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A probabilistic analysis of some tree algorithms
 ANNALS OF APPLIED PROBABILITY
, 2005
"... In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily without resorting to the usual complex analysis techniques. Thi ..."
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Cited by 22 (6 self)
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In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily without resorting to the usual complex analysis techniques. This approach gives a unified probabilistic treatment of these questions. It simplifies and extends some of the results known in this domain.
Digital Trees and Memoryless Sources: from Arithmetics to Analysis
 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA’10), Discrete Math. Theor. Comput. Sci. Proc
, 2010
"... Digital trees, also known as “tries”, are fundamental to a number of algorithmic schemes, including radixbased searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstr ..."
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Digital trees, also known as “tries”, are fundamental to a number of algorithmic schemes, including radixbased searching and sorting, lossless text compression, dynamic hashing algorithms, communication protocols of the tree or stack type, distributed leader election, and so on. This extended abstract develops the asymptotic form of expectations of the main parameters of interest, such as tree size and path length. The analysis is conducted under the simplest of all probabilistic models; namely, the memoryless source, under which letters that data items are comprised of are drawn independently from a fixed (finite) probability distribution. The precise asymptotic structure of the parameters’ expectations is shown to depend on fine singular properties in the complex plane of a ubiquitous Dirichlet series. Consequences include the characterization of a broad range of asymptotic regimes for error terms associated with trie parameters, as well as a classification that depends on specific arithmetic properties, especially irrationality measures, of the sources under consideration.
Renewal theory in analysis of tries and strings
, 2009
"... We give a survey of a number of simple applications of renewal theory to problems on random strings and tries: insertion depth, size, insertion mode and imbalance of tries; variations for btries and Patricia tries; Khodak and Tunstall codes. ..."
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Cited by 4 (1 self)
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We give a survey of a number of simple applications of renewal theory to problems on random strings and tries: insertion depth, size, insertion mode and imbalance of tries; variations for btries and Patricia tries; Khodak and Tunstall codes.
The total path length of split trees
, 2011
"... We consider the model of random trees introduced by Devroye [SIAM J Comput 28, 409– 432, 1998]. The model encompasses many important randomized algorithms and data structures. The pieces of data (items) are stored in a randomized fashion in the nodes of a tree. The total path length (sum of depths o ..."
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We consider the model of random trees introduced by Devroye [SIAM J Comput 28, 409– 432, 1998]. The model encompasses many important randomized algorithms and data structures. The pieces of data (items) are stored in a randomized fashion in the nodes of a tree. The total path length (sum of depths of the items) is a natural measure of the efficiency of the algorithm/data structure. Using renewal theory, we prove convergence in distribution of the total path length towards a distribution characterized uniquely by a fixed point equation. Our result covers, using a unified approach, many data structures such as binary search trees, mary search trees, quad trees, medianof(2k + 1) trees, and simplex trees. 1
Renewal theory for the analysis of tries and strings (Extended Abstract)
, 2010
"... We give a survey of a number of simple applications of renewal theory to problems on random strings, in particular to tries and Khodak and Tunstall codes. ..."
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We give a survey of a number of simple applications of renewal theory to problems on random strings, in particular to tries and Khodak and Tunstall codes.
On the Maximum Stable Throughput of Tree Algorithms with Free Access
"... A simple numerical procedure is presented to determine the maximum stable throughput (MST) of various tree algorithms with free access by defining an associated multitype branching process such that the criticality of the branching process corresponds to the stability of the tree algorithm. More pr ..."
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A simple numerical procedure is presented to determine the maximum stable throughput (MST) of various tree algorithms with free access by defining an associated multitype branching process such that the criticality of the branching process corresponds to the stability of the tree algorithm. More precisely, a bisection algorithm is proposed that only requires the computation of the dominant eigenvalue of the expectation matrix of the branching process, the size of which is typically below 20, at each step. Using this novel approach, many existing results on free access tree algorithms are reproduced without much effort (e.g., channels with/without noise, variable length packets, interference cancellation, etc.). Furthermore, in an almost plugandplay manner, the MST of the free access equivalent of many existing results on tree algorithms with blocked access is established (e.g., channels with capture, control signals, multi reception capabilities, etc.). The method can be applied to the class of independent and identically distributed arrival processes, which includes the Poisson process as a special case. Apart from determining the MST, the probability that a size i collision is resolved in a finite amount of time when the arrival rate exceeds the MST, can also be computed. Some limitations of the proposed methodology are discussed as well.
3.1. Measurements and Mathematical Modeling 2 3.1.1. Traffic Modeling 2
"... c t i v it y e p o r t 2007 Table of contents 1. Team.................................................................................... 1 ..."
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c t i v it y e p o r t 2007 Table of contents 1. Team.................................................................................... 1