The Gapped $k$-Deck Problem

TL;DR

This work initiates the study of reconstructing binary strings from s-gapped k-decks, introducing the central quantities $G_s(k)$ and $G(k)$ and showing that a padded Morse-Thue construction yields a constructive upper bound $G_2(k) \le 4(2^k - 1) - 2$. It provides a general bound for $G_s(k)$ and demonstrates why naive ungapped-deck techniques fail for the gapped setting. Building on Dudik and Schulman, the authors develop a counting framework with wildcards that leads to a significantly tighter asymptotic bound on $G(k)$, namely $G(k) \le 1.482\cdot 1.26^k k^3 \log_3(k/3) - 2$ for large $k$, and related refinements. Overall, the paper advances both constructive and analytic upper bounds for gapped k-deck reconstruction, with implications for readout models in molecular storage where gaps occur in subsequence data.

Abstract

The $k$-deck problem is concerned with finding the smallest positive integer $S(k)$ such that there exist at least two strings of length $S(k)$ that share the same $k$-deck, i.e., the multiset of subsequences of length $k$. We introduce the new problem of gapped $k$-deck reconstruction: For a given gap parameter $s$, we seek the smallest positive integer $G_s(k)$ such that there exist at least two distinct strings of length $G_s(k)$ that cannot be distinguished based on a "gapped" set of $k$-subsequences. The gap constraint requires the elements in the subsequences to be at least $s$ positions apart within the original string. Our results are as follows. First, we show how to construct sequences sharing the same $2$-gapped $k$-deck using a nontrivial modification of the recursive Morse-Thue string construction procedure. This establishes the first known constructive upper bound on $G_2(k)$. Second, we further improve this bound using the approach by Dudik and Schulman.