Reconstruction of Strings from their Substrings Spectrum

TL;DR

This paper investigates reconstructing strings from their substrings spectrum in noisy conditions, proposing coding strategies that ensure accurate reconstruction despite missing or erroneous substrings.

Contribution

It introduces novel code constructions and efficient encoding/decoding methods for string reconstruction under both missing and error-prone substring scenarios.

Findings

Codes with high rates approaching 1 for reliable reconstruction

Efficient encoding and decoding algorithms developed

Guaranteed reconstruction despite missing or erroneous substrings

Abstract

This paper studies reconstruction of strings based upon their substrings spectrum. Under this paradigm, it is assumed that all substrings of some fixed length are received and the goal is to reconstruct the string. While many existing works assumed that substrings are received error free, we follow in this paper the noisy setup of this problem that was first studied by Gabrys and Milenkovic. The goal of this study is twofold. First we study the setup in which not all substrings in the multispectrum are received, and then we focus on the case where the read substrings are not error free. In each case we provide specific code constructions of strings that their reconstruction is guaranteed even in the presence of failure in either model. We present efficient encoding and decoding maps and analyze the cardinality of the code constructions, while studying the cases where the rates of our codes approach 1.