Run Compressed Rank/Select for Large Alphabets

Abstract

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