Range LCP

In this paper, we define the Range LCP problem as follows. Preprocess a string S, of length n, to enable efficient solutions of the following query: Given [i, j], 0 < i ≤ j ≤ n, compute max`,k∈{i,...,j} LCP (S`, Sk), where LCP (S`, Sk) is the length of the longest common prefix of the suffixes of S starting at locations ` and k. This is a natural generalization of the classical LCP problem. Surprisingly, while it is known how to preprocess a string in linear time to enable LCP computation of two suffixes in constant time, this seems quite difficult in the Range LCP problem. It is trivial to answer such queries in time O(|j− i|) after a linear-time preprocessing and easy to show an O(1) query algorithm after an O(|S|) time preprocessing. We provide algorithms that solve the problem with the following complexities: 1. Preprocessing Time: O(|S|), Space: O(|S|), Query Time: O(|j−i| log log n). 2. Preprocessing Time: no preprocessing, Space: O(|j − i| log |j − i|), Query Time: O(|j − i| log |j − i|). However, the query just gives the pairs with the longest LCP, not the LCP itself. ? Partly supported by NSF grant CCR-09-04581, ISF grant 347/09, and BSF grant 2008217. ?? Partly supported by BSF grant 2008217. ? ? ? Partly supported by the National Science Foundation Award 0904246, Israel Science Foundation grant 347/09, Yahoo, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG. † Partly supported by the Israel Science Foundation grant 347/09 ‡ Partly supported by BSF grant 2006334, ISF grant 1484/08, and Google award. 3. Preprocessing Time: O(|S| log |S|), Space: O(|S| log |S|) for arbitrary small constant ε, Query Time: O(log log |S|)