ar X iv : 0 80 9 . 35 77 v 1 [ m at h . PR ] 2 1 Se p 20 08 DYNAMIC TREE ALGORITHMS
نویسنده
چکیده
In this paper, a general tree algorithm processing a random flow of arrivals is analyzed. Capetanakis-Tsybakov-Mikhailov's protocol in the context of communication networks with random access is an example of such an algorithm. In computer science, this corresponds to a trie structure with a dynamic input. Mathematically, it is related to a stopped branching process with exogeneous arrivals (immigration). Under quite general assumptions on the distribution of the number of arrivals and on the branching procedure, it is shown that there exists a positive constant λc so that if the arrival rate is smaller than λc, then the algorithm is stable under the flow of requests, i.e. that the total size of an associated tree is integrable. At the same time a gap in the earlier proofs of stability of the literature is fixed. When the arrivals are Poisson, an explicit characterization of λc is given. Under the stability condition, the asymptotic behavior of the average size of a tree starting with a large number of individuals is analyzed. The results are obtained with the help of a probabilistic rewriting of the functional equations describing the dynamic of the system. The proofs use extensively this stochastic background throughout the paper. In this analysis, two basic limit theorems play a key role: the renewal theorem and the convergence to equilibrium of an auto-regressive process with moving average. This paper is dedicated to Philippe Flajolet on the occasion of his 60th birthday.
منابع مشابه
ar X iv : 0 80 9 . 47 83 v 1 [ m at h . C A ] 2 7 Se p 20 08 HEAT - FLOW MONOTONICITY OF STRICHARTZ NORMS
Most notably we prove that for d = 1, 2 the classical Strichartz norm ‖ef‖ L 2+4/d s,x (R×Rd) associated to the free Schrödinger equation is nondecreasing as the initial datum f evolves under a certain quadratic heat-flow.
متن کاملar X iv : 0 80 9 . 26 64 v 1 [ m at h . A P ] 1 6 Se p 20 08 Balance laws with integrable unbounded sources ∗
We consider the Cauchy problem for a n×n strictly hyperbolic system of balance laws
متن کاملar X iv : 0 80 9 . 42 29 v 1 [ m at h - ph ] 2 4 Se p 20 08 Thermodynamic Limit for Spin Glasses . Beyond the Annealed Bound
Using a correlation inequality of Contucci and Lebowitz for spin glasses, we demonstrate existence of the thermodynamic limit for short-ranged spin glasses, under weaker hypotheses than previously available, namely without the assumption of the annealed bound.
متن کاملar X iv : 0 80 9 . 40 53 v 1 [ m at h . C A ] 2 3 Se p 20 08 SOME EXTREMAL FUNCTIONS IN FOURIER ANALYSIS , III
We obtain the best approximation in L 1 (R), by entire functions of exponential type, for a class of even functions that includes e −λ|x| , where λ > 0, log |x| and |x| α , where −1 < α < 1. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.
متن کاملar X iv : 0 80 9 . 22 94 v 1 [ m at h . FA ] 1 2 Se p 20 08 ASYMPTOTIC UNCONDITIONALITY
We show that a separable real Banach space embeds almost isometrically in a space Y with a shrinking 1-unconditional basis if and only if limn→∞ ‖x ∗ + x∗ n ‖ = limn→∞ ‖x ∗ − x∗ n ‖ whenever x∗ ∈ X∗, (x∗ n )∞ n=1 is a weak∗-null sequence and both limits exist. If X is reflexive then Y can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng.
متن کامل