Space-Efficient Preprocessing Schemes for Range Minimum Queries on Static Arrays

Given a static array of n totally ordered object, the range minimum query problem is to build an additional data structure that allows to answer subsequent on-line queries of the form “what is the position of a minimum element in the sub-array ranging from i to j?” efficiently. We focus on two settings, where (1) the input array is available at query time, and (2) the input array is only available at construction time. In setting (1), we show new data structures (a) of size n c(n) (2 + o(1)) bits and query time O(c(n)), or (b) with O(nHk) + o(n) bits and O(1) query time, where Hk denotes the empirical entropy of k’th order of the input array. In setting (2), we give a data structure of optimal size 2n+ o(n) bits and query time O(1). All data structures can be constructed in linear time and almost in-place.