RMQ : Best Time Complexity

What is the best time complexity one can achieve in finding LCA using RMQ(Static)??

I know RMQ using Sparse Table==>Preprocessing Time: O{NlogN) Query:O(1)

Segment Tree.Preprocessing Time: O{N) Query:O(logN).

Is there method with preprocessing time O(N) and Query : O(1)??

here it is:Topcoder