Randomization of search trees by subtree size

Abstract

In this paper we present randomized algorithms over binary search trees such that: a) the insertion of a set of keys in any fixed order into an initially empty tree always produces a random binary search tree; b) the deletion of any key from a random binary search tree results in a random binary search tree; c) the random choices made by the algorithms are based upon the sizes of the subtrees of the tree; this will imply that we will be able to support accesses by rank without additional storage requirements or modification of the data structures; and d) the cost of any elementary operation, measured as the number of visited nodes, is the same as the expected cost of its standard deterministic counterpart; hence, all operations have thus guaranteed expected cost O(log n), but now irrespective of any assumption on the input distribution.