Robert P. Dobrow

Total Path Length for Random Recursive Trees

Total path length, or search cost, for a rooted tree is defined as the sum of all root-to-node distances. Let Tn be the total path length for a random recursive tree of order n. Mahmoud [10] showed that Wn := (Tn − E[Tn])/n converges almost surely and in L2 to a nondegenerate limiting random variable W. Here we give recurrence relations for the moments of Wn and of W and show that Wn converges to W in Lp for each 0 < p < ∞. We confirm the conjecture that the distribution of W is not normal. We also show that the distribution of W is characterized among all distributions having zero mean and finite variance by the distributional identity***** Insert equation here *****where E(x) := − x ln x − (1 minus; x) ln(1 − x) is the binary entropy function, U is a uniform (0, 1) random variable, W* and W have the same distribution, and U, W and W* are mutually independent. Finally, we derive an approximation for the distribution of W using a Pearson curve density estimator.

The Move-To-Front Rule for Self-Organizing Lists with Markov Dependent Requests

We consider the move-to-front self-organizing linear search heuristic where the sequence of record requests is a Markov chain. Formulas are derived for the transition probabilities and stationary distribution of the permutation chain. The spectral structure of the chain is presented explicitly. Bounds on the discrepancy from stationarity for the permutation chain are computed in terms of the corresponding discrepancy for the request chain, both for separation and for total variation distance.

On the distribution of distances in recursive trees

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree T n with n labeled nodes is recursive tree if n = 1, or n >1 and T n can be constructed by joining node n to a node of some recursive tree T n-1 . For arbitrary nodes i < n in a random recursive tree we give the exact distribution of X i,n , the distance between nodes i and n. We characterize this distribution as the convolution or the law of X i,i+1 and n - i - 1 Bernoulli distributions. We further characterize the law of X i,i+1 as a mixture of sums of Bernoulli. For i = i n growing as a function of n, we show that X in,n is asymptotically normal in several settings.

Potential role of imatinib mesylate (Gleevec, STI-571) in the treatment of vestibular schwannoma.

Hypothesis: To determine the expression of the tyrosine kinases platelet-derived growth factor receptor (PDGFR) and c-Kit in vestibular schwannoma (VS) and to determine the potential role of imatinib mesylate (Gleevec) in regulating the growth and cell death of this tumor. Background: Protein tyrosine kinases are transmembrane tyrosine kinase receptors that transduce signals from inside and outside the cell and function as relay points for signaling pathways. They have a key role in numerous processes that affect cell proliferation, tumorigenesis, cancer invasion, metastasis, and modulation of apoptosis. A few of these kinases have been demonstrated to be overexpressed and dysregulated in many carcinomas, sarcomas, and benign tumors. Methods: Immunohistochemical staining was used to investigate the expression of PDGFR and c-Kit in archived acoustic neuroma tissue. Clinical data including size of tumors, age, sex, and symptoms were correlated with kinase expression, whereas Western blot analysis and immunofluorescence were performed to demonstrate the expression and localization of PDGFR and c-Kit in HEI193, an immortalized VS cell line. Clonogenic survival assays were performed to assess proliferation inhibition by Gleevec. Gleevecs effect on the cell cycle profile also was investigated via flow cytometry analysis. Results: Expression of PDGFR in the formalin-fixed VS tumor tissue was observed in 23 (67.5%) of the 34 samples. C-kit was expressed in 18 (52.9%) of the 34 samples. Western blot analysis demonstrates positive expression of c-Kit and PDGFR-Q in HEI193 and a primary VS culture. Western blot analysis showed downregulation of phospho-c-kit and phospho-PDGFR-Q with 5 and 10 uM Gleevec. Immunofluorescent staining of this cell line also reveals that PDGFR-&bgr; is localized primarily in the cytoplasm, whereas c-Kit is both nuclear and cytoplasmic. Cell cycle analysis of HEI193 96 hours after incubation with Gleevec indicates a dose-dependent increase in G1 from 61.6% to 70.7% and 74% at 5 and 10 uM of Gleevec, respectively. Colony formation assays demonstrate dose-dependent growth inhibition by Gleevec, in the HEI193 cell line as well as in a VS cell culture derived from a fresh tumor. Conclusion: The expression of PDGFR-Q and c-Kit in VS tissue may indicate novel molecular targets involved in the development of this tumor. Direct inhibition of these molecules by Gleevec may have relevant therapeutic applications.

Multiway Trees of Maximum and Minimum Probability under the Random Permutation Model

Multiway trees, also known as m-ary search trees, are data structures generalizing binary search trees. A common probability model for anlayzing the behavior of these structures is the random permutation model. The probability mass function Q on the set of m-ary search trees under the random permutation model is the distribution induced by sequentially inserting the records of a uniformly random permutation into an initially empty m-ary search tree. We study some basic properties of the functional Q, which serves as a measure of the “shape” of the tree. In particular, we determine exact and asymptotic expressions for the maximum and minimum values of Q and identify and count the trees achieving those values. Research was carried out while the first author was a postdoctoral research associate at the National Institute of Standards and Technology, Statistical Engineering Division. The first author’s institution will change its name to Truman State University in July, 1996. Research for the second author was supported by NSF grant DMS-9311367. AMS 1991 subject classifications. Primary 60C05; secondary 68P10, 68P05.

The Number of m -ary Search Trees on n Keys

Problems associated with m-ary trees have been studied by computer scientists and combinatorialists. It is well known that a simple generalization of the Catalan numbers counts the number of m-ary trees on n nodes. In this paper we consider τm, n, the number of m-ary search trees on n keys, a quantity that arises in studying the space of m-ary search trees under the uniform probability model. We prove an exact formula for τm, n, both by analytic and by combinatorial means. We use uniform local approximations for sums of i.i.d. random variables to study the asymptotic development of τm, n for fixed m as n→∞.

Speeding up the FMMR perfect sampling algorithm: a case study revisited

In a previous paper by the second author, two Markov chain Monte Carlo perfect sampling algorithms--one called coupling from the past (CFTP) and the other (FMMR) based on rejection sampling--are compared using as a case study the move-to-front (MTF) self-organizing list chain. Here we revisit that case study and, in particular, exploit the dependence of FMMR on the user-chosen initial state. We give a stochastic monotonicity result for the running time of FMMR applied to MTF and thus identify the initial state that gives the stochastically smallest running time; by contrast, the initial state used in the previous study gives the stochastically largest running time. By changing from worst choice to best choice of initial state we achieve remarkable speedup of FMMR for MTF; for example, we reduce the running time (as measured in Markov chain steps) from exponential in the length n of the list nearly down to n when the items in the list are requested according to a geometric distribution. For this same example, tile running time for CFTP grows exponentially in n.

Applied Stochastic Modelling

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Modeling, Analysis, Design, and Control of Stochastic Systems

Readership: This book is meant to be used as a textbook in a junior or senior level undergraduate course in stochastic models. Students are expected to be undergraduate students in engineering, operations research, computer science, mathematics, statistics, business administration, public policy, or any other discipline with a mathematical core. Students are expected to be familiar with elementary matrix operations (additions, multiplications, solving systems of linear equations; but not eigenvalues, eigenvectors), first-year calculus (derivatives and integrals of simple functions; but not differential equations), and probability (which is reviewed in Chapters I to 4 of this book).