Internal Shortest Absent Word Queries in Constant Time and Linear Space

TL;DR

This paper introduces a data structure that preprocesses a string to enable constant-time retrieval of the shortest absent substring within any range, using linear space and efficient construction.

Contribution

It presents a novel linear-space data structure that answers shortest absent word queries in constant time for any substring range.

Findings

Answers shortest absent word queries in constant time

Uses linear space for the data structure

Constructed in near-linear time proportional to string length

Abstract

Given a string $T$ of length $n$ over an alphabet $\Sigma\subset \{1,2,\ldots,n^{O(1)}\}$ of size $\sigma$, we are to preprocess $T$ so that given a range $[i,j]$, we can return a representation of a shortest string over $\Sigma$ that is absent in the fragment $T[i]\cdots T[j]$ of $T$. We present an $O(n)$-space data structure that answers such queries in constant time and can be constructed in $O(n\log_\sigma n)$ time.