Range minimum queries in minimal space

Abstract

We consider the problem of computing a sequence of range minimum queries. We assume a sequence of commands that contains values and queries. Our goal is to quickly determine the minimum value that exists between the current position and a previous position i. Range minimum queries are used as a sub-routine of several algorithms, namely related to string processing. We propose a data structure that can process these command sequences. We obtain efficient results for several variations of the problem, in particular we obtain O(1) time per command for the offline version and O(α(n)) amortized time for the online version, where α(n) is the inverse Ackermann function and n the number of values in the sequence. This data structure also has very small space requirements, namely O(ℓ) where ℓ is the maximum number of active i positions. We implemented our data structure and show that it is competitive against existing alternatives. We obtain comparable processing time, in the nanosecond range, and much smaller space requirements.