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Fringe Analysis Revisited
"... Fringe analysis is a technique used to study the average behavior of search trees. In this paper we survey the main results regarding this technique, and we improve a previous asymptotic theorem. At the same time we present new developments and applications of the theory which allow improvements in ..."
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Cited by 12 (6 self)
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Fringe analysis is a technique used to study the average behavior of search trees. In this paper we survey the main results regarding this technique, and we improve a previous asymptotic theorem. At the same time we present new developments and applications of the theory which allow improvements in several bounds on the behavior of search trees. Our examples cover binary search trees, AVL trees, 23 trees, and Btrees. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity ]: Nonnumerical Algorithms and Problems  computations on discrete structures; sorting and searching; E.1 [Data Structures]; trees. Contents 1 Introduction 2 2 The Theory of Fringe Analysis 4 3 Weakly Closed Collections 9 4 Including the Level Information 11 5 Fringe Analysis, Markov Chains, and Urn Processes 13 This work was partially funded by Research Grant FONDECYT 930765. email: rbaeza@dcc.uchile.cl 1 Introduction Search trees are one of the most used data structures t...
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 6 (3 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.
Massively Parallel and Distributed Dictionaries on AVL and Brother Trees
 In Parallel and Distributed Computing Systems
, 1996
"... 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 ..."
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Cited by 4 (4 self)
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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 23 trees. Therefore, it was possible (in insertions and deletions) to reconstruct the tree bottomup 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...
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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Cited by 3 (0 self)
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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...
Parallel Dictionaries with Local Rules on AVL and Brother Trees
 Information Processing Letters
, 1998
"... 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 23 ..."
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Cited by 1 (1 self)
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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 23 trees. Therefore, it was possible (in insertions and deletions) the bottomup 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...
Improved Bounds for the Expected Behaviour of AVL Trees
 BIT
, 1992
"... In this paper we improve previous bounds on expected measures of AVL trees by using fringe analysis. A new way of handling larger tree collections that are not closed is presented. An inherent difficulty posed by the transformations necessary to keep the AVL tree balanced makes its analysis difficul ..."
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Cited by 1 (1 self)
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In this paper we improve previous bounds on expected measures of AVL trees by using fringe analysis. A new way of handling larger tree collections that are not closed is presented. An inherent difficulty posed by the transformations necessary to keep the AVL tree balanced makes its analysis difficult when using fringe analysis methods. We derive a technique to cope with this difficulty obtaining the exact solution for fringe parameters even when unknown probabilities are involved. We show that the probability of a rotation in an insertion is between 0.37 and 0.73 (and seems to be less than 0.56), that the fraction of balanced nodes is between 0.56 and 0.78, and that the expected number of comparisons in a search seems to be at most 12% more than in the complete balanced tree. CR Categories: E.1, F.2.2, G.2.1. 1 Introduction Balanced tree structures are efficient ways of storing information. They provide an excellent solution for the dictionary data structure problem. For N elements t...
ObjectBased and ImageBased Object Representations
 ACM Computing Surveys
, 2004
"... An overview is presented of objectbased and imagebased representations of objects by their interiors. The representations are distinguished by the manner in which they can be used to answer two fundamental queries in database applications: (1) Feature query: given an object, determine its constitu ..."
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Cited by 1 (0 self)
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An overview is presented of objectbased and imagebased representations of objects by their interiors. The representations are distinguished by the manner in which they can be used to answer two fundamental queries in database applications: (1) Feature query: given an object, determine its constituent cells (i.e., their locations in space). (2) Location query: given a cell (i.e., a location in space), determine the identity of the object (or objects) of which it is a member as well as the remaining constituent cells of the object (or objects). Regardless of the representation that is used, the generation of responses to the feature and location queries is facilitated by building an index (i.e., the result of a sort) either on the objects or on their locations in space, and implementing it using an access structure that correlates the objects with the locations. Assuming the presence of an access structure, implicit (i.e., imagebased) representations are described that are good for finding the objects associated with a particular location or cell (i.e., the location query), while requiring that all cells be examined when determining the locations associated with a particular object (i.e., the feature query). In contrast, explicit (i.e., objectbased) representations are good for the feature query,