Compressed range minimum queries

Given a string S of n integers in [0,σ), a range minimum query RMQ(i, j) asks for the index of the smallest integer in S[i…j]. It is well known that the problem can be solved with a succinct data structure of size 2n + o(n) and constant query-time. In this paper we show how to preprocess S into a compressed representation that allows fast range minimum queries. This allows for sublinear size data structures with logarithmic query time. The most natural approach is to use string compression and construct a data structure for answering range minimum queries directly on the compressed string. We investigate this approach using grammar compression. We then consider an alternative approach. Even if S is not compressible, its Cartesian tree necessarily is. Therefore, instead of compressing S using string compression, we compress the Cartesian tree of S using tree compression. We show that this approach can be exponentially better than the former, and is never worse by more than an O(σ) factor (i.e. for constant alphabets it is never asymptotically worse).