Near-Optimal Search Time in $\delta$-Optimal Space, and Vice Versa

TL;DR

This paper demonstrates how to achieve near-optimal search times in repetitive sequence compression within the best possible space bounds, combining techniques to support efficient pattern matching and counting.

Contribution

It combines two advanced parsing techniques to achieve optimal search and counting times within $ ext{delta}$-optimal space bounds for repetitive sequences.

Findings

Achieves $O(m+(occ+1) ext{log}^ ext{epsilon} n)$ search time in $ ext{delta}$-optimal space.

Provides $O(m+ ext{log}^{2+ ext{epsilon}} n)$ time for occurrence counting.

Shows that additional sublogarithmic space enables optimal search and counting times.

Abstract

Two recent lower bounds on the compressibility of repetitive sequences, $\delta \le \gamma$, have received much attention. It has been shown that a length-$n$ string $S$ over an alphabet of size $\sigma$ can be represented within the optimal $O(\delta\log\tfrac{n\log \sigma}{\delta \log n})$ space, and further, that within that space one can find all the $occ$ occurrences in $S$ of any length-$m$ pattern in time $O(m\log n + occ \log^\epsilon n)$ for any constant $\epsilon>0$. Instead, the near-optimal search time $O(m+({occ+1})\log^\epsilon n)$ has been achieved only within $O(\gamma\log\frac{n}{\gamma})$ space. Both results are based on considerably different locally consistent parsing techniques. The question of whether the better search time could be supported within the $\delta$-optimal space remained open. In this paper, we prove that both techniques can indeed be combined to obtain the best of both worlds: $O(m+({occ+1})\log^\epsilon n)$ search time within $O(\delta\log\tfrac{n\log \sigma}{\delta \log n})$ space. Moreover, the number of occurrences can be computed in $O(m+\log^{2+\epsilon}n)$ time within $O(\delta\log\tfrac{n\log \sigma}{\delta \log n})$ space. We also show that an extra sublogarithmic factor on top of this space enables optimal $O(m+occ)$ search time, whereas an extra logarithmic factor enables optimal $O(m)$ counting time.