Showing papers by "Mark Daniel Ward published in 2010"

Asymptotic Rational Approximation To Pi: Solution of an ``Unsolved Problem'' Posed By Herbert Wilf

Abstract: The webpage of Herbert Wilf describes eight Unsolved Problems Here, we completely resolve the third of these eight problems The task seems innocent: find the first term of the asymptotic behavior of the coefficients of an ordinary generating function, whose coefficients naturally yield rational approximations to $\pi$ Upon closer examination, however, the analysis is fraught with difficulties For instance, the function is the composition of three functions, but the innermost function has a non-zero constant term, so many standard techniques for analyzing function compositions will completely fail Additionally, the signs of the coefficients are neither all positive, nor alternating in a regular manner The generating function involves both a square root and an arctangent The complex-valued square root and arctangent functions each rely on complex logarithms, which are multivalued and fundamentally depend on branch cuts These multiple values and branch cuts make the function extremely tedious to visualize using Maple We provide a complete asymptotic analysis of the coefficients of Wilf's generating function The asymptotic expansion is naturally additive (not multiplicative); each term of the expansion contains oscillations, which we precisely characterize The proofs rely on complex analysis, in particular, singularity analysis (which, in turn, rely on a Hankel contour and transfer theorems)

Sux Trees: A Survey, and Future Challenges

Abstract: Sux trees are one of the most fundamental structures for data compression and pattern matching algorithms. They are retrieval trees (a.k.a. tries) built from the suxes of a string (i.e., a data sequence). A rich combinatorial theory about overlapping strings has been known and utilized for three decades. Some of the combinatorial methods of analysis have been extended, to encompass sux trees built over random strings whose distribution follows a Bernoulli model or a Markov model. Many opportunities exist for an extended, robust theory to much richer stochastic models that are applicable in practice, such as high-order Markov models and Hidden Markov Models. We will survey some recent results about sux trees, derived by analytic, combinatorial, and probabilistic analysis in tandem. We will also outline some challenges for the future analysis of sux