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Reconstruction of multiple strings of constant weight from prefix-suffix compositions

Yaoyu Yang, Zitan Chen

TL;DR

This work provides necessary and sufficient conditions for the multiset of strings so that unique reconstruction up to reversal is possible.

Abstract

Motivated by studies of data retrieval in polymer-based storage systems, we consider the problem of reconstructing a multiset of binary strings that have the same length and the same weight from the compositions of their prefixes and suffixes of every possible length. We provide necessary and sufficient conditions for which unique reconstruction up to reversal of the strings is possible. Additionally, we present two algorithms for reconstructing strings from the compositions of prefixes and suffixes of constant-length constant-weight strings.

Reconstruction of multiple strings of constant weight from prefix-suffix compositions

TL;DR

This work provides necessary and sufficient conditions for the multiset of strings so that unique reconstruction up to reversal is possible.

Abstract

Motivated by studies of data retrieval in polymer-based storage systems, we consider the problem of reconstructing a multiset of binary strings that have the same length and the same weight from the compositions of their prefixes and suffixes of every possible length. We provide necessary and sufficient conditions for which unique reconstruction up to reversal of the strings is possible. Additionally, we present two algorithms for reconstructing strings from the compositions of prefixes and suffixes of constant-length constant-weight strings.

Paper Structure

This paper contains 11 sections, 19 theorems, 34 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Necessary and sufficient conditions for unique reconstruction
  4. Necessity
  5. Sufficiency
  6. Reconstruction algorithms
  7. An algorithm that outputs a multiset of strings compatible with $M$
  8. An algorithm that outputs all multisets of strings compatible with $M$
  9. Concluding remarks
  10. Proof of Lemma \ref{['le:half-diff-0']}
  11. Proof of Lemma \ref{['le:half-bar-w-0']}

Key Result

Proposition 1

Let $[k_1,k_2]\subset\llbracket n\rrbracket$ be a maximal interval between $f_{m}$ and $f_{m^*}$. If $k_2 + 1 < n-k_2$, i.e., $k_2 < \lfloor n/2\rfloor$, then $[n-k_2,n-k_1]$ is another maximal interval between $f_{m}$ and $f_{m^*}$. Similarly, if $k_1 > \lceil n/2 \rceil$, then $[n-k_2,n-k_1]$ is a

Theorems & Definitions (56)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 46 more