Flajolet, Philippe, Gourdon, Xavier, Martínez Parra, Conrado · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · 1996
In a randomly grown binary search tree (BST) of size n, any fixed\n\t\t\t\t pattern occurs with a frequency that is on average proportional to\n\t\t\t\t n. Deviations from the average case are highly unlikely and well\n\t\t\t\t quantified by a Gaussian law. Trees with forbidden patterns occur with\n\t\t\t\t an exponentially small probability that is characterized in terms of\n\t\t\t\t Bessel functions. The results obtained extend to BSTs a type of\n\t\t\t\t property otherwise known for strings and combinatorial tree\n\t\t\t\t models. They apply to paged trees or to quicksort with halting on\n\t\t\t\t short subfiles. As a consequence, various pointer saving strategies\n\t\t\t\t for maintaining trees obeying the random BST model can be precisely\n\t\t\t\t quantified. The methods used are based on analytic models, especially\n\t\t\t\t bivariate generating functions subjected to singularity perturbation\n\t\t\t\t asymptotics.
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Tomi Pasanen · Theoretical Computer Science · 2010
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Manuel Eberl, Max W. Haslbeck, Tobias Nipkow · Lecture notes in computer science · 2018
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Manuel Eberl, Max W. Haslbeck, Tobias Nipkow · Journal of Automated Reasoning · 2020
Abstract This work is a case study of the formal verification and complexity analysis of some famous probabilistic algorithms and data structures in the proof assistant Isabelle/HOL. In particular, we consider the expected number of comparisons in randomised quicksort, the relationship between randomised quicksort and average-case deterministic quicksort, the expected shape of an unbalanced random Binary Search Tree, the randomised binary search trees described by Martínez and Roura, and the expected shape of a randomised treap. The last three have, to our knowledge, not been analysed using a theorem prover before and the last one is of particular interest because it involves continuous distributions.
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Alois Panholzer · 2004
This paper deals with the size of the spanning tree of p randomly chosen nodes in a binary search tree. It is shown via generating functions methods, that for fixed p, the (normalized) spanning tree size converges in law to the Normal distribution. The special case p = 2 reproves the recent result (obtained by the contraction method by Mahmoud and Neininger [Ann. Appl. Probab. 13 (2003) 253–276]), that the distribution of distances in random binary search trees has a Gaussian limit law. In the proof we use the fact that the spanning tree size is closely related to the number of passes in Multiple Quickselect. This parameter, in particular, its first two moments, was studied earlier by Panholzer and Prodinger [Random Structures Algorithms 13 (1998) 189–209]. Here we show also that this normalized parameter has for fixed p-order statistics a Gaussian limit law. For p = 1 this gives the well-known result that the depth of a randomly selected node in a random binary search tree converges in law to the
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Jean‐Jil Duchamps, Amaury Lambert · The Annals of Applied Probability · 2018
Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure (infinitely-many-allele model) on the skeleton of the tree. The partition of the boundary into distinct alleles is the so-called allelic partition. In this paper, we are interested in the infinite trees generated by supercritical, possibly time-inhomogeneous, binary branching processes, and in their boundary, which is the set of particles “coexisting at infinity”. We prove that any such tree can be mapped to a random, compact ultrametric tree called the coalescent point process, endowed with a “uniform” measure on its boundary which is the limit as $t\to\infty$ of the properly rescaled counting measure of the population at time $t$. We prove that the clonal (i.e., carrying the same allele as the root) part of the boundary is a regenerative set that we characterize. We then study the allelic partition of the boundary through the measures of its blocks. We also study the dynamics of the clonal subtree, which is a Markovian increasing tree process as mutations are removed.
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F. Redig, W. M. Ruszel, E. Saada · Journal of Statistical Physics · 2012
We study the abelian sandpile model on a random binary tree. Using a transfer matrix approach introduced by Dhar and Majumdar, we prove exponential decay of correlations, and in a small supercritical region (i.e., where the branching process survives with positive probability) exponential decay of avalanche sizes. This shows a phase transition phenomenon between exponential decay and power law decay of avalanche sizes. Our main technical tools are: (1) A recursion for the ratio between the numbers of weakly and strongly allowed configurations which is proved to have a well-defined stochastic solution; (2) quenched and annealed estimates of the eigenvalues of a product of n random transfer matrices.
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Alan Stacey · Combinatorics Probability Computing · 1999
Each edge of the standard rooted binary tree is equipped with a random weight ; weights are independent and identically distibuted. The value of a vertex is the sum of the weights on the path from the root to the vertex. We wish to search the tree to find a vertex of large weight. A very natural conjecture of Aldous states that, in the sense of stochastic domination, an obvious greedy algorithm is best possible. We show that this conjecture is false. We prove, however, that in a weaker sense there is no significantly better algorithm.
