Range LCP queries revisited

The Range LCP problem is to preprocess a string S[1…n], to enable efficient solutions of the following query: given a range [l, r] as the input, report max_{i,j∈{l,…,r}} |LCP(S_i, S_j)|. Here LCP(S_i, S_j) is the longest common prefix of the suffixes of S starting at locations i and j and |LCP(S_i, S_j)| is its length. We study a natural extension of this problem, where the query consists of two ranges. Additionally, we allow a bounded number (say k ≥ 0) of mismatches in the LCP computation. Specifically, our task is to report the following when two ranges [ℓ₁, r₁] and [ℓ₂, r₂] comes as input: max _{{ℓ1≤i≤r1,ℓ2≤j≤r2}} |LCP_k(S_i, S_j)| Here LCP_k(S_i, S_j) is the longest prefix of S_i and S_j with at most k mismatches allowed. We show that the queries can be answered in O(k) time using an O(n²/w) space data structure, where w is the word size. We also present space efficient data structures for k = 0 and k = 1. For k = 0, we obtain a linear space data structure with query time O(√n/w log^ϵ n), where w is the word size and ϵ > 0 is an arbitrarily small constant. For the case k = 1 we obtain an O(n log n) space data structure with query time O(√ n log n). Finally, we give a reduction from Set Intersection to Range LCP queries, suggesting that it will be very difficult to improve our upper bound by more than a factor of O(log^ϵ n).