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.