On Succinct Representations of Binary Trees

Lecture Notes in Computer Science, 2012

We provide two succinct representations of binary trees that can be used to represent the Cartesian tree of an array A of size n. Both the representations take the optimal 2n + o(n) bits of space in the worst case and support range minimum queries (RMQs) in O(1) time. The first one is a modification of the representation of Farzan and Munro (SWAT 2008); a consequence of this result is that we can represent the Cartesian tree of a random permutation in 1.92n + o(n) bits in expectation. The second one uses a well-known transformation between binary trees and ordinal trees, and ordinal tree operations to effect operations on the Cartesian tree. This provides an alternative, and more natural, way to view the 2D-Min-Heap of Fischer and Huen (SICOMP 2011). Furthermore, we show that the pre-processing needed to output the data structure can be performed in linear time using o(n) bits of extra working space, improving the result of Fischer and Heun who use n + o(n) bits working space.

2007

There exist two well-known succinct representations of ordered trees: BP (balanced parenthesis) and DFUDS (depth first unary degree sequence) . Both have size 2n + o(n) bits for n-node trees, which asymptotically matches the information-theoretic lower bound. Many fundamental operations on trees can be done in constant time on word RAM, for example finding the parent, the first child, the next sibling, the number of descendants, etc. However there has been no single representation supporting every existing operation in constant time; BP does not support i-th child, while DFUDS does not support lca (lowest common ancestor).

ArXiv, 2021

We present a new universal source code for distributions of unlabeled binary and ordinal trees that achieves optimal compression to within lower order terms for all tree sources covered by existing universal codes. At the same time, it supports answering many navigational queries on the compressed representation in constant time on the word-RAM; this is not known to be possible for any existing tree compression method. The resulting data structures, “hypersuccinct trees”, hence combine the compression achieved by the best known universal codes with the operation support of the best succinct tree data structures. We apply hypersuccinct trees to obtain a universal compressed data structure for range-minimum queries. It has constant query time and the optimal worst-case space usage of 2n + o(n) bits, but the space drops to 1.736n + o(n) bits on average for random permutations of n elements, and 2 lg ( n r ) + o(n) for arrays with r increasing runs, respectively. Both results are optima...

Lecture Notes in Computer Science, 2006

Ordinal trees are arbitrary rooted trees where the children of each node are ordered. We consider succinct, or highly space-efficient, representations of (static) ordinal trees with n nodes that use 2n + o(n) bits of space to represent ordinal trees. There are a number of such representations: each supports a different set of tree operations in O(1) time on the RAM model. In this paper we focus on the practical performance the fundamental Level-Order Unary Degree Sequence (LOUDS) representation [Jacobson, Proc. 30th FOCS, 549-554, 1989]. Due to its conceptual simplicity, LOUDS would appear to be a representation with good practical performance. A tree can also be represented succinctly as a balanced parenthesis sequence [Munro and Raman,

Combinatorial Pattern Matching, 2011

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. As part of our review preliminary work, we also give an overview of the, sometimes redundant, terminology relative to succinct data-structures and indices.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2012

We consider succinct, or space-efficient, representations of ordinal trees. Representations exist that take 2n + o(n) bits to represent a static n-node ordinal tree-close to the information-theoretic minimum-and support navigational operations in O(1) time on a RAM model; and some implementations have good practical performance. The situation is different for dynamic ordinal trees. Although there is theoretical work on succinct dynamic ordinal trees, there is little work on the practical performance of these data structures. Motivated by applications to representing XML documents, in this paper, we report on a preliminary study on dynamic succinct data structures. Our implementation is based on representing the tree structure as a sequence of balanced parentheses, with navigation done using the min-max tree of Sadakane and Navarro (SODA '10). Our implementation shows promising performance for update and navigation, and our findings highlight two issues that we believe will be important to future implementations: the difference between the finger model of (say) Farzan and Munro (ICALP '09) and the parenthesis model of Sadakane and Navarro, and the choice of the balanced tree used to represent the min-max tree.

ACM Transactions on Algorithms, 2012

Various methods have been used to represent a tree on n nodes in essentially the informationtheoretic minimum space while supporting various navigational operations in constant time, but different representations usually support different operations. Our main contribution is a succinct representation of ordinal trees, based on that of Geary et al. , that supports all the navigational operations supported by various succinct tree representations while requiring only 2n + o(n) bits. It also supports efficient level-order traversal, a useful ordering previously supported only with a very limited set of operations [10]. Our second contribution expands on the notion of a single succinct representation supporting more than one traversal ordering, by showing that our method supports two other encoding schemes as abstract data types. In particular, it supports extracting a word (O(lg n) bits) 4 of the balanced parenthesis sequence [13] or depth first unary degree sequence in O(f (n)) time, using at most n/f (n) + o(n) additional bits, for any f (n) in O(lg n) and Ω(1).

Lecture Notes in Computer Science, 2013

We survey succinct representations of ordinal, or rooted, ordered trees. Succinct representations use space that is close to the appropriate information-theoretic minimum, but support operations on the tree rapidly, usually in O(1) time.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2009

We consider the succinct representation of ordinal and cardinal trees on the RAM with logarithmic word size. Given a tree T , our representations support the following operations in O(1) time: (i) BP-substring(i, b), which reports the substring of length b bits (b is at most the wordsize) beginning at position i of the balanced parenthesis representation of T , (ii) DFUDS-substring(i, b), which does the same for the depth first unary degree sequence representation, and (iii) a similar operation for tree-partition based representations of T . We give:

Algorithmica, 2005

This paper focuses on space efficient representations of rooted trees that permit basic navigation in constant time. While most of the previous work has focused on binary trees, we turn our attention to trees of higher degree. We consider both cardinal trees (or k-ary tries), where each node has k slots, labelled {1, . . . , k}, each of which may have a reference to a child, and ordinal trees, where the children of each node are simply ordered. Our representations use a number of bits close to the information theoretic lower bound and support operations in constant time.