Combined Data Structure for Previous- and Next-Smaller-Values

TL;DR

This paper introduces a space-efficient data structure that supports constant-time previous and next smaller value queries, as well as range minimum queries, on static arrays, with applications to suffix tree storage.

Contribution

It presents an optimal-size data structure enabling constant-time queries for previous/next smaller values and range minima without array access.

Findings

Supports constant-time queries for previous/next smaller values

Uses at most n log_2(3+2√2)+o(n) bits of space

Facilitates space-efficient suffix tree storage

Abstract

Let $A$ be a static array storing $n$ elements from a totally ordered set. We present a data structure of optimal size at most $n\log_2(3+2\sqrt{2})+o(n)$ bits that allows us to answer the following queries on $A$ in constant time, without accessing $A$: (1) previous smaller value queries, where given an index $i$, we wish to find the first index to the left of $i$ where $A$ is strictly smaller than at $i$, and (2) next smaller value queries, which search to the right of $i$. As an additional bonus, our data structure also allows to answer a third kind of query: given indices $i<j$, find the position of the minimum in $A[i..j]$. Our data structure has direct consequences for the space-efficient storage of suffix trees.