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

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.