Performance of height-balanced trees

Abstract

This paper presents the results of simulations that investigate the performance of height-balanced (HB[k]} trees. It is shown that the only statistic of HB[1] trees (AVL trees) that is a function of the size of the tree is the time to search for an item in the tree. For sufficiently large trees, the execution times of all procedures for maintaining HB[1] trees are independent of the size of the tree. In particular, an average of .465 restructures are required per insertion, with an average of 2.78 nodes revisited to restore the HB[1] property; an average of .214 restructures are required per deletion, with an average of 1.91 nodes revisited to restore the HB[1] property. Moreover, the execution times of procedures for maintaining HB[k] trees, for k > 1, are also independent of the size of the tree except for the average number of nodes revisited on a delete operation in order to restore the HB[k] property on traceback. The cost of maintaining HB[k] trees drops sharply as the allowable imbalance (k) increases. Both analytical and experimental results that show the cost of maintaining HB [k ] trees as a function of k are discussed.

A new class of balanced search trees : half-balanced binary search tress

“…The number or rebalancing opérations due to an insertion or a deletion is constant on the average as has been shown experimentally for AVL-trees [5] and SBB-trees [14], and analytically for BB (1 -N /2/2)-trees [8,4]. The maximal number of rebalancing opérations due to a deletion is O (lg n) for the three classes, and for an insertion it is also 0 (lg n) for SBB-trees and BB (1-^^-trees, but only 1 for AVL-trees [12,3].…”

“…We can firstly try to obtain a tree where h i2 < &n by a single left rotation in subtree 7~l 5 as illustrated in figure 6. …”

Rebalancing operations for deletions in AVL-trees

“…This measure is well established in the literature (Mehlhorn [7]) and led us to stronger results (our results hold for arbitrary not just random séquence of deletions) and suggested to use combinatorial (not probabilistic) methods of analysis. More precisely we show, that the total number of rebalancing opérations in processing a séquence of n arbitrary deletions from an AVL-tree with n leaves is bounded by 1,618 n. Expérimental data (Karlton et al [4]) suggests that the expected number of rebalancing opérations for n random deletions is 1,126 n and hence only slightly less than the amortized number. …”

“…2 or a. 3 the respective contribution of d x to the number of the unbalanced nodes will be chargea to the variables X 2 , X 3 or X 4 . Analogously a deletion in a subtree of type III destroys a balanced node and leads to an opération b (Absorption) and thus the contribution of d 3 to the mentioned number will be charged to the variable X 5 .…”

“…• Experiments in [4] show that the average size of X x is 0,912 and of X 3 +X 4 is 0,214 for a random deletion.…”