The Gapped $k$-Deck Problem

TL;DR

This paper introduces the gapped $k$-deck problem, exploring the minimal string length where two distinct strings share the same gapped $k$-deck, and provides new bounds using recursive and combinatorial methods.

Contribution

It defines the gapped $k$-deck problem and offers the first constructive upper bound on $G_2(k)$, improving previous bounds with novel recursive constructions.

Findings

Constructed sequences sharing the same 2-gapped $k$-deck using Morse-Thue modifications.

Established the first known upper bound on $G_2(k)$.

Improved bounds on $G_2(k)$ using Dudik and Schulman's approach.

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.