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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 ..."
Abstract

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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.
Abstract Analytic Combinatorics— A Calculus of Discrete Structures
"... The efficiency of many discrete algorithms crucially depends on quantifying properties of large structured combinatorial configurations. We survey methods of analytic combinatorics that are simply based on the idea of associating numbers to atomic elements that compose combinatorial structures, then ..."
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The efficiency of many discrete algorithms crucially depends on quantifying properties of large structured combinatorial configurations. We survey methods of analytic combinatorics that are simply based on the idea of associating numbers to atomic elements that compose combinatorial structures, then examining the geometry of the resulting functions. In this way, an operational calculus of discrete structures emerges. Applications to basic algorithms, data structures, and the theory of random discrete structures are outlined. 1 Algorithms and Random Structures A prime factor in choosing the best algorithm for a given computational task is efficiency with respect to the resources consumed, for instance, auxiliary storage, execution time, amount of communication needed. For a given algorithm A, such a complexity measure being fixed, what is of interest is the relation Size of the problem instance (n) − → Complexity of the algorithm (C), which serves to define the complexity function C(n) ≡ CA(n) of algorithm A. Precisely, this complexity function can be specified in several ways. (i) Worstcase analysis takes C(n) to be the maximum of C over all inputs of size n. This corresponds to a pessimistic scenario, one which is of relevance in critical systems and realtime computing. (ii) Averagecase analysis takes C(n) to be the expected value (average) of C over inputs of size n. The aim is to capture the “typical ” cost of a computational task observed when the algorithm is repeatedly applied to various kinds of data. (iii) Probabilistic analysis takes C(n) to be an indicator of the most likely values of C. Its more general aim is to obtain fine estimates on the probability distribution of C, beyond averagecase analysis.