Trace Reconstruction with Bounded Edit Distance

Abstract

The trace reconstruction problem studies the number of noisy samples needed to recover an unknown string x ∈ {0, 1} n with high probability, where the samples are independently obtained by passing x through a random deletion channel with deletion probability p. The problem is receiving significant attention recently due to its applications in DNA sequencing and DNA storage. Yet, there is still an exponential gap between upper and lower bounds for the trace reconstruction problem. In this paper we study the trace reconstruction problem when x is confined to an edit distance ball of radius k, which is essentially equivalent to distinguishing two strings with edit distance at most k. It is shown that n O(k) samples suffice to achieve this task with high probability.

Near-Optimal Average-Case Approximate Trace Reconstruction from Few Traces

“…Sima and Bruck [SB21] have recently studied exact trace reconstruction under an edit distance constraint. They showed that n O(k) traces suffice to distinguish between two (known) worst-case n-bit strings that are promised to have edit distance at most k from each other.…”

The Input and Output Entropies of the $k$-Deletion/Insertion Channel

“…This transmission results in several noisy copies of x, and the goal is to find the required minimum number of these noisy copies that enables the reconstruction of x with high probability or in the worst case. Theoretical bounds and other results for this problem were proved in several works such as [1], [8], [13], [17], [24], and other works also studied algorithms for the sequence reconstruction problem for channels that introduce deletion and insertion errors; see e.g., [10], [12], [26], [47].…”