Arithmetic for Rooted Trees

We propose a new arithmetic for non-empty rooted unordered trees simply called trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication and stretch, prove their properties, and show that all trees can be generated from a starting tree of one vertex. We then show how a given tree can be obtained as the sum or product of two trees, thus defining prime trees with respect to addition and multiplication. In both cases we show how primality can be decided in time polynomial in the number of vertices and we prove that factorization is unique. We then define negative trees and suggest dealing with tree equations, giving some preliminary results. Finally we comment on how our arithmetic might be useful, and discuss preceding studies that have some relations with our. To the best of our knowledge our approach and results are completely new aside for an earlier version of this work submitte as an arXiv manuscript.Comment: 18 pages, 8 figure

Generation, Ranking and Unranking of Ordered Trees with Degree Bounds

We study the problem of generating, ranking and unranking of unlabeled ordered trees whose nodes have maximum degree of $\Delta$. This class of trees represents a generalization of chemical trees. A chemical tree is an unlabeled tree in which no node has degree greater than 4. By allowing up to $\Delta$ children for each node of chemical tree instead of 4, we will have a generalization of chemical trees. Here, we introduce a new encoding over an alphabet of size 4 for representing unlabeled ordered trees with maximum degree of $\Delta$. We use this encoding for generating these trees in A-order with constant average time and O(n) worst case time. Due to the given encoding, with a precomputation of size and time O(n^2) (assuming $\Delta$ is constant), both ranking and unranking algorithms are also designed taking O(n) and O(nlogn) time complexities.Comment: In Proceedings DCM 2015, arXiv:1603.0053

On lexicographic enumeration of regular and context-free languages

We show that it is possible to efficiently enumerate the words of a regular language in lexicographic order. The time needed for generating the next word is O(n) when enumerating words of length n. We also define a class of context-free languages for which efficient enumeration is possible

Compressing Sets and Multisets of Sequences

This is the accepted manuscript for a paper published in IEEE Transactions on Information Theory, Vol. 61, No. 3, March 2015, doi: 10.1109/TIT.2015.2392093. © 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.This paper describes lossless compression algorithms for multisets of sequences, taking advantage of the multiset’s unordered structure. Multisets are a generalization of sets, where members are allowed to occur multiple times. A multiset can be encoded naïvely by simply storing its elements in some sequential order, but then information is wasted on the ordering. We propose a technique that transforms the multiset into an order-invariant tree representation, and derive an arithmetic code that optimally compresses the tree. Our method achieves compression even if the sequences in the multiset are individually incompressible (such as cryptographic hash sums). The algorithm is demonstrated practically by compressing collections of SHA-1 hash sums, and multisets of arbitrary, individually encodable objects.This work was supported in part by the Engineering and Physical Sciences Research Council under Grant EP/I036575 and in part by a Google Research Award. This paper was presented at the 2014 Data Compression Conferenc

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

Generation of Neuronal Trees by a New Three Letters Encoding

A neuronal tree is a rooted tree with n leaves whose each internal node has at least two children; this class not only is defined based on the structure of dendrites in neurons, but also refers to phylogenetic trees or evolutionary trees. More precisely, neuronal trees are rooted-multistate phylogenetic trees whose size is defined as the number of leaves. In this paper, a new encoding over an alphabet of size 3 (minimal cardinality) is introduced for representing the neuronal trees with a given number of leaves. This encoding is used for generating neuronal trees with n leaves in A-order with constant average time and O(n) time complexity in the worst case. Also, new ranking and unranking algorithms are presented in time complexity of O(n) and O(n log n), respectively

Time-Optimal Tree Computations on Sparse Meshes

The main goal of this work is to fathom the suitability of the mesh with multiple broadcasting architecture (MMB) for some tree-related computations. We view our contribution at two levels: on the one hand, we exhibit time lower bounds for a number of tree-related problems on the MMB. On the other hand, we show that these lower bounds are tight by exhibiting time-optimal tree algorithms on the MMB. Specifically, we show that the task of encoding and/or decoding n-node binary and ordered trees cannot be solved faster than Ω(log n) time even if the MMB has an infinite number of processors. We then go on to show that this lower bound is tight. We also show that the task of reconstructing n-node binary trees and ordered trees from their traversais can be performed in O(1) time on the same architecture. Our algorithms rely on novel time-optimal algorithms on sequences of parentheses that we also develop