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11
Universal Limit Laws for Depths in Random Trees
 SIAM Journal on Computing
, 1998
"... Random binary search trees, bary search trees, medianof(2k+1) trees, quadtrees, simplex trees, tries, and digital search trees are special cases of random split trees. For these trees, we o#er a universal law of large numbers and a limit law for the depth of the last inserted point, as well as a ..."
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Cited by 50 (8 self)
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Random binary search trees, bary search trees, medianof(2k+1) trees, quadtrees, simplex trees, tries, and digital search trees are special cases of random split trees. For these trees, we o#er a universal law of large numbers and a limit law for the depth of the last inserted point, as well as a law of large numbers for the height.
An asymptotic theory for CauchyEuler differential equations with applications to the analysis of algorithms
, 2002
"... CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We ..."
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Cited by 22 (10 self)
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CauchyEuler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We study in this paper the most general framework for CauchyEuler equations and propose an asymptotic theory that covers almost all applications where CauchyEuler equations appear. Our approach is very general and requires almost no background on differential equations. Indeed the whole theory can be stated in terms of recurrences instead of functions. Old and new applications of the theory are given. New phase changes of limit laws of new variations of quicksort are systematically derived. We apply our theory to about a dozen of diverse examples in quicksort, binary search trees, urn models, increasing trees, etc.
Limit laws for local counters in random binary search trees
 Random Structures and Algorithms
, 1991
"... Limit laws for several quantities in random binary search trees that are related to the local shape of a tree around each node can be obtained very simply by applying central limit theorems for rndependent random variables. Examples include: the number of leaves (L a), the number of nodes with k de ..."
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Cited by 20 (2 self)
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Limit laws for several quantities in random binary search trees that are related to the local shape of a tree around each node can be obtained very simply by applying central limit theorems for rndependent random variables. Examples include: the number of leaves (L a), the number of nodes with k descendants (k fixed), the number of nodes with no left child, the number of nodes with k left descendants. Some of these results can also be obtained via the theory of urn models, but the present method seems easier to apply.
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...
Large Deviations for the Weighted Height of an Extended Class of Trees
 Algorithmica
, 2006
"... We use large deviations to prove a general theorem on the asymptotic edgeweighted height H ⋆ n of a large class of random trees for which H ⋆ n ∼ c log n for some positive constant c. A graphical interpretation is also given for the limit constant c. This unifies what was already known for binary ..."
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Cited by 12 (6 self)
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We use large deviations to prove a general theorem on the asymptotic edgeweighted height H ⋆ n of a large class of random trees for which H ⋆ n ∼ c log n for some positive constant c. A graphical interpretation is also given for the limit constant c. This unifies what was already known for binary search trees [11], [13], random recursive trees [12] and plane oriented trees [23] for instance. New applications include the heights of some random lopsided trees [19] and of the intersection of random trees.
Transitional Behaviors of the Average Cost of Quicksort With Medianof(2t + 1)
, 2001
"... A fine analysis is given of the transitional behavior of the average cost of quicksort with medianofthree. Asymptotic formulae are derived for the stepwise improvement of the average cost of quicksort when iterating medianofthree k rounds for all possible values of k. The methods used are genera ..."
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Cited by 11 (6 self)
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A fine analysis is given of the transitional behavior of the average cost of quicksort with medianofthree. Asymptotic formulae are derived for the stepwise improvement of the average cost of quicksort when iterating medianofthree k rounds for all possible values of k. The methods used are general enough to apply to quicksort with medianof(2t + 1) and to explain in a precise manner the transitional behaviors of the average cost from insertion sort to quicksort proper. Our results also imply nontrivial bounds on the expected height, "saturation level", and width in a random locally balanced binary search tree.
On the Number of Descendants and Ascendants in Random Search Trees
, 1997
"... We consider here the probabilistic analysis of the number of descendants and the number of ascendants of a given internal node in a random search tree. The performance of several important algorithms on search trees is closely related to these quantities. For instance, the cost of a successful searc ..."
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Cited by 7 (2 self)
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We consider here the probabilistic analysis of the number of descendants and the number of ascendants of a given internal node in a random search tree. The performance of several important algorithms on search trees is closely related to these quantities. For instance, the cost of a successful search is proportional to the number of ascendants of the sought element. On the other hand, the probabilistic behavior of the number of descendants is relevant for the analysis of paged data structures and for the analysis of the performance of quicksort, when recursive calls are not made on small subfiles. We also consider the number of ascendants and descendants of a random node in a random search tree, i.e., the grand averages of the quantities mentioned above. We address these questions for standard binary search trees and for locally balanced search trees. These search trees were introduced by Poblete and Munro and are binary search trees such that each subtree of size 3 is balanced; in oth...
Partial match queries in random kd trees
 SIAM Journal on Computing
, 2005
"... Abstract. We solve the open problem of characterizing the leading constant in the asymptotic approximation to the expected cost used for random partial match queries in random kd trees. Our approach is new and of some generality; in particular, it is applicable to many problems involving differenti ..."
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Cited by 3 (0 self)
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Abstract. We solve the open problem of characterizing the leading constant in the asymptotic approximation to the expected cost used for random partial match queries in random kd trees. Our approach is new and of some generality; in particular, it is applicable to many problems involving differential equations (or difference equations) with polynomial coefficients. Key words. kd trees, partialmatch queries, differential equations, averagecase analysis of algorithms, method of linear operators, asymptotic analysis. AMS subject classifications. 68W40 68P05 68P10 68U05 1. Introduction. Multidimensional
Hairy Search Trees
, 1995
"... Random search trees have the property that their depth depends on the order in which they are built. They have to be balanced in order to obtain a more efficient storageandretrieval data structure. Balancing a search tree is time consuming. This explains the popularity of data structures which app ..."
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Cited by 1 (0 self)
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Random search trees have the property that their depth depends on the order in which they are built. They have to be balanced in order to obtain a more efficient storageandretrieval data structure. Balancing a search tree is time consuming. This explains the popularity of data structures which approximate a balanced tree but have lower amortised balancing costs, such as AVL trees, Fibonacci trees and 23 trees. The algorithms for maintaining these data structures efficiently are complex and hard to derive. This observation has led to insertion algorithms that perform local balancing around the newly inserted node, without backtracking on the search path. This is also called a fringe heuristic. The resulting class of trees is referred to as 1locally balanced trees, in this note referred to as hairy trees. In this note a simple analysis of their behaviour is povided. Keywords: search trees, heuristic balancing, local balancing, fringe heuristic, hairy trees. Classification: AMS 68P05,...
Applications of Steins method in the analysis of random binary search trees. Steins method and Applications
 Institute for Mathematical Sciences Lecture Notes Series
, 2005
"... Abstract. Under certain conditions, sums of functions of subtrees of a random binary search tree are asymptotically normal. We show how Stein’s method can be applied to study these random trees, and survey methods for obtaining limit laws for such functions of subtrees. ..."
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Cited by 1 (0 self)
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Abstract. Under certain conditions, sums of functions of subtrees of a random binary search tree are asymptotically normal. We show how Stein’s method can be applied to study these random trees, and survey methods for obtaining limit laws for such functions of subtrees.