SEQUITUR is an algorithm that infers a hierarchical structure from a sequence of discrete symbols by replacing repeated phrases with a grammatical rule that generates the phrase, and continuing this process recursively. The result is a hierarchical representation of the original sequence, which offers insights into its lexical structure. The algorithm is driven by two constraints that reduce the size of the grammar, and produce structure as a by-product. SEQUITUR breaks new ground by operating incrementally. Moreover, the method’s simple structure permits a proof that it operates in space and time that is linear in the size of the input. Our implementation can process 50,000 symbols per second and has been applied to an extensive range of real world sequences. 1.

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: Consider an alphabet \Sigma = fa 1 ; : : : ; ang with corresponding symbol probabilities p 1 ; : : : ; pn . The L\Gammarestricted prefix code is a prefix code where all the code lengths are not greater than L. The value L is a given integer such that L dlog ne. Define the average code length difference by ffl = P n i=1 p i :l i \Gamma P n i=1 p i :l i , where l 1 ; : : : ; l n are the code lengths of the optimal L-restricted prefix code for \Sigma and l 1 ; : : : ; l n are the code lengths of the optimal prefix code for \Sigma. Let / be the golden ratio 1,618. In this paper, we show that ffl ! 1=/ L\Gammadlog(n+dlog ne\GammaL)e\Gamma1 when L ? dlog ne. We also prove the sharp bound ffl ! dlog ne \Gamma 1, when L = dlog ne. By showing the lower bound 1 / L\Gammadlog ne+2+dlog n n\GammaL e \Gamma1 on the maximum value of ffl, we guarantee that our bound is asymptotically tight in the range dlog ne ! L n=2. Furthermore, we present an O(n) time and space 1=/ L\Gammadlo...

We present two short proofs of an identity found by Sylvester, and rediscovered by Louck. The first proof is an elementary version of Knuth's proof, and is analogous to Macdonald's proof of a related identity of Milne. The second is Sylvester's own proof of his identity.

Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forest-valued Markov process which describes the growth of the Brownian forest. The key result is a composition rule for binary Galton–Watson forests with i.i.d. exponential branch lengths. We give elementary proofs of this composition rule and explain how it is intimately linked with Williams ’ decomposition for Brownian motion with drift. 1. Introduction. Given 0 ≤ λ<µ, the binary(λ, µ) forest is defined as a collection of independent binary Galton–Watson trees, with branching probability (µ − λ)/2µ, branch lengths independent exponential(2µ), planted into the positive real line at the points of a homogeneous Poisson process of rate µ − λ. We call the vertices on the positive real line roots. The unique branch connecting

We describe algorithms for constructing optimal binary search trees, in which the access cost of a key depends on the k preceding keys which were reached in the path to it. This problem has applications to searching on secondary memory and robotics. Two kinds of optimal trees are considered, namely optimal worst case trees and weighted average case trees. The time and space complexities of both algorithms are O(n ) and O(n respectively. The algorithms are based on a convenient decomposition and characterizations of sequences of keys which are paths of special kinds in binary search trees. Finally, using generating functions, we present an exact analysis of the number of steps performed by the algorithms.

FIFO, First-In-First-Out, and LIFO, Last-In-First-Out, are well-known techniques for handling sequences.

FIFO, first-in-first-out, and LIFO, last-in-last-out,are well-known techniques for handling sequences.

Both statistical ecology and environmental statistics have numer-60 10 ous challenges and opportunities in the waiting for the twenty-first century, calling for increasing numbers of nontraditional statistical approaches. Both theoretical and applied ecology are using advancing data analytical and interpretational software and hardware to satisfy public policy and discovery research, variously incorporating geospatial information, site-specific data and remote sensing imagery. We discuss a declared need for geoinformatic surveillance for spatial critical area detection. We explore, for ecological and environmental use, an innovation of the circle-based spatial scan statistic

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