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In a randomly grown binary search tree BST of size n, any fixed pattern occurs with a frequency that is on average proportional to n. Deviations from the average case are highly unlikely and well quantified by a Gaussian law. Trees with forbidden patterns occur with an exponentially small probability that is characterized in terms of Bessel functions. The results obtained extend to BSTs a type of property otherwise known for strings and combinatorial tree models. They apply to paged trees or to quicksort with halting on short subfiles. As a consequence, various pointer saving strategies for maintaining trees obeying the random BST model can be precisely quantified. The methods used are based on analytic models, especially bivariate generating function subjected to singularity perturba-Ž .
A "hybrid method", dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions-this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.Date: June 5, 2006.
A unified treatment of parameters relevant to factoring polynomials over finite fields is given. The framework is based on generating functions for describing parameters of interest and on singularity analysis for extracting asymptotic values. An outcome is a complete analysis of the standard polynomial factorization chain that is based on the elimination of repeated factors, distinct degree factorization, and equal degree separation. Several basic statistics on polynomials over finite fields are obtained in the course of the analysis.
INTRODUCTIONFactoring polynomials over finite fields intervenes in many areas of computer science and computational mathematics, like symbolic computaw x w x tion at large 24 , polynomial factorization over integers 12, 40 , cryptograw x wx wx phy 10, 44, 48 , number theory 5 , and coding theory 4 . The implications include finding complete partial fraction decompositions, designing cyclic redundancy codes, computing the number of points on elliptic curves, and building arithmetic public key cryptosystems. In particular, the factorization of random polynomials over finite fields is directly needed in the randomized index calculus method for computing discrete logarithms over w x finite fields 48 .
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