Abstract
A weighted string, also known as a position weight matrix, is a sequence of probability distributions over some alphabet. We revisit the Weighted Shortest Common Supersequence (WSCS) problem, introduced by Amir et al. [SPIRE 2011], that is, the SCS problem on weighted strings. In the WSCS problem, we are given two weighted strings \(W_1\) and \(W_2\) and a threshold \(\tfrac{1}{z} \) on probability, and we are asked to compute the shortest (standard) string S such that both \(W_1\) and \(W_2\) match subsequences of S (not necessarily the same) with probability at least \(\tfrac{1}{z} \). Amir et al. showed that this problem is NP-complete if the probabilities, including the threshold \(\tfrac{1}{z} \), are represented by their logarithms (encoded in binary).
We present an algorithm that solves the WSCS problem for two weighted strings of length n over a constant-sized alphabet in \(\mathcal {O}(n^2\sqrt{z} \log {z})\) time. Notably, our upper bound matches known conditional lower bounds stating that the WSCS problem cannot be solved in \(\mathcal {O}(n^{2-\varepsilon })\) time or in \(\mathcal {O}^*(z^{0.5-\varepsilon })\) with time, where the \(\mathcal {O}^*\) notation suppresses factors polynomial with respect to the instance size (with numeric values encoded in binary), unless there is a breakthrough improving upon long-standing upper bounds for fundamental NP-hard problems (CNF-SAT and Subset Sum, respectively).
We also discover a fundamental difference between the WSCS problem and the Weighted Longest Common Subsequence (WLCS) problem, introduced by Amir et al. [JDA 2010]. We show that the WLCS problem cannot be solved in \(\mathcal {O}(n^{f(z)})\) time, for any function f(z), unless \(\mathrm {P}=\mathrm {NP}\).
Tomasz Kociumaka was supported by ISF grants no. 824/17 and 1278/16 and by an ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064).
Jakub Radoszewski and Juliusz Straszyński were supported by the “Algorithms for text processing with errors and uncertainties” project carried out within the HOMING program of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund.
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Notes
- 1.
Note that in general \(z\notin \mathcal {O}^*(1)\) unless z is encoded in unary.
- 2.
We consider the case of \(|\varSigma |=\mathcal {O}(1)\) just for simplicity. For a general alphabet, our algorithm can be modified to work in \(\mathcal {O}(n^2 |\varSigma |\sqrt{z} \log {z})\) time.
- 3.
For any two integers \(\ell \le r\), we use \([\ell \mathinner {.\,.}r]\) to denote the integer range \(\{\ell ,\ldots ,r\}\).
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Charalampopoulos, P. et al. (2019). Weighted Shortest Common Supersequence Problem Revisited. In: Brisaboa, N., Puglisi, S. (eds) String Processing and Information Retrieval. SPIRE 2019. Lecture Notes in Computer Science(), vol 11811. Springer, Cham. https://doi.org/10.1007/978-3-030-32686-9_16
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