String Reconstruction from Substring Compositions

Reconstructing Mixtures of Coded Strings from Prefix and Suffix Compositions

“…It is currently not known which combinations of coded binary strings can be distinguished from each other based on prefixsufix masses and for which code rates is it possible to perform unique multistring reconstruction. In a related research direction, the problem of reconstructing a string from its MS/MS output was considered in [7], under the name of string reconstruction from its substring composition multiset. The composition of any binary string is the number of 0s and the number of 1s in the string.…”

“…The substring composition multiset, Cpsq of a string s is the multiset of compositions of all possible substrings of the string s. As an illustration, the set of all substrings of 001 is t0, 0, 1, 00, 01, 001u, and the substring composition multiset of 001 equals t0 1 , 0 1 , 1 1 , 0 2 , 0 1 1 1 , 0 2 1 1 u. Two modeling assumptions are used in [7] and subsequent works [8]- [10]: a) Using MS/MS measurements, one can uniquely infer the composition of a polymer substring from its mass; and b) When a polymer is broken down for mass spectrometry analysis, the masses of all its substrings are observed with identical frequency.…”

“…Under the above modeling assumptions, the authors of [7] established that strings are uniquely reconstructable up to reversal provided that the length of the strings n is one less than a prime or twice a prime, or n ď 7. The work [8]- [10] demonstrated that logarithmic code redundancy can ensure unique reconstruction of single strings drawn from codebooks based on Bertrand-Catalan paths or Reed-Solomon-like constructions.…”

Reconstruction and Error-Correction Codes for Polymer-Based Data Storage

“…In an earlier line of work, the authors of [7] introduced the problem of binary string reconstruction from its substring composition multiset to address the issue of MS/MS readout analysis. The substring composition multiset of a binary string is obtained by writing out all substrings of the string of all possible length and then representing each substring by its composition.…”

“…Note that composition multisets ignore information about the actual order of the bits and may hence be seen as only capturing the information about the "mass" or "weight" of the string. The problem addressed in [7] was to determine for which string lengths may one guarantee unique reconstruction from an error-free composition multiset up to string reversal. The main results of [7,Theorem 17,18,20] asserts that binary strings of length ≤ 7, one less than a prime, or one less than twice a prime are uniquely reconstructable up to reversal.…”