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89
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.
A MultiLevel WDM Access Protocol for an Optically Interconnected Multiprocessor System
 IEEE/OSA Journal of Lightwave Technology
, 1999
"... Scalable, hierarchical, alloptical WDM networks for processor interconnection in multiprocessor systems have been recently considered. The principal objective of this paper is to introduce an access protocol for this type of network which supports a distributed shared memory(DSM) environment. The o ..."
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Cited by 35 (14 self)
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Scalable, hierarchical, alloptical WDM networks for processor interconnection in multiprocessor systems have been recently considered. The principal objective of this paper is to introduce an access protocol for this type of network which supports a distributed shared memory(DSM) environment. The objectives of the protocol are reduced averagelatency per packet, support of broadcast/multicast, collisionless communication, and exploitation of inherent DSM traffic characteristics. The protocol is based on a hybrid approach that combines reservation access and preallocated reception channels for a WDM system. The proposed approach trades maximum capacity for reduced communication latency to improve system response. The performance of the protocol is analyzed through semimarkov analytic and simulation models with varying system parameters such as number of nodes and channels. The performance of the new protocol is compared to a TDMbased protocol and their relative merits are examined. ...
Singularity Analysis, Hadamard Products, and Tree Recurrences
, 2003
"... We present a toolbox for extracting asymptotic information on the coecients of combinatorial generating functions. This toolbox notably includes a treatment of the eect of Hadamard products on singularities in the context of the complex Tauberian technique known as singularity analysis. As a consequ ..."
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Cited by 28 (9 self)
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We present a toolbox for extracting asymptotic information on the coecients of combinatorial generating functions. This toolbox notably includes a treatment of the eect of Hadamard products on singularities in the context of the complex Tauberian technique known as singularity analysis. As a consequence, it becomes possible to unify the analysis of a number of divideandconquer algorithms, or equivalently random tree models, including several classical methods for sorting, searching, and dynamically managing equivalence relations.
Analysis of Shellsort and related algorithms
 ESA ’96: Fourth Annual European Symposium on Algorithms
, 1996
"... This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellso ..."
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Cited by 26 (0 self)
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This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellsort and Shellsortbased networks; averagecase results; proposed probabilistic sorting networks based on the algorithm; and a list of open problems. 1 Shellsort The basic Shellsort algorithm is among the earliest sorting methods to be discovered (by D. L. Shell in 1959 [36]) and is among the easiest to implement, as exhibited by the following C code for sorting an array a[l],..., a[r]: shellsort(itemType a[], int l, int r) { int i, j, h; itemType v;
The path length of random skip lists
 Acta Informatica
, 1994
"... Abstract. The skip list is a recently introduced data structure that may be seen as an alternative to (digital) tries. In the present paper we analyze the path length of random skip lists asymptotically, i.e. we study the cumulated successful search costs. In particular we derive a precise asymptoti ..."
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Cited by 24 (6 self)
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Abstract. The skip list is a recently introduced data structure that may be seen as an alternative to (digital) tries. In the present paper we analyze the path length of random skip lists asymptotically, i.e. we study the cumulated successful search costs. In particular we derive a precise asymptotic result on the variance, being of order n 2 (which isincontrast to tries under the symmetric Bernoulli model, where it is only of order n). We also intend to present some sort of technical toolkit for the skilful manipulation and asymptotic evaluation of generating functions that appear in this context.
Generalized Digital Trees and their Differencedifferential Equations
, 1992
"... . Consider a tree partitioning process in which n elements are split into b at the root of a tree (b a design parameter), the rest going recursively into two subtrees with a binomial probability distribution. This extends some familiar tree data structures of computer science like the digital trie ..."
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Cited by 24 (5 self)
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. Consider a tree partitioning process in which n elements are split into b at the root of a tree (b a design parameter), the rest going recursively into two subtrees with a binomial probability distribution. This extends some familiar tree data structures of computer science like the digital trie and the digital search tree. The exponential generating function for the expected size of the tree satisfies a difference differential equation of order b, d b dz b f(z) = e z + 2e z=2 f( z 2 ): The solution involves going to ordinary (rather than exponential) generating functions, analyzing singularities by means of Mellin transforms and contour integration. The method is of some general interest since a large number of related problems on digital structures can be treated in this way via singularity analysis of ordinary generating functions. Work of this author was supported in part by the Basic Research Action of the E.C. under contract No. 3075 (Project ALCOM). y The resea...
Randomized Binary Search Trees
 Journal of the ACM
, 1997
"... In this paper we present randomized algorithms over binary search trees such that: a) the insertion of a set of keys, in any fixed order, into an initially empty tree always produces a random binary search tree; b) the deletion of any key from a random binary search tree results in a random binary s ..."
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Cited by 22 (2 self)
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In this paper we present randomized algorithms over binary search trees such that: a) the insertion of a set of keys, in any fixed order, into an initially empty tree always produces a random binary search tree; b) the deletion of any key from a random binary search tree results in a random binary search tree; c) the random choices made by the algorithms are based upon the sizes of the subtrees of the tree; this implies that we can support accesses by rank without additional storage requirements or modification of the data structures; and d) the cost of any elementary operation, measured as the number of visited nodes, is the same as the expected cost of its standard deterministic counterpart; hence, all search and update operations have guaranteed expected cost O(log n), but now irrespective of any assumption on the input distribution. 1. Introduction Given a binary search tree (BST, for short), common operations are the search of an item given its key and the retrieval of the inform...
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.