Practical Entropy-Bounded Schemes for O(1)-Range Minimum Queries

The Range Minimum Query (RMQ) Problem is to preprocess an array A of length n in 0(n) time such that subsequent on-line queries asking for the position of a minimal element between two specified indices can be obtained in constant time. Several solutions to this problem have been proposed, starting with Berkman and Vishkin's linear-space solution [6], and leading to a succinct solution using only 2n + o(n) bits in addition to the input array [12]. The theoretical contribution of this article is to push this latter approach one step further and show that for compressible input arrays the RMQ-information can be compressed as well. In particular, we show that information for 0(l)-RMQs can be stored within the same entropy bounds that are achieved by the currently best schemes for storing A itself in compressed form, while still being able to access O(logn) contiguous bits in O(l) time [10]. Two such entropy- bounded schemes for 0(l)-RMQs are developed, each with its own practical advantage. We test these two methods extensively and compare them to three other schemes for RMQs: the currently best non-succinct solution [2], and two succinct approaches: Sadakane's 4n + o(n)-bit solution [22], and our own 2n + o(n)-bit solution. The results of this practical evaluation are (1) the practical space consumptions of the non-compressed schemes scale surprisingly well with their theoretical guarantees, and (2) for compressible input arrays our new compressed schemes can indeed reduce the space, with little or no slowdown in query time.