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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 ..."
Abstract

Cited by 20 (0 self)
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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.
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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Cited by 11 (1 self)
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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.
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.
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.