scholarly journals On the Diameter of Tree Associahedra

10.37236/7762 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Jean Cardinal ◽  
Stefan Langerman ◽  
Pablo Pérez-Lantero
Keyword(s):  
Computer Science ◽  
Search Tree ◽  
Binary Search ◽  
The Other ◽  
Discrete Mathematics ◽  
Binary Search Trees ◽  
Search Trees ◽  
Worst Case ◽  
Natural Notion ◽  
Minimum Number

We consider a natural notion of search trees on graphs, which we show is ubiquitous in various areas of discrete mathematics and computer science. Search trees on graphs can be modified by local operations called rotations, which generalize rotations in binary search trees. The rotation graph of search trees on a graph $G$ is the skeleton of a polytope called the graph associahedron of $G$.We consider the case where the graph $G$ is a tree. We construct a family of trees $G$ on $n$ vertices and pairs of search trees on $G$ such that the minimum number of rotations required to transform one search tree into the other is $\Omega (n\log n)$. This implies that the worst-case diameter of tree associahedra is $\Theta (n\log n)$, which answers a question from Thibault Manneville and Vincent Pilaud. The proof relies on a notion of projection of a search tree which may be of independent interest.

scholarly journals Comparing Leaf and Root Insertion

2010 ◽  
Vol 44 ◽  
Author(s):  
Jaco Geldenhuys ◽  
Brink Van der Merwe
Keyword(s):  
Standard Method ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Worst Case ◽  
Average Case ◽  
Tree Leaf ◽  
Leaf Insertion

We consider two ways of inserting a key into a binary search tree: leaf insertion which is the standard method, and root insertion which involves additional rotations. Although the respective cost of constructing leaf and root insertion binary search trees trees, in terms of comparisons, are the same in the average case, we show that in the worst case the construction of a root insertion binary search tree needs approximately 50% of the number of comparisons required by leaf insertion.

DYNAMIC OPTIMAL BINARY SEARCH TREE

1990 ◽  
Vol 01 (04) ◽  
pp. 449-463 ◽  
Author(s):  
A. P. KORAH ◽  
M. R. KAIMAL
Keyword(s):  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Basic Operation ◽  
Binary Search Trees ◽  
Search Trees ◽  
Worst Case ◽  
Insertion And Deletion ◽  
Optimal Tree

In this paper we present a strategy to maintain a dynamic optimal binary search tree. The algorithms for insertion and deletion use swapping as the basic operation. Since in average situations the tree reorganization is limited to local changes, it can be favourably compared with the local balancing algorithms. The present algorithms dynamically maintain the optimal tree with an amortized time of O(log2 n), where n is the total number of nodes in the tree. In the worst case situations, the algorithms take only O(n) time. This is significant when they are compared to the algorithms producing static optimal binary search trees.

scholarly journals On the Subtree Size Profile of Binary Search trees

2010 ◽  
Vol 19 (4) ◽  
pp. 561-578 ◽  
Author(s):  
FLORIAN DENNERT ◽  
RUDOLF GRÜBEL
Keyword(s):  
Qualitative Difference ◽  
Search Tree ◽  
Binary Search ◽  
Random Trees ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Joint Distributions ◽  
Size Profile ◽  
Process View

For random trees T generated by the binary search tree algorithm from uniformly distributed input we consider the subtree size profile, which maps k ∈ ℕ to the number of nodes in T that root a subtree of size k. Complementing earlier work by Devroye, by Feng, Mahmoud and Panholzer, and by Fuchs, we obtain results for the range of small k-values and the range of k-values proportional to the size n of T. In both cases emphasis is on the process view, i.e., the joint distributions for several k-values. We also show that the dynamics of the tree sequence lead to a qualitative difference between the asymptotic behaviour of the lower and the upper end of the profile.

Red-black trees in a functional setting

1999 ◽  
Vol 9 (4) ◽  
pp. 471-477 ◽  
Author(s):  
CHRIS OKASAKI
Keyword(s):  
Computer Science ◽  
Binary Search ◽  
Binary Search Trees ◽  
Search Trees ◽  
Introductory Computer Science ◽  
Science Classes ◽  
Functional Setting

Everybody learns about balanced binary search trees in their introductory computer science classes, but even the stouthearted tremble at the thought of actually implementing such a beast. The details surrounding rebalancing are usually just too messy. To show that this need not be the case, we present an algorithm for insertion into red-black trees (Guibas and Sedgewick, 1978) that any competent programmer should be able to implement in fifteen minutes or less.

scholarly journals Supernode Binary Search Trees

2003 ◽  
Vol 14 (03) ◽  
pp. 465-490 ◽  
Author(s):  
Haejae Jung ◽  
Sartaj Sahni
Keyword(s):  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Tree Structures

Balanced binary search tree structures such as AVL, red-black, and splay trees store exactly one element per node. We propose supernode versions of these structures in which each node may have a large number of elements. Some properties of supernode binary search tree structures are established. Experiments oonducted by us show that the supernode structures proposed by us use less space than do the corresponding one-element-per-node versions and also take less time for the standard dictionary operations: search, insert and delete.

