Dynamic "Succincter"

Augmented B-trees (aB-trees) are a broad class of data structures. The seminal work "succincter" by Patrascu showed that any aB-tree can be stored using only two bits of redundancy, while supporting queries to the tree in time proportional to its depth. It has been a versatile building block for constructing succinct data structures, including rank/select data structures, dictionaries, locally decodable arithmetic coding, storing balanced parenthesis, etc. In this paper, we show how to "dynamize" an aB-tree. Our main result is the design of dynamic aB-trees (daB-trees) with branching factor two using only three bits of redundancy (with the help of lookup tables that are of negligible size in applications), while supporting updates and queries in time polynomial in its depth. As an application, we present a dynamic rank/select data structure for $n$-bit arrays, also known as a dynamic fully indexable dictionary (FID). It supports updates and queries in $O(\log n/\log\log n)$ time, and when the array has $m$ ones, the data structure occupies $$\log\binom{n}{m} + O(n/2^{\log^{0.199}n})$$ bits. Note that the update and query times are optimal even without space constraints due to a lower bound by Fredman and Saks. Prior to our work, no dynamic FID with near-optimal update and query times and redundancy $o(n/\log n)$ was known. We further show that a dynamic sequence supporting insertions, deletions and rank/select queries can be maintained in (optimal) $O(\log n/\log\log n)$ time and with $O(n \cdot \text{poly}\log\log n/\log^2 n)$ bits of redundancy.Comment: 33 pages, 1 figure; in FOCS 202

LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations

LRM-Trees are an elegant way to partition a sequence of values into sorted consecutive blocks, and to express the relative position of the first element of each block within a previous block. They were used to encode ordinal trees and to index integer arrays in order to support range minimum queries on them. We describe how they yield many other convenient results in a variety of areas, from data structures to algorithms: some compressed succinct indices for range minimum queries; a new adaptive sorting algorithm; and a compressed succinct data structure for permutations supporting direct and indirect application in time all the shortest as the permutation is compressible.Comment: 13 pages, 1 figur