Reconstruction of a Single String from a Part of its Composition Multiset

TL;DR

This paper investigates the problem of reconstructing strings from their composition multisets, characterizes non-uniquely reconstructable strings, and develops codes for error correction and partial information scenarios, with applications in polymer data storage.

Contribution

It provides a complete structural description of strings not uniquely reconstructable from prefix-suffix compositions and constructs optimal and error-correcting composition codes.

Findings

Existence of non-uniquely reconstructable strings for all lengths n ≥ 6.

Explicit construction of all uniquely reconstructable strings for each length n ≥ 6.

Development of composition codes capable of correcting errors and handling partial composition information.

Abstract

Motivated by applications in polymer-based data storage, we study the problem of reconstructing a string from part of its composition multiset. We give a full description of the structure of the strings that cannot be uniquely reconstructed (up to reversal) from their multiset of all of their prefix-suffix compositions. Leveraging this description, we prove that for all $n\ge 6$, there exists a string of length $n$ that cannot be uniquely reconstructed up to reversal. Moreover, for all $n\ge 6$, we explicitly construct the set consisting of all length $n$ strings that can be uniquely reconstructed up to reversal. As a by product, we obtain that any binary string can be constructed using Dyck strings and Catalan-Bertrand strings. For any given string $\bm{s}$, we provide a method to explicitly construct the set of all strings with the same prefix-suffix composition multiset as $\bm{s}$, as well as a formula for the size of this set. As an application, we construct a composition code of maximal size. Furthermore, we construct two classes of composition codes which can respectively correct composition missing errors and mass reducing substitution errors. In addition, we raise two new problems: reconstructing a string from its composition multiset when at most a constant number of substring compositions are lost; reconstructing a string when only given its compositions of substrings of length at most $r$. For each of these setups, we give suitable codes under some conditions.