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Sequence Reconstruction for Sticky Insertion/Deletion Channels

Van Long Phuoc Pham, Yeow Meng Chee, Kui Cai, Van Khu Vu

TL;DR

This work addresses sequence reconstruction over the $(t,s)$-sticky-insdel channel, where at most $t$ sticky insertions and $s$ sticky deletions are possible while the run structure is preserved. It develops the asymmetric error ball framework via $\mathcal{A}_{t,s}({\boldsymbol u})$ and $\mathcal{B}_{t,s}({\boldsymbol x})$, derives a recursive and a closed-form expression for the minimal output count $N_{t,s}(r)$ needed for unique reconstruction, and provides a polynomial-time algorithm to recover the original sequence from $N_{t,s}(r)$ outputs. The results generalize the known $s=0$ case and yield exact reconstruction guarantees, with explicit per-run statistics guiding the algorithm. The findings have practical impact for reliable data retrieval in storage systems modeled by sticky insertions/deletions, such as racetrack memories and DNA-based storage.

Abstract

The sequence reconstruction problem for insertion/deletion channels has attracted significant attention owing to their applications recently in some emerging data storage systems, such as racetrack memories, DNA-based data storage. Our goal is to investigate the reconstruction problem for sticky-insdel channels where both sticky-insertions and sticky-deletions occur. If there are only sticky-insertion errors, the reconstruction problem for sticky-insertion channel is a special case of the reconstruction problem for tandem-duplication channel which has been well-studied. In this work, we consider the $(t, s)$-sticky-insdel channel where there are at most $t$ sticky-insertion errors and $s$ sticky-deletion errors when we transmit a message through the channel. For the reconstruction problem, we are interested in the minimum number of distinct outputs from these channels that are needed to uniquely recover the transmitted vector. We first provide a recursive formula to determine the minimum number of distinct outputs required. Next, we provide an efficient algorithm to reconstruct the transmitted vector from erroneous sequences.

Sequence Reconstruction for Sticky Insertion/Deletion Channels

TL;DR

This work addresses sequence reconstruction over the -sticky-insdel channel, where at most sticky insertions and sticky deletions are possible while the run structure is preserved. It develops the asymmetric error ball framework via and , derives a recursive and a closed-form expression for the minimal output count needed for unique reconstruction, and provides a polynomial-time algorithm to recover the original sequence from outputs. The results generalize the known case and yield exact reconstruction guarantees, with explicit per-run statistics guiding the algorithm. The findings have practical impact for reliable data retrieval in storage systems modeled by sticky insertions/deletions, such as racetrack memories and DNA-based storage.

Abstract

The sequence reconstruction problem for insertion/deletion channels has attracted significant attention owing to their applications recently in some emerging data storage systems, such as racetrack memories, DNA-based data storage. Our goal is to investigate the reconstruction problem for sticky-insdel channels where both sticky-insertions and sticky-deletions occur. If there are only sticky-insertion errors, the reconstruction problem for sticky-insertion channel is a special case of the reconstruction problem for tandem-duplication channel which has been well-studied. In this work, we consider the -sticky-insdel channel where there are at most sticky-insertion errors and sticky-deletion errors when we transmit a message through the channel. For the reconstruction problem, we are interested in the minimum number of distinct outputs from these channels that are needed to uniquely recover the transmitted vector. We first provide a recursive formula to determine the minimum number of distinct outputs required. Next, we provide an efficient algorithm to reconstruct the transmitted vector from erroneous sequences.
Paper Structure (10 sections, 5 theorems, 43 equations, 5 algorithms)

This paper contains 10 sections, 5 theorems, 43 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximum size of sticky insdel ball
  4. Minimum number of distinct outputs
  5. Lower bound
  6. Upper Bound
  7. Reconstruction Algorithm
  8. Naive Approach
  9. Optimized Approach
  10. Conclusion

Key Result

Lemma 1

For ${\boldsymbol u} =u_1u_2\ldots u_r \in \mathbb{Z}_+^r$, let ${\boldsymbol u} + s = (u_1+s, u_2+s, ..., u_r+s)$. The following inequality holds,

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Lemma 4
  • Claim 2
  • proof
  • Claim 3
  • proof