Relaxed balance through standard rotations

Abstract

We consider binary search trees, where rebalancing transformations need not be connected with updates but may be delayed. For standard AVL tree rebalancing, we prove that even though the rebalancing operations are uncoupled from updates, their total number is bounded by O(Mlog(M + N)), where M is the number of updates to an AVL tree of initial size N. Hence, relaxed balancing of AVL trees comes at no extra cost asymptotically. Furthermore, our scheme differs from most other relaxed balancing schemes in an important aspect: No rebalancing transformation can be done in the wrong direction, i.e., no performed rotation can make the tree less balanced. Moreover, each performed rotation indeed corresponds to a real imbalance situation in the tree.

Our results are important in designing efficient concurrency control strategies for main-memory databases. Main-memory search structures have gained new applications in large embedded systems, such as switching systems for mobile telephones.

Some of the work was done while this author was visiting the Department of Computer Sciences, University of Wisconsin at Madison. The work of this author was supported in part by SNF (Denmark), in part by NSF (U.S.) grant CCR-9510244, and in part by the ESPRIT Long Term Research Programme of the EU under project number 20244 (ALCOM-IT).