Parallel dictionaries with local rules on AVL and brother trees

We present a set of local rules to deal with dictionaries, having as a main advantage their possible scheduling in a highly synchronized way to get parallel dictionaries on AVLs. Up to now trees used in massively parallel dictionaries needed to have all the leaves at the same depth, such as 2--3 trees. Therefore, it was possible (in insertions and deletions) to reconstruct the tree bottom-up in a very regular fashion, as a pipeline of plane waves moving up. On AVL trees the situation looks different because leaves can have different depth, therefore any wave in a pipeline is highly irregular. To solve this problem we define {\it virtual} plane waves allowing us to develop an EREW dictionary for $k$ keys with $k$ processors and time $O (\log n + \log k)$. Later on we generalize the sePostprint (published version

A Unified approach to concurrent and parallel algorithms on balanced data structures

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.Postprint (published version

Concurrent rebalancing on hyperred-black trees

The HyperRed-Black trees are a relaxed version of Red-Black trees accepting high degree of concurrency. In the Red-Black trees consecutive red nodes are forbidden. This restriction has been withdrawn in the Chromatic trees. They have been introduced by O.~Nurmi and E.~Soisalon-Soininen to work in a concurrent environment. A Chromatic tree can have big clusters of red nodes surrounded by black nodes. Nevertheless, concurrent rebalancing of Chromatic trees into Red-Black trees has a serious drawback: in big cluster of red nodes only the top node can be updated. Direct updating inside the cluster is forbidden. This approach gives us limited degree of concurrency. The HyperRed-Black trees has been designed to solve this problem. It is possible to update red nodes in the inside of a red cluster. In a HyperRed-Black tree nodes can have a multiplicity of colors; they can be red, black or hyper-red.Postprint (published version

Fringe analysis for parallel MacroSplit insertion algorithms in 2--3 trees

We extend the fringe analysis (used to study the expected behavior of balanced search trees under sequential insertions) to deal with synchronous parallel insertions on 2--3 trees. Given an insertion of k keys in a tree with n nodes, the fringe evolves following a transition matrix whose coefficients take care of the precise form of the algorithm but does not depend on k or n. The derivation of this matrix uses the binomial transform recently developed by P. Poblete, J. Munro and Th. Papadakis. Due to the complexity of the preceding exact analysis, we develop also two approximations. A first one based on a simplified parallel model, and a second one based on the sequential model. These two approximated analysis prove that the parallel insertions case does not differ significantly from the sequential case, namely on the terms O(1/n^2).Postprint (published version