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(M log(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.

Abstract. Relaxed balancing means that, in a dictionary stored as a balanced tree, the necessary rebalancing after updates may be delayed. This is in contrast to strict balancing meaning that rebalancing is performed immediately after the update. Relaxed balancing is important for efficiency in highly dynamic applications where updates can occur in bursts. The rebalancing tasks can be performed gradually after all urgent updates, allowing the concurrent use of the dictionary even though the underlying tree structure is not completely in balance. In this paper we propose a new scheme of how to make known rebalancing techniques relaxed in an efficient way. The idea is applied to the red-black trees, but can be applied to any class of balanced trees. The key idea is to accumulate insertions and deletions such that they can be settled in arbitrary order using the same rebalancing operations as for standard balanced search trees. As a result it can be shown that the number of needed rebalancing operations known from the strict balancing scheme carry over to relaxed balancing. 1

This paper describes the first complete implementation of a non-blocking binary search tree in an asynchronous shared-memory system using single-word compare-and-swap operations. The implementationis linearizable andtolerates anynumberofcrash failures. InsertandDeleteoperations thatmodify different parts of the tree do not interfere with one another, so they can run completely concurrently. Find operations only perform reads of shared memory. 1

We consider the use of search trees to represent the dictionary aspects of a main-memory database in a concurrent environment. Efficiency considerations require that the trees be balanced and that operations on a search tree should not block too large a part of the tree for too long a time. We propose a new, elegant solution for this problem based on the notion of relaxed AVL trees and the decoupling of updates and rebalancing. The main advantage of our solution is that the implementation of the dictionary operations in a concurrent environment is as simple as their implementation in a sequential environment, whereas previous concurrent solutions are more descriptively complex. x This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada, from the Information Technology Research Centre of Ontario, and from the Research Grants Committee of Hong Kong, and partially by the Academy of Finland. The Hong Kong University of Science ...