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General balanced trees
 Journal of Algorithms
, 1999
"... We show that, in order to achieve efficient maintenance of a balanced binary search tree, no shape restriction other than a logarithmic height is required. The obtained class of trees, general balanced trees, may be maintained at a logarithmic amortized cost with no balance information stored in the ..."
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We show that, in order to achieve efficient maintenance of a balanced binary search tree, no shape restriction other than a logarithmic height is required. The obtained class of trees, general balanced trees, may be maintained at a logarithmic amortized cost with no balance information stored in the nodes. Thus, in the case when amortized bounds are sufficient, there is no need for sophisticated balance criteria. The maintenance algorithms use partial rebuilding. This is important for certain applications and has previously been used with weightbalanced trees. We show that the amortized cost incurred by general balanced trees is lower than what has been shown for weightbalanced trees. � 1999 Academic Press 1.
Fast updating of wellbalanced trees
 In SWAT 90, 2nd Scandinavian Workshop on Algorithm Theory
, 1990
"... Trees of optimal and nearoptimal height may be represented as a pointerfree structure in an array of size O(n). In this way we obtain an array implementation of a dictionary with O(log n) search cost and O(log2 n) update cost, allowing interpolation search to improve the expected search time. 1 In ..."
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Trees of optimal and nearoptimal height may be represented as a pointerfree structure in an array of size O(n). In this way we obtain an array implementation of a dictionary with O(log n) search cost and O(log2 n) update cost, allowing interpolation search to improve the expected search time. 1 Introduction The binary search tree is a fundamental and well studied data structure, commonly used in computer applications to implement the abstract data type dictionary. In a comparisonbased model of computation, the lower bound on the three basic operations insert, delete and search is dlog(n + 1)e comparisons per operation. This bound may be achieved by storing the set in a binary search tree of optimal height. Definition 1 A binary tree has optimal height if and only if the height of the tree is dlog(n + 1)e. A special case of a tree of optimal height is an optimally balanced tree, as defined below. Definition 2 A binary tree is optimally balanced if and only if the difference in length between the longest and shortest paths is at most one.
Binary Search Trees of Almost Optimal Height
 ACTA INFORMATICA
, 1990
"... First we present a generalization of symmetric binary Btrees, SBB(k) trees. The obtained structure has a height of only \Sigma (1 + 1k) log(n + 1)\Upsilon, where k may be chosen to be any positive integer. The maintenance algorithms require only a constant number of rotations per updating operati ..."
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First we present a generalization of symmetric binary Btrees, SBB(k) trees. The obtained structure has a height of only \Sigma (1 + 1k) log(n + 1)\Upsilon, where k may be chosen to be any positive integer. The maintenance algorithms require only a constant number of rotations per updating operation in the worst case. These properties together with the fact that the structure is relatively simple to implement makes it a useful alternative to other search trees in practical applications. Then, by using an SBB(k)tree with a varying k we achieve a structure with a logarithmic amortized cost per update and a height of log n + o(log n). This result is an improvement of the upper bound on the height of a dynamic binary search tree. By maintaining two trees simultaneously the amortized cost is transformed into a worstcase cost. Thus, we have improved the worstcase complexity of the dictionary problem.
Concurrent Perfect Balancing of Binary Search Trees
"... When a balanced data structure is updated and searched concurrently, updating and balancing should be decoupled so as to make updating faster. The balancing is done by special maintenance processes that run concurrently with the search and update tasks. We show that it is not necessary to use a wea ..."
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When a balanced data structure is updated and searched concurrently, updating and balancing should be decoupled so as to make updating faster. The balancing is done by special maintenance processes that run concurrently with the search and update tasks. We show that it is not necessary to use a weak balance condition like AVL or redblack condition, since balancing a binary tree perfectly so that the search paths become as short as possible is not much more expensive, that is, a process must lock only 5 nodes at a time even when perfect balance is desired. In contrast to other algorithms that perfectly balance a binary search tree, our algorithm keeps the tree (weakly) balanced during the further balancing. This is important if the data structure is used by concurrent search and update processes.
Puneet Kumar
"... Exponential Tree in the form of forest is proposed in such a manner that (a) it provides faster access of a node and, (b) it becomes more compatible with the parallel environment. Empirically, it has been show that the proposed method decreases the total internal path length of an Exponential Tree ..."
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Exponential Tree in the form of forest is proposed in such a manner that (a) it provides faster access of a node and, (b) it becomes more compatible with the parallel environment. Empirically, it has been show that the proposed method decreases the total internal path length of an Exponential Tree quite considerably. The experiments were conducted by creating three different data structures using the same input a conventional binary tree, a forest of hashed binary trees and a forest of hashed exponential trees. It has been shown that a forest of hashed exponential trees so produced has lesser internal path length and height in comparison of other two. It also increases the degree of parallelism.
Running head General Balanced Trees
"... 2 Abstract We show that, in order to achieve efficient maintenance of a balanced binary search tree, no shape restriction other than a logarithmic height is required. The obtained class of trees, general balanced trees, may be maintained at a logarithmic amortized cost with no balance information st ..."
