The Gapped $k$-Deck Problem
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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2022
- DOI:
- 10.48550/arXiv.2201.12671
- arXiv:
- arXiv:2201.12671
- Bibcode:
- 2022arXiv220112671G
- Keywords:
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- Mathematics - Combinatorics