The Gapped $k$-Deck Problem

Abstract

The k-deck problem is concerned with finding the smallest value Spkq of a positive integer n such that there exist at least two strings of length n that share the same k-deck, i.e., the same 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 possible value of n, G s pkq, such that there exist at least two distinct strings of length n that cannot be distinguished based on a "gapped" set of ksubsequences. The gap constraint requires the elements in the subsequences to be at least s positions apart in 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 pkq. Second, we further improve this upper bound in a nonconstructive manner using the approach by Dudik and Schulman [6]. Third, we comment on the general case s ě 2 and present a number of open problems.