Practical Range Minimum Queries Revisited

Abstract

Finding the position of the minimal element in a subarray A[i..j] of an array A of size n is a fundamental operation in many applications. In 2011, Fischer and Heun presented the first index of size 2n+o(n) bits which answers the operation in constant time for any subarray. The index can be computed in linear time and queries can be answered without consulting the original array. The most recent and currently fastest practical index is due to Ferrada and Navarro (DCC'16). It reduces the range minimum query (RMQ) to more fundamental and well studied queries on binary vectors, namely rank and select, and a RMQ query on an array of sublinear size derived from A. A range min-max tree is employed to solve this recursive RMQ call. In this paper, we review their practical design and suggest a series of changes which result in consistently faster query times. Specifically, we provide a customized select implementation, switch to two levels of recursion, and use the sparse table solution for the recursion base case instead of a range min-max tree. We provide an extensive empirical evaluation of our new implementation and also compare it to the state of the art. Our experimental study shows that our proposal significantly outperforms the previous solutions on established benchmarks (up to a factor of three) and furthermore accelerates real world applications such as traversing a succinct tree or listing all distinct elements in an interval of an array.