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, 2-3 trees, and B-trees. 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 93-0765. e-mail: rbaeza@dcc.uchile.cl 1 Introduction Search trees are one of the most used data structures t...

We present a fourth-order fringe analysis for the expected behavior of 2-3 trees, which includes 97% of the elements in the tree. It is accomplished by exploiting the structure of the transition matrix. Our results improve a number of bounds, in particular the bounds on the expected number of nodes and the expected space utilization. We also study 2-3 trees built by using overflow techniques. 1 Introduction Fringe analysis was formally introduced by Yao in 1974 [Yao74, Yao78] as a method to analyze search trees that considers only the bottom part or fringe of the tree. From the behavior of the subtrees in the fringe, it is possible to obtain bounds on most complexity measures for the complete tree, as well as some exact results. Classical fringe analysis considers only insertions. The model assumes that the n! possible permutations of the n keys used as input are equally likely. A search tree built under this model is called a random tree. This is equivalent to saying that the n-th in...

We consider the use of search trees to represent the dictionary aspects of a main-memory database in a concurrent environment. Efficiency considerations require that the trees be balanced and that operations on a search tree should not block too large a part of the tree for too long a time. We propose a new, elegant solution for this problem based on the notion of relaxed AVL trees and the decoupling of updates and rebalancing. The main advantage of our solution is that the implementation of the dictionary operations in a concurrent environment is as simple as their implementation in a sequential environment, whereas previous concurrent solutions are more descriptively complex. x This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada, from the Information Technology Research Centre of Ontario, and from the Research Grants Committee of Hong Kong, and partially by the Academy of Finland. The Hong Kong University of Science ...