Multi-strand Reconstruction from Substrings

TL;DR

This paper studies the problem of reconstructing multiple strings simultaneously from their shared substrings of fixed length, providing bounds and constructions for codes that enable accurate reconstruction.

Contribution

It introduces the concept of multi-strand -reconstruction codes, establishes lower bounds on substring length for reconstruction, and presents two near-optimal code constructions.

Findings

Lower bounds on  for successful reconstruction

Two code constructions with rates approaching 1

Asymptotic analysis of code rates and bounds

Abstract

The problem of string reconstruction based on its substrings spectrum has received significant attention recently due to its applicability to DNA data storage and sequencing. In contrast to previous works, we consider in this paper a setup of this problem where multiple strings are reconstructed together. Given a multiset $S$ of strings, all their substrings of some fixed length $\ell$, defined as the $\ell$-profile of $S$, are received and the goal is to reconstruct all strings in $S$. A multi-strand $\ell$-reconstruction code is a set of multisets such that every element $S$ can be reconstructed from its $\ell$-profile. Given the number of strings~$k$ and their length~$n$, we first find a lower bound on the value of $\ell$ necessary for existence of multi-strand $\ell$-reconstruction codes with non-vanishing asymptotic rate. We then present two constructions of such codes and show that their rates approach~$1$ for values of $\ell$ that asymptotically behave like the lower bound.