scholarly journals Cache Oblivious Search Trees via Binary Trees of Small Height

2001 ◽  
Vol 8 (36) ◽  
Author(s):  
Gerth Stølting Brodal ◽  
Rolf Fagerberg ◽  
Riko Jacob
Keyword(s):  
Data Structure ◽  
Binary Tree ◽  
Search Tree ◽  
Search Trees ◽  
Worst Case ◽  
Complexity Bounds ◽  
Element Size ◽  
Small Height ◽  
Data Elements ◽  
Cache Oblivious

We propose a version of cache oblivious search trees which is simpler than the previous proposal of Bender, Demaine and Farach-Colton and has the same complexity bounds. In particular, our data structure avoids the use of weight balanced B-trees, and can be implemented as just a single array of data elements, without the use of pointers. The structure also improves space utilization.<br /> <br />For storing n elements, our proposal uses (1+epsilon)n times the element size of memory, and performs searches in worst case O(log_B n) memory transfers, updates in amortized O((log^2 n)/(epsilon B)) memory transfers, and range queries in worst case O(log_B n + k/B) memory transfers, where k is the size of the output.<br /> <br />The basic idea of our data structure is to maintain a dynamic binary tree of height log n + O(1) using existing methods, embed this tree in a static binary tree, which in turn is embedded in an array in a cache oblivious fashion, using the van Emde Boas layout of Prokop.<br /> <br />We also investigate the practicality of cache obliviousness in the area of search trees, by providing an empirical comparison of different methods for laying out a search tree in memory.<br /> <br />The source code of the programs, our scripts and tools, and the data we present here are available online under ftp.brics.dk/RS/01/36/Experiments/.

Query-Optimal Partially Persistent B-Trees with Constant Worst-Case Update Time

2017 ◽  
Vol 28 (02) ◽  
pp. 141-169
Author(s):  
George Lagogiannis
Keyword(s):  
Data Structure ◽  
Data Structures ◽  
Search Tree ◽  
Worst Case ◽  
B Trees

In this paper we present the first data structure for partially persistent B-trees with constant worst-case update time, where deletions are handled in a symmetrical to insertions manner. Our structure matches the query times of optimal partially persistent B-trees that do not support constant update time thus, we have managed to reduce the worst-case update time to a constant, without a penalty in the query times. The new data structure is produced by mixing two other data structures, (a) the partially persistent B-tree and (b) the balanced search tree with constant worst-case update time.

scholarly journals DEGREE PROFILE OF m-ARY SEARCH TREES: A VEHICLE FOR DATA STRUCTURE COMPRESSION

2015 ◽  
Vol 30 (1) ◽  
pp. 113-123
Author(s):  
Ravi Kalpathy ◽  
Hosam Mahmoud
Keyword(s):  
Phase Transition ◽  
Data Structure ◽  
Search Tree ◽  
Behavior Changes ◽  
Search Trees ◽  
Asymptotically Normal ◽  
Space Saving ◽  
Individual Node

We revisit the random m-ary search tree and study a finer profile of its node outdegrees with the purpose of exploring possibilities of data structure compression. The analysis is done via Pólya urns. The analysis shows that the number of nodes of each individual node outdegree has a phase transition: Up to m = 26, the number of nodes of outdegree k, for k = 0, 1, …, m, is asymptotically normal; that behavior changes at m = 27. Based on the analysis, we propose a compact m-ary tree that offers significant space saving.

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.

A SIMPLE BALANCED SEARCH TREE WITH O(1) WORST-CASE UPDATE TIME

1996 ◽  
Vol 07 (02) ◽  
pp. 137-149 ◽  
Author(s):  
RUDOLF FLEISCHER
Keyword(s):  
Data Structure ◽  
Slight Modification ◽  
Search Tree ◽  
Optimal Solutions ◽  
Worst Case ◽  
Finger Search

In this paper we show how a slight modification of (a, 2b)-trees allows us to perform member and neighbor queries in O( log n) time and updates in O(1) worst-case time (once the position of the inserted or deleted key is known). Our data structure is quite natural and much simpler than previous worst-case optimal solutions. It is based on two techniques : 1) bucketing, i.e., storing an ordered list of 2 log n keys in each leaf of an (a, 2b) tree, and 2) preventive splitting, i.e., splitting nodes before they can grow bigger than allowed. If only insertions are allowed, it can also be used as a finger search tree with O( log * n) worst-case update time.

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 Rotation Algorithms for Balancing Search Trees

1970 ◽  
Vol 10 (1) ◽  
pp. 26-32
Author(s):  
Chumphol Bunkhumpornpat ◽  
Varin Chouvatut ◽  
Saowaluk Rattanaudomsawat
Keyword(s):  
Data Structure ◽  
Hierarchical Structure ◽  
Time Complexity ◽  
Clustering Algorithm ◽  
Minimum Spanning Tree ◽  
Search Tree ◽  
Optimal Time ◽  
Worst Case ◽  
Cluster Membership ◽  
The Hierarchical Structure

A search tree is a data structure constructed from a minimum spanning tree. This data structure is used for determining the cluster membership of a query instance clustered by a similarity-guaranteed clustering algorithm. According to the line topology of a search tree in the worst case, the time complexity of tree traversing is O(n) where n is the number of nodes of the tree. Unfortunately, the AVL tree algorithm cannot solve this problem because the algorithm is unable to maintain the hierarchical structure of a search tree. From the definition of balance factor, our proposed algorithm is designed to rotate nodes until the tree becomes balanced while maintaining the hierarchical structure. Consequently, the balanced search tree achieves the optimal time complexity of O(log n) for the searching purpose.

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.

Improving the Red-Black tree delete algorithm

2021 ◽  
Author(s):  
ZEGOUR Djamel Eddine
Keyword(s):  
Data Structure ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Search Performance ◽  
Running Time ◽  
Standard Algorithm ◽  
Color Changes ◽  
Red Black Tree ◽  
Symbol Tables

Abstract Today, Red-Black trees are becoming a popular data structure typically used to implement dictionaries, associative arrays, symbol tables within some compilers (C++, Java …) and many other systems. In this paper, we present an improvement of the delete algorithm of this kind of binary search tree. The proposed algorithm is very promising since it colors differently the tree while reducing color changes by a factor of about 29%. Moreover, the maintenance operations re-establishing Red-Black tree balance properties are reduced by a factor of about 11%. As a consequence, the proposed algorithm saves about 4% on running time when insert and delete operations are used together while conserving search performance of the standard algorithm.

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