Run Compressed Rank/Select for Large Alphabets

Given a string of length n that is composed of r runs of letters from the alphabet {0, 1, . . . , σ−1} such that 2 ≤ σ ≤ r, we describe a data structure that, provided r ≤ n/ log n, stores the string in r log nσ r + o(r log nσ r ) bits and supports select and access queries in O(log log(n/r) log logn ) time and rank queries in O(log log(nσ/r) log logn ) time. We show that r log n(σ−1) r bits are necessary for any such data structure and, thus, our solution is succinct. We also describe a data structure that uses (1+ ǫ)r log nσ r +O(r) bits, where ǫ > 0 is an arbitrary constant, with the same query times but without the restriction r ≤ n/ log n. By simple reductions to the colored predecessor problem, we show that the query times are optimal in the important case r ≥ 2log δ n, 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.