Efficient Reorganization of Binary Search Trees

Abstract.

We consider the problem of maintaining a binary search tree ({bst}) that minimizes the average access cost needed to satisfy randomly generated requests. We analyze scenarios in which the accesses are generated according to a vector of fixed probabilities which is unknown . Our approach is statistical.

We devise policies for modifying the tree structure dynamically, using rotations of accessed records. The aim is to produce good approximations of the optimal structure of the tree, while keeping the number of rotations as small as possible. The heuristics that we propose achieve a close approximation to the optimal BST, with lower organization costs than any previously studied.

We introduce the MOVE_ONCE rule. The average access cost to the tree under this rule is shown to equal the value achieved by the common rule Move to the Root (MTR). The advantage of MOVE_ONCE over MTR and similar rules is that it relocates each of the items in the tree at most once. We show that the total expected cost of modifying the tree by the MOVE_ONCE rule is bounded from above by 2(n+1)H _n -4n rotations (in a tree with n records), where H _n is the n th harmonic number. Extensive experiments show that this value is an overestimate, and in fact the number of rotations is linear for all the access probability vectors we tested. An approximate analysis is shown to match the experimental results, producing the expected number n((π ² /3)-2) - 2\ln n+0.1354 .

Next we combine the MOVE_ONCE rule with reference counters, one per record, that provide estimates of the reference probabilities. We call the resulting reorganization rule MOUCS. We show that, for any δ , α >0 and sufficiently large n , it achieves a cost that approaches the optimum up to an absolute difference of δ with probability higher than 1- α , within a number of accesses that is proportional to n (\lg n) ² /(αδ ² ) .