A New Algebraic Approach for String Reconstruction From Substring Compositions

In this paper, we propose a new algorithm for the problem of string reconstruction from its substring composition multiset. Motivated by applications in polymer-based data storage for recovering strings from tandem mass-spectrometry sequencing, the proposed algorithm leverages the equivalent polynomial formulation of the problem which facilitates efficient parallel implementation. The computational complexity of the proposed reconstruction algorithm is upper bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$6.5n^{2}$ </tex-math></inline-formula> finite field operations, where the field size is upper bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10n$ </tex-math></inline-formula>, implying that the computational complexity is upper bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$6.5n^{2}(3.22+\log {n})$ </tex-math></inline-formula> binary operations. Furthermore, it allows parallelization leading to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n \log n)$ </tex-math></inline-formula> reconstruction latency. We characterize sufficient conditions for a length n binary string that guarantee the string’s reconstruction time complexity to be bounded polynomially. Moreover, the sufficient conditions on binary strings that guarantee reconstruction in polynomial time are more general than the conditions for the algorithm by Acharya et al. This is used to construct new codebooks of reconstruction codes that have efficient encoding procedures, and are larger, by at least a linear factor in size, compared to the previously best known construction by Pattabiraman et al., (2023).