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2 Abstract We show that, in order to achieve efficient maintenance of a balanced binary search tree, no shape restriction other than a logarithmic height is required. The obtained class of trees, general balanced trees, may be maintained at a logarithmic amortized cost with no balance information stored in the nodes. Thus, in the case when amortized bounds are sufficient, there is no need for sophisticated balance criteria. The maintenance algorithms use partial rebuilding. This is important for certain applications, and has previously been used with weightbalanced trees. We show that the amortized cost incurred by general balanced trees is lower than what has been shown for weightbalanced trees. 3 1 Introduction One of the fundamental data structures in computer science is the binary search tree. New methods to maintain data in search trees have been developed and thoroughly analyzed all through the history of the discipline. Attention has mainly been focused on trees with bounded height or balanced trees. The reason for this is obvious since worstcase access time is proportional to the height of a tree. The traditional way to maintain balance at a low cost is by means of some more os less sophisticated balance criterion. To illustrate the large variation in the world of balance criteria some examples are given below. We use jT j to denote the number of leaves (weight) of the tree T.
A Forest of Hashed Binary Search Trees with Reduced Internal Path Length and better Compatibility with the Concurrent Environment
"... We propose to maintain a Binary Search Tree in the form of a forest in such a way that – (a) it provides faster node access and, (b) it becomes more compatible with the concurrent environment. Using a small array, the stated goals were achieved without applying any restructuring algorithm. Empirical ..."
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We propose to maintain a Binary Search Tree in the form of a forest in such a way that – (a) it provides faster node access and, (b) it becomes more compatible with the concurrent environment. Using a small array, the stated goals were achieved without applying any restructuring algorithm. Empirically, we have shown that the proposed method brings down the total internal pathlength of a Binary Search Tree quite considerably. The experiments were conducted by creating two different data structures using the same input a conventional binary search tree, and a forest of hashed trees. Our empirical results suggest that the forest so produced has lesser internal path length and height in comparison to the conventional tree. A binary search tree is not a wellsuited data structure for concurrent processing. The evidence also shows that maintaining a large tree in form of multiple smaller trees (forest) increases the degree of parallelism.
with minimum of one Rotation for Greater Elements from BST
"... Tree is a best data structure for data storage and retrieval of data whenever it could be accommodated in the memory. At the same time, this is true only when the tree is heightbalanced and lesser depth from the root. In this paper, we propose a sorting based new algorithm to construct the Balanced ..."
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Tree is a best data structure for data storage and retrieval of data whenever it could be accommodated in the memory. At the same time, this is true only when the tree is heightbalanced and lesser depth from the root. In this paper, we propose a sorting based new algorithm to construct the Balanced search tree from Binary Search Tree with minimum of one rotation for the given elements n>14. If the given elements n < 14 then the algorithm automatically constructs the Balanced Search tree without needs any rotations. To maintain the tree in shape and depth, we apply two strategies in the input data. The first one is to apply sorting on the given input data. And the second one is to find the multiples of two positions on the sorted input data. Then, we compare the 3 positions of multiples of two and rewrite it by descending order and repeat this for the entire elements and the rest of the positions also on the sorted data. After, a new input data is formed. Then construct the Binary search tree on the given input data. At last, we will find the output as; a height balanced BST (AVL) with lesser depth from the root for the smaller data such as N < 14, for the greater element N>14, it requires one rotation from the BST., and the search cost is minimum as possible. In this paper, few case studies have been carried out and analyzed in terms of height and space requirement. Hence, the height of the output BST, normally obtain by
Maintaining alphabalanced Trees by Partial Rebuilding
"... The balance criterion defining the class of ffbalanced trees states that the ratio between the shortest and longest paths from a node to a leaf be at least ff. We show that a straightforward use of partial rebuilding for maintenance of ffbalanced trees requires an amortized cost of \Omega (pn) p ..."
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The balance criterion defining the class of ffbalanced trees states that the ratio between the shortest and longest paths from a node to a leaf be at least ff. We show that a straightforward use of partial rebuilding for maintenance of ffbalanced trees requires an amortized cost of \Omega (pn) per update. By slight modifications of the maintenance algorithms the cost can be reduced to O(log n) for any value of ff, 0! ff! 1.
Binary Search Tree Balancing Methods: A Critical Study
"... Binary search tree is a bestsuited data structure for data storage and retrieval when entire tree could be accommodated in the primary memory. However, this is true only when the tree is heightbalanced. Lesser the height faster the search will be. Despite of the wide popularity of Binary search tr ..."
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Binary search tree is a bestsuited data structure for data storage and retrieval when entire tree could be accommodated in the primary memory. However, this is true only when the tree is heightbalanced. Lesser the height faster the search will be. Despite of the wide popularity of Binary search trees there has been a major concern to maintain the tree in proper shape. In worst case, a binary search tree may reduce to a linear link list, thereby reducing search to be sequential. Unfortunately, structure of the tree depends on nature of input. If input keys are not in random order the tree will become higher and higher on one side. In addition to that, the tree may become unbalanced after a series of operations like insertions and deletions. To maintain the tree in optimal shape many algorithms have been presented over the years. Most of the algorithms are static in nature as they take a whole binary search tree as input to create a balanced version of the tree. In this paper, few techniques have been discussed and analyzed in terms of time and space requirement. Key words: