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Sequence Reconstruction under Channels with Multiple Bursts of Insertions or Deletions

Zhaojun Lan, Yubo Sun, Wenjun Yu, Gennian Ge

TL;DR

In contrast to burst-insertion balls, it is proved that the size of a burst-deletion ball is dependent on its chosen center, and a reconstruction algorithm with linear runtime complexity is proposed to reconstruct the correct transmitted sequence.

Abstract

The sequence reconstruction problem involves a model where a sequence is transmitted over several identical channels. This model investigates the minimum number of channels required for the unique reconstruction of the transmitted sequence. Levenshtein established that this number exceeds the maximum size of the intersection between the error balls of any two distinct transmitted sequences by one. In this paper, we consider channels subject to multiple bursts of insertions and multiple bursts of deletions, respectively, where each burst has an exact length of value b. We provide a complete solution for the insertion case while partially addressing the deletion case.

Sequence Reconstruction under Channels with Multiple Bursts of Insertions or Deletions

TL;DR

In contrast to burst-insertion balls, it is proved that the size of a burst-deletion ball is dependent on its chosen center, and a reconstruction algorithm with linear runtime complexity is proposed to reconstruct the correct transmitted sequence.

Abstract

The sequence reconstruction problem involves a model where a sequence is transmitted over several identical channels. This model investigates the minimum number of channels required for the unique reconstruction of the transmitted sequence. Levenshtein established that this number exceeds the maximum size of the intersection between the error balls of any two distinct transmitted sequences by one. In this paper, we consider channels subject to multiple bursts of insertions and multiple bursts of deletions, respectively, where each burst has an exact length of value b. We provide a complete solution for the insertion case while partially addressing the deletion case.
Paper Structure (15 sections, 31 theorems, 47 equations, 2 algorithms)

This paper contains 15 sections, 31 theorems, 47 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Bursts of Insertions
  3. Bursts of Deletions
  4. Problem Descriptions
  5. Our Contributions
  6. Notations
  7. Burst-Insertion Channels
  8. The Size of a Radius-$t$ $b$-Burst-Insertion Ball
  9. The Maximum Intersection Size Between Two Radius-$t$ $b$-Burst-Insertion Balls
  10. The Reconstruction Algorithm for Burst-Insertion Errors
  11. Burst-Deletion Channels
  12. The Size of a Radius-$t$ $b$-Burst-Deletion Ball
  13. The Maximum Intersection Size Between Two Radius-$t$ $b$-Burst-Deletion Balls
  14. The Reconstruction Algorithm for Burst-Deletion Errors
  15. Conclusion

Key Result

Theorem 3.2

For $n,t\geq 0$, $q\geq2$, $b\geq 1$, and $\boldsymbol{x}\in\Sigma_q^n$, we have $|\mathcal{I}_{t,b}(\boldsymbol{x})|= \delta_{q,b}(n,t)$, where Then, we get $I_{q,b}(n,t)= \delta_{q,b}(n,t)$, where $I_{q,b}(n,t) := \max\{|\mathcal{I}_{t,b}(\boldsymbol{x})|: \boldsymbol{x} \in \Sigma_q^n\}$.

Theorems & Definitions (41)

  • Claim 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Corollary 3.5
  • Remark 3.6
  • Remark 3.7
  • Theorem 3.8
  • Remark 3.9
  • Lemma 3.10
  • ...and 31 more