A Fast Contention-Friendly Binary Search Tree

2016 ◽  
Vol 26 (03) ◽  
pp. 1650015 ◽  
Author(s):  
Tyler Crain ◽  
Vincent Gramoli ◽  
Michel Raynal
Keyword(s):  
High Performance ◽  
State Of The Art ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Structural Adaptation ◽  
Tree Algorithm ◽  
Benchmark Suite

This paper presents a fast concurrent binary search tree algorithm. To achieve high performance under contention, the algorithm divides update operations within an eager abstract access that returns rapidly for efficiency reason and a lazy structural adaptation that may be postponed to diminish contention. To achieve high performance under read-only workloads, it features a rebalancing mechanism and guarantees that read-only operations searching for an element execute lock-free. We evaluate the contention-friendly binary search tree using Synchrobench, a benchmark suite to compare synchronization techniques. More specifically, we compare its performance against five state-of-the-art binary search trees that use locks, transactions or compare-and-swap for synchronization on Intel Xeon, AMD Opteron and Oracle SPARC. Our results show that our tree is more efficient than other trees and double the throughput of existing lock-based trees under high contention.

scholarly journals Profiles of random trees: correlation and width of random recursive trees and binary search trees

2005 ◽  
Vol 37 (02) ◽  
pp. 321-341 ◽  
Author(s):  
Michael Drmota ◽  
Hsien-Kuei Hwang
Keyword(s):  
Asymptotic Estimate ◽  
Correlation Coefficients ◽  
Singularity Analysis ◽  
Binary Search ◽  
Random Trees ◽  
The Other ◽  
Binary Search Trees ◽  
Search Trees ◽  
Expected Width ◽  
Recursive Trees

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

STRONGER QUICKHEAPS

2011 ◽  
Vol 22 (04) ◽  
pp. 945-969
Author(s):  
GONZALO NAVARRO ◽  
RODRIGO PAREDES ◽  
PATRICIO V. POBLETE ◽  
PETER SANDERS
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Binary Search ◽  
Priority Queues ◽  
Binary Search Trees ◽  
Search Trees ◽  
Worst Case ◽  
Tree Algorithms ◽  
Amortized Complexity ◽  
Time Bounds

The Quickheap (QH) is a recent data structure for implementing priority queues which has proved to be simple and efficient in practice. It has also been shown to offer logarithmic expected amortized complexity for all of its operations. Yet, this complexity holds only when keys inserted and deleted are uniformly distributed over the current set of keys. This assumption is in many cases difficult to verify, and does not hold in some important applications such as implementing some minimum spanning tree algorithms using priority queues. In this paper we introduce an elegant model called a Leftmost Skeleton Tree (LST) that reveals the connection between QHs and randomized binary search trees, and allows us to define Randomized QHs. We prove that these offer logarithmic expected amortized complexity for all operations regardless of the input distribution. We also use LSTs in connection to α-balanced trees to achieve a practical α-Balanced QH that offers worst-case amortized logarithmic time bounds for all the operations. Both variants are much more robust than the original QHs. We show experimentally that randomized QHs behave almost as efficiently as QHs on random inputs, and that they retain their good performance on inputs where that of QHs degrades.

scholarly journals On the Number of Descendants and Ascendants in Random Search Trees

10.37236/1358 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Conrado Martínez ◽  
Alois Panholzer ◽  
Helmut Prodinger
Keyword(s):  
Generating Functions ◽  
Random Search ◽  
Natural Extension ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Higher Moments ◽  
Binary Search Trees ◽  
Search Trees ◽  
Explicit Formulae

The number of descendants of a node in a binary search tree (BST) is the size of the subtree having this node as a root; the number of ascendants is the number of nodes on the path connecting this node with the root. Using a purely combinatorial approach (generating functions and differential equations) we are able to extend previous results. For the number of descendants we get explicit formulæ for all moments; for the number of ascendants, which is harder, we get the variance. A natural extension of binary search trees occurs when performing local reorganisations. Poblete and Munro have already analyzed some aspects of these locally balanced binary search trees (LBSTs). Here, we relate these structures with the performance of median–of–three Quicksort. We get as new results the variances for ascendants and descendants in this setting. If the rank of the node itself is picked at random ("grand averages"), the corresponding parameters only depend on the size $n$. In this instance, we get all the moments for the descendants (BST and LBST), as well as the probabilities. For ascendants (LBST), we get the variance and (in principle) the higher moments, as well as the (normal) limiting distribution. The emphasis is on explicit formulæ, and these are sometimes quite involved. Thus, in some instances, we have decided to state abridged versions in the paper and collect the long forms into an appendix that can be downloaded from the URLs http://info.tuwien.ac.at/theoinf/abstract/abs_120.htm and http://www.lsi.upc.es/~conrado/research/.

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