Run Compressed Rank/Select for Large Alphabets

TL;DR

This paper introduces a space-efficient data structure for rank/select queries on large alphabet strings with runs, achieving near-optimal space and query times, and demonstrates its effectiveness through implementation and comparison.

Contribution

The paper presents a novel succinct data structure for compressed rank/select operations on large alphabets with runs, improving space efficiency and query performance.

Findings

Uses r log(nσ/r) + o(r log(nσ/r)) bits of space.

Supports select and access in polylogarithmic time.

Achieves 31-46% space savings over previous methods.

Abstract

Given a string of length $n$ that is composed of $r$ runs of letters from the alphabet $\{0,1,\ldots,\sigma{-}1\}$ such that $2 \le \sigma \le r$, we describe a data structure that, provided $r \le n / \log^{\omega(1)} n$, stores the string in $r\log\frac{n\sigma}{r} + o(r\log\frac{n\sigma}{r})$ bits and supports select and access queries in $O(\log\frac{\log(n/r)}{\log\log n})$ time and rank queries in $O(\log\frac{\log(n\sigma/r)}{\log\log n})$ time. We show that $r\log\frac{n(\sigma-1)}{r} - O(\log\frac{n}{r})$ bits are necessary for any such data structure and, thus, our solution is succinct. We also describe a data structure that uses $(1 + \epsilon)r\log\frac{n\sigma}{r} + O(r)$ bits, where $\epsilon > 0$ is an arbitrary constant, with the same query times but without the restriction $r \le n / \log^{\omega(1)} n$. By simple reductions to the colored predecessor problem, we show that the query times are optimal in the important case $r \ge 2^{\log^\delta n}$, for an arbitrary constant $\delta > 0$. We implement our solution and compare it with the state of the art, showing that the closest competitors consume 31-46% more space.