{j'}\) is longer than \(I_i\).
- 5.
Intuitively, \({\texttt {pred}}\) defines the shortcuts so that we can skip some intervals in \(I_{<i}\) to compute \(t_i\).
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Hon, WK., Thankachan, S.V., Xu, B. (2015). An In-place Framework for Exact and Approximate Shortest Unique Substring Queries. In: Elbassioni, K., Makino, K. (eds) Algorithms and Computation. ISAAC 2015. Lecture Notes in Computer Science(), vol 9472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48971-0_63
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DOI: https://doi.org/10.1007/978-3-662-48971-0_63
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