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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 11 (5 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...
Statistics on Random Trees
, 1991
"... In this paper we give a survey of the symbolic operator methods to do statistics on random trees. We present some examples and apply the techniques to find their asymptotic behaviour. 1 Introduction Let us consider a class E of combinatorial objects, let A be an algorithm defined over the class ..."
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Cited by 2 (0 self)
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In this paper we give a survey of the symbolic operator methods to do statistics on random trees. We present some examples and apply the techniques to find their asymptotic behaviour. 1 Introduction Let us consider a class E of combinatorial objects, let A be an algorithm defined over the class E, and let denote the complexity measure we are interested in. Such a class E of combinatorial objects consists on a set, usually denoted by the same name as the class, and a size measure j \Delta j E : E \Gamma! IN. The subscript E in j \Delta j E will be dropped whenever it is clear from the context. We shall denote by E n the set of objects in E of size n. To analyze the average behaviour of A on an input e 2 E n with respect to measure means to compute A (n) = EfA (e) j e 2 E n g; (1:1) where EfXg denotes the expectation of the random variable X [Knu68, VF90]. By definition of expectation, Equation (1.1) can be written as A (n) = X k k PrfA (e) = k j e 2 E n g = X e2En Prfeg ...
Average Case Analysis of Gosper's Algorithm
, 2003
"... Abstract Gosper's algorithm is an automatic procedure that provides an answer to the question if a sum of hypergeometric terms can be expressed as the difference of a hypergeometric term and a constant. In order to asses the practical applicability of the algorithm, it is important to know its ..."
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Abstract Gosper's algorithm is an automatic procedure that provides an answer to the question if a sum of hypergeometric terms can be expressed as the difference of a hypergeometric term and a constant. In order to asses the practical applicability of the algorithm, it is important to know its expected running time. In this paper, an asymptotic average case analysis of one of the most important steps of Gosper's algorithm is performed. The space of input functions of the algorithm is described in terms of urn models, and the analysis is performed by using specialized probabilistic transform techniques. Among the obtained results are asymptotic expressions for the complexity of the socalled cancelling procedure and asymptotic expressions for the expected size of the system of linear equations that determines the existence of a solution. The results show that the average complexity of Gosper's algorithm is of the same order for several different types of probabilistic input models.