Efficient reorganization of binary search trees

Abstract

We consider the problem of maintaining a binary search tree that minimizes the average access cost with respect to randomly generated requests. We analyze scenarios, in which the accesses are generated according to a vector of probabilities, which is fixed but unknown.

In this paper we devise policies for modifying the tree structure dynamically, using rotations of accessed elements towards the root. Our aim is to produce good approximations of the optimal order of the tree, while minimizing the amount of rotations.

We first introduce the Move Once (mo) rule, under which the average access cost to the tree is shown to equal the average access cost under the commonly used Move to the Root (mtr), at each reference. The advantage of mo over other rules is that mo relocates each of the items in the tree at most once. Consequently, modifying the tree by the mo rule results in O(nlgn) rotations (with n the number of items) for any infinite sequence of accesses.

Then we propose to combine the mo with the usage of counters (accumulating the reference history for each item), that provide approximations of the reference probabilities. We show, that for any δ, α>0, this rule (which we call moucs) approaches the optimal cost to within a difference of δ with probability higher than 1−α, after a number of accesses, which is linear in n times 1/α times 1/δ ².

Author supported in part by the Technion V.P.R. Fund — E. and J. Bishop Research Fund and by the Fund for the Promotion of Research at the Technion.