Succinct Indices for Range Queries with applications to Orthogonal Range Maxima

TL;DR

This paper introduces a space-efficient data structure for 2D range maximum queries with priorities, improving query time over previous methods by leveraging succinct indices and effective entropy concepts.

Contribution

It presents a novel succinct index structure that preprocesses points with priorities to answer range maximum queries more efficiently in both space and time.

Findings

Uses O(N) words of space

Answers queries in O(log N log log N) time

Improves upon Chazelle's 1985 result

Abstract

We consider the problem of preprocessing $N$ points in 2D, each endowed with a priority, to answer the following queries: given a axis-parallel rectangle, determine the point with the largest priority in the rectangle. Using the ideas of the \emph{effective entropy} of range maxima queries and \emph{succinct indices} for range maxima queries, we obtain a structure that uses O(N) words and answers the above query in $O(\log N \log \log N)$ time. This is a direct improvement of Chazelle's result from FOCS 1985 for this problem -- Chazelle required $O(N/\epsilon)$ words to answer queries in $O((\log N)^{1+\epsilon})$ time for any constant $\epsilon > 0$.