Tight Bounds for the Number of Absent Subsequences

TL;DR

This paper establishes tight bounds on the number of absent subsequences of a given length in words with a fixed universality index, providing exact counts and efficient enumeration algorithms.

Contribution

It introduces tight bounds and exact formulas for absent subsequences and develops optimal algorithms for enumerating subsequences efficiently.

Findings

Derived tight bounds for absent subsequences of length k.

Provided exact count formulas for minimal absent subsequences.

Developed linear delay and constant delay enumeration algorithms.

Abstract

A {\em subsequence} of a word $w$ is a word $u$ that can be obtained by deleting some letters from $w$ while maintaining the relative order of the remaining letters, e.g., $\mathtt{lala}$ is a subsequence of $\mathtt{alfalfa}$. A word, over some alphabet $\Sigma$, which has all possible words of length $\iota$ over $\Sigma$ as subsequences is called $\iota$-universal, and the largest $\iota$ for which this holds is called the universality index of $w$, and denoted $\iota(w)$. Moreover, words that are not subsequences of $w$ are called absent subsequences (AS) of $w$, and their investigation was started in (Kosche et al., 2022). In this paper, we present tight bounds on the number of AS of a given length $k$ among all words with the same universality index $\iota$. For both the lower and upper bound, we construct words that have, respectively, a minimal and maximal number of absent subsequences of the respective length $k$, and, in the case of the lower bound, we provide the exact number of missing subsequences as a closed form. Finally, we present efficient enumeration algorithms for the set of subsequences of given length of a word: we give a novel, optimal enumeration algorithm with output linear delay of this set of subsequences, with preprocessing time $O(|w|)$, which is further improved to an incremental enumeration algorithm with $O(1)$ delay of this set of subsequences, with preprocessing time $O(|w|)$.