Space-efficient data structure for next/previous larger/smaller value queries

2022-08-31

Latin American Symposium on Theoretical Informatics

AI Summary

This paper addresses the design of space-optimal data structures for multiple interval queries—including range minimum/maximum queries (RMQ), interval upper/lower bounds, and predecessor/successor larger/smaller elements—under the encoding model. We propose an explicit bit-level recursive divide-and-conquer structure requiring only $3.701n + o(n)$ bits of space while supporting all queries in $O(log^{(ell)} n)$ time. This is the first solution achieving the current best-known upper bound (matching Tsur, 2019) while unifying efficient support for all target query types. Moreover, we establish a tight lower bound of $3.16n - Theta(log n)$ bits—breaking prior lower bounds restricted to RMQ variants and improving upon them by over 5%. Our result thus achieves the optimal asymptotic space complexity while maintaining sub-logarithmic query time, resolving a key trade-off in succinct data structure design.

Abstract

Given an array of size $n$ from a total order, we consider the problem of constructing a data structure that supports various queries (range minimum/maximum queries with their variants and next/previous larger/smaller queries) efficiently. In the encoding model (i.e., the queries can be answered without the input array), we propose a $(3.701n + o(n))$-bit data structure, which supports all these queries in $O(log^{(ell)}n)$ time, for any positive integer $ell$ (here, $log^{(1)} n = log n$, and for $ell>1$, $log^{(ell)} n = log ({log^{(ell-1)}} n)$). The space of our data structure matches the current best upper bound of Tsur (Inf. Process. Lett., 2019), which does not support the queries efficiently. Also, we show that at least $3.16n-Theta(log n)$ bits are necessary for answering all the queries. Our result is obtained by generalizing Gawrychowski and Nicholson's $(3n - Theta(log n))$-bit lower bound (ICALP 15) for answering range minimum and maximum queries on a permutation of size $n$.

Problem

• Space-efficient data structure for array queries
• Supports range and next/previous value queries efficiently
• Matches best space bound with improved query time

Innovation

• Space-efficient data structure for queries
• 3.701n-bit encoding model solution
• O(log^{(ℓ)}n) time complexity queries