Near-Optimal Search Time in $δ$-Optimal Space

Two recent lower bounds on the compressiblity of repetitive sequences, $\delta \le \gamma$, have received much attention. It has been shown that a string $S[1..n]$ can be represented within the optimal $O(\delta\log\frac{n}{\delta})$ space, and further, that within that space one can find all the $occ$ occurrences in $S$ of any pattern of length $m$ in time $O(m\log n + occ \log^\epsilon n)$ for any constant $\epsilon>0$. Instead, the near-optimal search time $O(m+(occ+1)\log^\epsilon n)$ was 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 obtained within the $\delta$-optimal space was open. In this paper we prove that both techniques can indeed be combined in order to obtain the best of both worlds, $O(m+(occ+1)\log^\epsilon n)$ search time within $O(\delta\log\frac{n}{\delta})$ space.