Reconstruction of Sets of Strings From Prefix/Suffix Compositions
Ryan Gabrys, Srilakshmi Pattabiraman, Olgica Milenković
Abstract
The problem of reconstructing strings from substring information has found many applications due to its importance in genomic data sequencing and DNA- and polymer-based data storage. One important paradigm requires reconstructing mixtures of strings based on the union of compositions of their prefixes and suffixes, generated by mass spectrometry devices. We describe new coding methods that allow for unique joint reconstruction of subsets of strings selected from a code and provide upper and lower bounds on the asymptotic rate of the underlying codebooks. Our code constructions combine properties of binary <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B_{h}$ </tex-math></inline-formula> and Dyck strings that can be extended to accommodate missing substrings in the pool. As auxiliary results, we present simple entropy upper bounds for binary <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B_{h}$ </tex-math></inline-formula> codes and an improved bound for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h=4$ </tex-math></inline-formula> , and also describe errors that arise during mass spectrometry.