Succinct Representations of Binary Trees for Range Minimum Queries

We provide two succinct representations of binary trees that can be used to represent the Cartesian tree of an array A of size n. Both the representations take the optimal 2n + o(n) bits of space in the worst case and support range minimum queries (RMQs) in O(1) time. The first one is a modification of the representation of Farzan and Munro (SWAT 2008); a consequence of this result is that we can represent the Cartesian tree of a random permutation in 1.92n + o(n) bits in expectation. The second one uses a well-known transformation between binary trees and ordinal trees, and ordinal tree operations to effect operations on the Cartesian tree. This provides an alternative, and more natural, way to view the 2D-Min-Heap of Fischer and Huen (SICOMP 2011). Furthermore, we show that the pre-processing needed to output the data structure can be performed in linear time using o(n) bits of extra working space, improving the result of Fischer and Heun who use n + o(n) bits working space.