Longest Common Extensions in Sublinear Space

Abstract

The longest common extension problem (LCE problem) is to construct a data structure for an input string $T$ of length $n$ that supports ${\mathrm {LCE}}(i,j)$ queries. Such a query returns the length of the longest common prefix of the suffixes starting at positions $i$ and $j$ in $T$. This classic problem has a well-known solution that uses $\mathcal {O}(n)$ space and $\mathcal {O}(1)$ query time. In this paper we show that for any trade-off parameter $1 \le \tau \le n$, the problem can be solved in $\mathcal {O}(\frac{n}{\tau })$ space and $\mathcal {O}(\tau )$ query time. This significantly improves the previously best known time-space trade-offs, and almost matches the best known time-space product lower bound.

P. Bille— Supported by the Danish Research Council and the Danish Research Council under the Sapere Aude Program (DFF 4005-00267).

I. L. Gørtz— Research partly supported by Mikkel Thorup’s Advanced Grant from the Danish Council for Independent Research under the Sapere Aude research career programme and the FNU project AlgoDisc - Discrete Mathematics, Algorithms, and Data Structures.

H. W. Vildhøj— This research was supported by a Grant from the GIF, the German-Israeli Foundation for Scientific Research and Development, and by a BSF grant 2010437.