We present a set of local rules to deal with distributed dictionaries, having as a main advantage their possible scheduling in a highly synchronized way to get massively parallel dictionaries on AVLs. Recall that, 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 straight plane weaves moving up. On AVL trees the situation looks different because leaves can have different depth, therefore any weave in a pipeline is highly irregular. To solve this problem we define 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 sequential algorithms on brother trees presented by T. Ottmann and D. Wood in the same way. 1 AVL Trees They are a basic data structure [1, 5]. However it left open how to desi...

We present a new type of search trees, called Skip trees, which are a generalization of Skip lists. To be precise, there is a one-to-one mapping between the two data types which commutes with the sequential update algorithms. A Skip list is a data structure used to manage data bases which stores values in a sorted way and in which it is insured that the form of the Skip list is independent of the order of updates by using randomization techniques. Skip trees inherit all the properties of Skip lists, including the time bounds of sequential algorithms. The algorithmic improvement of the Skip tree type is that a concurrent algorithm on the fly approach can be designed. Among other advantages, this algorithm is more compressive than the one designed by Pugh for Skip lists and accepts a higher degree of concurrence because it is based on a set of local updates. From a practical point of view, although the Skip list should be in the main memory, Skip trees can be registered into a secondary...

We present a set of local rules to deal with dictionaries. Their main advantage is that they can be scheduled in a highly synchronized way to get parallel dictionaries on AVL trees. 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) the bottom-up reconstruction of the tree 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 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 sequential algorithms on brother trees presented by Ottmann and Wood in the same way. Key words: Parallel algorithms, Parallel dictionaries, AVL trees, Brother trees. 1 Introduction AVL trees are an important bas...

. The fringe analysis studies the distribution of bottom subtrees or fringe of trees under the assumption of random selection of keys, yielding an average case analysis of the fringe of trees. We are interested in the fringe analysis of the synchronized parallel insertion algorithms of Paul, Vishkin, and Wagener (PVW) on 2--3 trees. This algorithm inserts k keys with k processors into a tree of size n with time O(log n+ log k). As the direct analysis of this algorithm is very difficult we tackle this problem by introducing a new family of algorithms, denoted MacroSplit algorithms, and our main theorem proves that two algorithms of this family, denoted MaxMacroSplit and MinMacroSplit, upper and lower bounds the fringe of the PVW algorithm. Published papers deal with the fringe analysis of sequential algorithms and it was an open problem for parallel algorithms on search trees. We extend the fringe analysis to parallel algorithms and we get a rich mathematical structure giving new interp...