Near-Optimal Search Time in δ-Optimal Space, and Vice Versa

Two recent lower bounds on the compressibility of repetitive sequences, delta <= 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 n log / 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 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 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 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.