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

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

Gap terminology and related combinatorial properties for AVL trees and Fibonacci-isomorphic trees

We introduce gaps that are edges or external pointers in AVL trees such that the height difference between the subtrees rooted at their two endpoints is equal to 2. Using gaps we prove the Basic-Theorem that illustrates how the size of an AVL tree (and its subtrees) can be represented by a series of powers of 2 of the heights of the gaps, this theorem is the first such simple formula to characterize the number of nodes in an AVL tree. Then, we study the extreme case of AVL trees, the perfectly unbalanced AVL trees, by introducing Fibonacci-isomorphic trees that are isomorphic to Fibonacci trees of the same height and showing that they have the maximum number of gaps in AVL trees. Note that two ordered trees (such as AVL trees) are isomorphic iff there exists a one-to-one correspondence between their nodes that preserves not only adjacency relations in the trees, but also the roots. In the rest of the paper, we study combinatorial properties of Fibonacci-isomorphic trees. (C) 2018 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/)

New Combinatorial Properties and Algorithms for AVL Trees

In this thesis, new properties of AVL trees and a new partitioning of binary search trees named core partitioning scheme are discussed, this scheme is applied to three binary search trees namely AVL trees, weight-balanced trees, and plain binary search trees. We introduce the core partitioning scheme, which maintains a balanced search tree as a dynamic collection of complete balanced binary trees called cores. Using this technique we achieve the same theoretical efficiency of modern cache-oblivious data structures by using classic data structures such as weight-balanced trees or height balanced trees (e.g. AVL trees). We preserve the original topology and algorithms of the given balanced search tree using a simple post-processing with guaranteed performance to completely rebuild the changed cores (possibly all of them) after each update. Using our core partitioning scheme, we simultaneously achieve good memory allocation, space-efficient representation, and cache-obliviousness. We also apply this scheme to arbitrary binary search trees which can be unbalanced and we produce a new data structure, called Cache-Oblivious General Balanced Tree (COG-tree). Using our scheme, searching a key requires O(log_B n) block transfers and O(log n) comparisons in the external-memory and in the cache-oblivious model. These complexities are theoretically efficient. Interestingly, the core partition for weight-balanced trees and COG-tree can be maintained with amortized O(log_B n) block transfers per update, whereas maintaining the core partition for AVL trees requires more than a poly-logarithmic amortized cost. Studying the properties of these trees also lead us to some other new properties of AVL trees and trees with bounded degree, namely, we present and study gaps in AVL trees and we prove Tarjan et al.'s conjecture on the number of rotations in a sequence of deletions and insertions

Finding Paths in the Rotation Graph of Binary Trees

A binary tree coding scheme is a bijection mapping a set of binary trees to a set of integer tuples called codewords. One problem considered in the literature is that of listing the codewords for n-node binary trees, such that successive codewords represent trees differing by a single rotation, a standard operation for rebalancing binary search trees. Then, the codeword sequence corresponds to an Hamiltonian path in the rotation graph Rn of binary trees, where each node is labelled with an n-node binary tree, and an edge connects two nodes when their trees differ by a single rotation. A related problem is finding a shortest path between two nodes in Rn, which reduces to the problem of transforming one binary tree into another using a minimum number of rotations. Yet a third problem is determining properties of the rotation graph. Our work addresses these three problems. A correspondence between n-node binary trees and triangulations of (n+2)-gons allows labelling nodes of Rn, with triangulations, where adjacent triangulations differ by a single diagonal flip. It has been proven, using properties of triangulations, that Rn is Hamiltonian, and that its diameter is bounded above by 2n-6 for n ≥ 11. In Chapter Three we use triangulations to show that the radius of Rn, is n-1; to characterize the n+2 nodes in the center; to show that Rn is the union of n+2 copies of Rn-1; and to prove that Rn is (n-1)-connected. We also introduce the skeleton graph RSn of Rn, and give additional properties of both graphs. In Chapter Four, we give an algorithm, OzLex, which, for each of many different coding schemes, generates 2n-1 different sequences of codewords for n-node binary trees. We also show that, for every n ≥ 4, all such sequences combined represent 2n Hamiltonian paths in Rn. In Appendix Two, we modify OzLex to create TransOx, an algorithm which generates (n+2)2n sequences of codewords from a single coding scheme, and prove that, for n ≥ 5, the sequences represent (n+2)2n-1 Hamiltonian paths. The distance between extreme nodes in Rn is the diameter of the graph. In Chapter Five, we give properties of extreme nodes in terms of their corresponding triangulations; Appendix One contains additional related information. We present two heuristics, based on flipping diagonals, that find a path between two nodes in Rn: Findpath-1, in O(n log n) time; and FindPath-2, in 0(n2 log n) time. Each computes paths with less than twice the minimum length. We also identify a class of triangulation pairs where Findpath-2 significantly outperforms FindPath-1

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum