On Succinct Representations of Binary Trees

We observe that a standard transformation between \emph{ordinal} trees (arbitrary rooted trees with ordered children) and binary trees leads to interesting succinct binary tree representations. There are four symmetric versions of these transformations. Via these transformations we get four succinct representations of $n$-node binary trees that use $2n + n/(\log n)^{O(1)}$ bits and support (among other operations) navigation, inorder numbering, one of pre- or post-order numbering, subtree size and lowest common ancestor (LCA) queries. The ability to support inorder numbering is crucial for the well-known range-minimum query (RMQ) problem on an array $A$ of $n$ ordered values. While this functionality, and more, is also supported in $O(1)$ time using $2n + o(n)$ bits by Davoodi et al.'s (\emph{Phil. Trans. Royal Soc. A} \textbf{372} (2014)) extension of a representation by Farzan and Munro (\emph{Algorithmica} \textbf{6} (2014)), their \emph{redundancy}, or the $o(n)$ term, is much larger, and their approach may not be suitable for practical implementations. One of these transformations is related to the Zaks' sequence (S.~Zaks, \emph{Theor. Comput. Sci.} \textbf{10} (1980)) for encoding binary trees, and we thus provide the first succinct binary tree representation based on Zaks' sequence. Another of these transformations is equivalent to Fischer and Heun's (\emph{SIAM J. Comput.} \textbf{40} (2011)) \minheap\ structure for this problem. Yet another variant allows an encoding of the Cartesian tree of $A$ to be constructed from $A$ using only $O(\sqrt{n} \log n)$ bits of working space.Comment: Journal version of part of COCOON 2012 pape

Simple and Efficient Fully-Functional Succinct Trees

The fully-functional succinct tree representation of Navarro and Sadakane (ACM Transactions on Algorithms, 2014) supports a large number of operations in constant time using $2n+o(n)$ bits. However, the full idea is hard to implement. Only a simplified version with $O(\log n)$ operation time has been implemented and shown to be practical and competitive. We describe a new variant of the original idea that is much simpler to implement and has worst-case time $O(\log\log n)$ for the operations. An implementation based on this version is experimentally shown to be superior to existing implementations

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

Compact Binary Relation Representations with Rich Functionality

Binary relations are an important abstraction arising in many data representation problems. The data structures proposed so far to represent them support just a few basic operations required to fit one particular application. We identify many of those operations arising in applications and generalize them into a wide set of desirable queries for a binary relation representation. We also identify reductions among those operations. We then introduce several novel binary relation representations, some simple and some quite sophisticated, that not only are space-efficient but also efficiently support a large subset of the desired queries.Comment: 32 page

Compressed Representations of Permutations, and Applications

We explore various techniques to compress a permutation $\pi$ over n integers, taking advantage of ordered subsequences in $\pi$, while supporting its application $\pi$(i) and the application of its inverse $\pi^{-1}(i)$ in small time. Our compression schemes yield several interesting byproducts, in many cases matching, improving or extending the best existing results on applications such as the encoding of a permutation in order to support iterated applications $\pi^k(i)$ of it, of integer functions, and of inverted lists and suffix arrays