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

Relaxed balancing has become a commonly used concept in the design of concurrent search tree algorithms. The idea of relaxed balancing is to uncouple the rebalancing from the updating in order to speed up the update operations and to allow a high degree of concurrency. Many different relaxed balancing algorithms have been proposed, especially for red-black trees and AVL trees, but their performance in concurrent environments is not yet well understood. This paper presents an experimental comparison of three relaxed balancing algorithms for red-black trees. Using the simulation of a multi processor environment we study the performance of chromatic trees, the algorithm that is got by applying the general method of how to make strict balancing schemes relaxed to red-black trees, and the relaxed redblack tree. Furthermore, we compare the relaxed balancing algorithms with the standard red-black tree, i.e. the strictly balanced red-black tree combined with the locking scheme of El...

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. The contribution of the present paper is that we introduce a new scheme for relaxed balancing, which is obtained by a simple generalization of strict balancing. Our approach implies a simple proof of the fact that the number of the needed rebalancing operations (to put the tree in balance) for relaxed balancing is the same as for strict balancing. 1 Introduction A dictionary is a scheme for storing a set of data such that the operations sea...