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The Performance of Concurrent RedBlack Tree Algorithms
 Lecture Notes in Computer Science
, 1998
"... 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 bala ..."
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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 redblack 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 redblack 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 redblack trees, and the relaxed redblack tree. Furthermore, we compare the relaxed balancing algorithms with the standard redblack tree, i.e. the strictly balanced redblack tree combined with the locking scheme of El...
Relaxed balanced redblack trees
 In Proc. 3rd Italian Conference on Algorithms and Complexity
, 1997
"... 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 high ..."
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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 redblack 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 Balance through Standard Rotations
 In Fifth International Workshop on Algorithms and Data Structures
, 1997
"... 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 ..."
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Cited by 14 (4 self)
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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 mainmemory databases. Mainmemory search structures have gained new applications in large embedded systems, such as switching systems for mobile telephones.
Heightrelaxed AVL rebalancing: A unified, finegrained approach to concurrent dictionaries
, 1998
"... We address the concurrent rebalancing of almost balanced binary search trees (AVL trees). Such a rebalancing may for instance be necessary after successive insertions and deletions of keys. We show that this problem can be studied through the selfreorganization of distributed systems of nodes contr ..."
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We address the concurrent rebalancing of almost balanced binary search trees (AVL trees). Such a rebalancing may for instance be necessary after successive insertions and deletions of keys. We show that this problem can be studied through the selfreorganization of distributed systems of nodes controlled by local evolution rules in the line of the approach of Dijkstra and Scholten. This yields a much simpler algorithm that the ones previously known. Based on the basic rebalancing framework, we describe algorithms to manage concurrent insertion and deletion of keys. Finally, this approach is used to emulate other well known concurrent AVL algorithms. As a byproduct, this solves in a very general setting an old question raised by H.T. Kung and P.L. Lehman: where should rotations take place to rebalance arbitrary search trees? Keywords: Concurrent algorithms, Search trees, AVL trees, Concurrent insertions and deletions, Concurrent generalized rotations, Safety and liveness proofs, Emula...
Amortized Constant Relaxed Rebalancing Using Standard Rotations
 Acta Informatica
, 1998
"... The idea of relaxed balance is to uncouple the rebalancing in search trees from the updating in order to speed up request processing in mainmemory databases during bursts of updates. This paper contains the first proof that amortized constant time rebalancing can be obtained in a relaxed binary sea ..."
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The idea of relaxed balance is to uncouple the rebalancing in search trees from the updating in order to speed up request processing in mainmemory databases during bursts of updates. This paper contains the first proof that amortized constant time rebalancing can be obtained in a relaxed binary search tree using only standard single and double rotations. 1 Introduction The idea of relaxed balance is to uncouple the rebalancing in search trees from the updating in order to speed up request processing in mainmemory databases. The motivation is twofold: If search and update requests for a search tree come in bursts (possibly from several external sources), the search tree may occasionally not be able to process the requests as fast as it might be desirable. For this reason, it would be convenient to be able to "turn off" rebalancing for a short period of time in order to speed up the request processing. However, when the burst is over, the tree should be rebalanced again, while search...
A Unified Approach to Concurrent and Parallel Algorithms on Balanced Data Structures
 IN IEEE, EDITOR, PROC. OF XVII INT. CONF. CHILEAN COMPUTER SOCIETY
, 1997
"... Concurrent and parallel algorithms are different. However, in the case of dictionaries, both kinds of algorithms share many common points. We present a unified approach emphasizing these points. It is based on a careful analysis of the sequential algorithm, extracting from it the more basic facts, e ..."
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Cited by 5 (2 self)
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Concurrent and parallel algorithms are different. However, in the case of dictionaries, both kinds of algorithms share many common points. We present a unified approach emphasizing these points. It is based on a careful analysis of the sequential algorithm, extracting from it the more basic facts, encapsulated later on as local rules. We apply the method to the insertion algorithms in AVL trees. All the concurrent and parallel insertion algorithms have two main phases. A percolation phase, moving the keys to be inserted down, and a rebalancing phase. Finally, some other algorithms and balanced structures are discussed.
Chromatic Search Trees Revisited
, 1997
"... Relaxed balancing of search trees was introduced with the aim of speeding up the updates and allowing a high degree of concurrency. In a relaxed version of a search tree the rebalancing operations are uncoupled from the updates and may be delayed. Using local transformations the rebalancing can be p ..."
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Relaxed balancing of search trees was introduced with the aim of speeding up the updates and allowing a high degree of concurrency. In a relaxed version of a search tree the rebalancing operations are uncoupled from the updates and may be delayed. Using local transformations the rebalancing can be performed concurrently with updates and search operations. In this paper we revisit the rebalancing of chromatic trees, a relaxed version of redblack trees. During the rebalancing of a chromatic tree it can occur that the rebalancing operations change the search structure of the tree in order to settle rebalancing requests from the history although the tree is already a balanced redblack tree. We propose how to perform the rebalancing task in such a way that each performed structural change indeed correspond to a real imbalance situation in the tree. The number of rebalancing operations needed to restore the redblack tree balance condition is still O(i+d) pointer changes and ...
View Serializable Updates of Concurrent Index Structures
 DBPL 2001 : 247262, LNCS 2397
, 2001
"... Abstract. We present new algorithms for concurrent reading and updating of B *trees and binary search trees. Our algorithms are based on the wellknown link technique, and improve previously proposed solutions in several respects. We prove formally that our algorithms are correct. We show that they ..."
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Abstract. We present new algorithms for concurrent reading and updating of B *trees and binary search trees. Our algorithms are based on the wellknown link technique, and improve previously proposed solutions in several respects. We prove formally that our algorithms are correct. We show that they satisfy a view serializability criterion, which fails for previous solutions. This stronger serializability criterion is central to the proof that several subtle (but essential) optimizations incorporated in our algorithms are correct. 1
Complexity of Layered Binary Search Trees with Relaxed Balance
, 1999
"... When search trees are made relaxed, balance constraints are weakened such that updates can be made without immediate rebalancing. This can lead to a speedup in some circumstances. However, the weakened balance constraints also make it more challenging to prove complexity results for relaxed str ..."
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When search trees are made relaxed, balance constraints are weakened such that updates can be made without immediate rebalancing. This can lead to a speedup in some circumstances. However, the weakened balance constraints also make it more challenging to prove complexity results for relaxed structures. In our opinion, one of the simplest and most intuitive presentations of balanced search trees has been given via layered trees. We show that relaxed layered trees are among the best of the relaxed structures. More precisely, rebalancing is worstcase logarithmic and amortized constant per update, and restructuring is worstcase constant per update. Introduction Usually, updating in a balanced search tree is carried out as follows: First, a search is carried out in order to determine the location of the update. Second, the update is performed. Third, local balance constraints are reconsidered. Supported in part by the Danish Natural Sciences Research Council (SNF). y Depart...