Tight Lower Bounds for the Longest Common Extension Problem

TL;DR

This paper establishes a fundamental lower bound trade-off between space and query time for the Longest Common Extension problem in a read-only string model, confirming the optimality of existing solutions.

Contribution

It proves a tight lower bound of $S(n)T(n) = ext{Omega}(n \log n)$ for space and time trade-offs in the non-uniform cell-probe model under specific conditions.

Findings

The trade-off $S(n)T(n) = ext{Omega}(n \log n)$ is proven to be tight.

The lower bound applies to read-only input strings with large alphabet sizes.

The result confirms the optimality of known data structures for LCE queries.

Abstract

The longest common extension problem is to preprocess a given string of length $n$ into a data structure that uses $S(n)$ bits on top of the input and answers in $T(n)$ time the queries $\mathit{LCE}(i,j)$ computing the length of the longest string that occurs at both positions $i$ and $j$ in the input. We prove that the trade-off $S(n)T(n) = \Omega(n\log n)$ holds in the non-uniform cell-probe model provided that the input string is read-only, each letter occupies a separate memory cell, $S(n) = \Omega(n)$, and the size of the input alphabet is at least $2^{8\lceil S(n) / n\rceil}$. It is known that this trade-off is tight.