Simultaneous encodings for range and next/previous larger/smaller value queries

TL;DR

This paper introduces space-efficient encodings for supporting range and next/previous larger or smaller value queries on arrays, achieving near-optimal space and constant query time improvements over previous methods.

Contribution

It presents new encodings that support multiple range and value queries simultaneously with improved space efficiency and constant query time, extending and combining existing data structures.

Findings

Supports all queries in 4.088n bits with non-constant time

Supports all queries in 4.585n bits with constant time

Improves upon previous 5.08n bits encoding by Fischer

Abstract

Given an array of $n$ elements from a total order, we propose encodings that support various range queries (range minimum, range maximum and their variants), and previous and next smaller/larger value queries. When query time is not of concern, we obtain a $4.088n + o(n)$-bit encoding that supports all these queries. For the case when we need to support all these queries in constant time, we give an encoding that takes $4.585n + o(n)$ bits, where $n$ is the length of input array. This improves the $5.08n+o(n)$-bit encoding obtained by encoding the colored $2d$-Min and Max heaps proposed by Fischer~[TCS, 2011]. We first extend the original DFUDS [Algorithmica, 2005] encoding of the colored 2d-Min (Max) heap that supports the queries in constant time. Then, we combine the extended DFUDS of $2d$-Min heap and $2d$-Max heap using the Min-Max encoding of Gawrychowski and Nicholson [ICALP, 2015] with some modifications. We also obtain encodings that take lesser space and support a subset of these queries.