# Entropy Trees and Range-Minimum Queries In Optimal Average-Case Space

## TL;DR

This paper introduces a new adaptive succinct data structure for range-minimum queries that achieves optimal average-case space, improving upon previous worst-case bounds and utilizing a novel compressed binary tree representation.

## Contribution

It constructs an average-case optimal space data structure for RMQ, resolving an open problem and introducing a new compressed binary tree structure with constant-time query support.

## Key findings

- Achieves 1.736n+o(n) bits of space on average for RMQ
- Supports constant-time queries in the new data structure
- Introduces a hyper-succinct binary tree encoding

## Abstract

The range-minimum query (RMQ) problem is a fundamental data structuring task with numerous applications. Despite the fact that succinct solutions with worst-case optimal $2n+o(n)$ bits of space and constant query time are known, it has been unknown whether such a data structure can be made adaptive to the reduced entropy of random inputs (Davoodi et al. 2014). We construct a succinct data structure with the optimal $1.736n+o(n)$ bits of space on average for random RMQ instances, settling this open problem. Our solution relies on a compressed data structure for binary trees that is of independent interest. It can store a (static) binary search tree generated by random insertions in asymptotically optimal expected space and supports many queries in constant time. Using an instance-optimal encoding of subtrees, we furthermore obtain a "hyper-succinct" data structure for binary trees that improves upon the ultra-succinct representation of Jansson, Sadakane and Sung (2012).