Longest Common Extensions with Recompression

Abstract

Given two positions $i$ and $j$ in a string $T$ of length $N$, a longest common extension (LCE) query asks for the length of the longest common prefix between suffixes beginning at $i$ and $j$. A compressed LCE data structure is a data structure that stores $T$ in a compressed form while supporting fast LCE queries. In this article we show that the recompression technique is a powerful tool for compressed LCE data structures. We present a new compressed LCE data structure of size $O(z \lg (N/z))$ that supports LCE queries in $O(\lg N)$ time, where $z$ is the size of Lempel-Ziv 77 factorization without self-reference of $T$. Given $T$ as an uncompressed form, we show how to build our data structure in $O(N)$ time and space. Given $T$ as a grammar compressed form, i.e., an straight-line program of size n generating $T$, we show how to build our data structure in $O(n \lg (N/n))$ time and $O(n + z \lg (N/z))$ space. Our algorithms are deterministic and always return correct answers.