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Kaan Ersahin · 2004
Synthetic Aperture Radar (SAR) systems are important active sensors for Earth observation systems. They enable continuous monitoring of the targets (e.g., sea ice, agricultural areas, ships, etc.) through clouds and independent of solar illumination. Space-borne polarimetric SAR missions which are planned for the near future draw our attention to the crucial task of information extraction from such data. The
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David Aldous · Combinatorics Probability Computing · 1992
There is a simple greedy algorithm for seeking large values of a function f defined on the vertices of the binary tree. Modeling f as a random function whose increments along edges are i.i.d., we show that (under a natural assumption) the values found by the greedy algorithm grow linearly in time, with rate specified in terms of a fixed-point identity for distributions.
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Kones, Ira · eScholarship@McGill (McGill) · 2018
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Huawei Huang, Zhao Dongfeng · 2002
This paper employs the average cycle analysis method and binary tree collision resolution algorithm to analyze the performance of a random access system. It studies the packet collision resolution in a random multiple access system. The formula in which the colliding packets are resolved successfully is obtained. Also, the formulae of the throughput of two systems are given, one system employs the binary tree collision resolution basic algorithm, the other employs a binary tree collision resolution advanced algorithm. Furthermore, the computer simulation results of packet collision resolution are given in this paper. We can show that, compared to the binary tree collision resolution basic algorithm, the binary tree collision resolution advanced algorithm is a remarkable improvement.
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Supervisor Julien Berestycki · 2014
A binary search tree is a binary tree in which each (internal) node is as-sociated to a key, where the keys are drawn from some totally ordered set, say distinct real values x1,..., xn. The first key is associated to the root. The next key is put to the left child of the root if it is smaller than the key of the root, and it is put to the right child of the root if it is larger than the key of the root. In this way we proceed further by inserting key by key. So starting from an array x1,..., xn we get a binary tree with n (internal) nodes such that the keys of the left subtree of any given node σ are smaller than the key of σ, and the keys of the right subtree are larger than the key of σ. Binary search trees are widely used to store (totally ordered) data, and many parameters have been discussed in the literature. It is usually assumed that x1,..., xn is a sample drawn in an i.i.d. fashion from a common, atom-less, distribution. In that case the binary search tree can be thought of as a Markov chain of binary trees (Tn, n ≥ 0) describing the evolution of a binary search tree as new keys are read. More precisely
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Hojune Shin, Jina Choi, Dain Lee et al. · Lecture notes in computer science · 2024
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Mariusz Jurkiewicz · International Journal of Electronics and Telecommunications · 2021
In this paper we construct and consider a new group-based digital signature scheme with evolving secret key, which is built using a bilinear map. This map is an asymmetric pairing of Type 3, and although, for the reason of this paper, it is treated in a completely abstract fashion it ought to be viewed as being actually defined over E(Fqn)[p] × E(Fqnk )[p] → Fqnk [p]. The crucial element of the scheme is the key updater algorithm. With the adoption of pairings and binary trees where a number of leaves is the same as a number of time periods, we are assured that an updated secret key can not be used to recover any of its predecessors. This, in consequence, means that the scheme is forward-secure. To formally justify this assertion, we conduct analysis in fu-cma security model by reducing the security of the scheme to the computational hardness of solving the Weak ℓ-th Bilinear Diffie-Hellman Inversion problem type. We define this problem and explain why it can be treated as a source of security for cryptographic schemes. As for the reduction itself, in general case, it could be possible to make only in the random oracle model.
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Akash Deep, Chris Monico, W. Brent Lindquist et al. · ArXiv.org · 2025
We propose a machine learning-based extension of the classical binomial option pricing model that incorporates key market microstructure effects. Traditional models assume frictionless markets, overlooking empirical features such as bid-ask spreads, discrete price movements, and serial return correlations. Our framework augments the binomial tree with path-dependent transition probabilities estimated via Random Forest classifiers trained on high-frequency market data. This approach preserves no-arbitrage conditions while embedding real-world trading dynamics into the pricing model. Using 46,655 minute-level observations of SPY from January to June 2025, we achieve an AUC of 88.25% in forecasting one-step price movements. Order flow imbalance is identified as the most influential predictor, contributing 43.2% to feature importance. After resolving time-scaling inconsistencies in tree construction, our model yields option prices that deviate by 13.79% from Black-Scholes benchmarks, highlighting the impact of microstructure on fair value estimation. While computational limitations restrict the model to short-term derivatives, our results offer a robust, data-driven alternative to classical pricing methods grounded in empirical market behavior.
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J. Crols, Guilherme Paim, Shirui Zhao et al. · 2024
Bayesian Neural Networks (BNNs) offer opportunities for greatly enhancing the trustworthiness of conventional neural networks by monitoring the uncertainties in decision-making. A significant drawback for BNN inference at the extreme edge, however, is the imperative need to incorporate Gaussian Random Number Generators (GRNG) within each neuron. State-of-the-art GRNG algorithms heavily depend on multiple arithmetic operations and the use of extensive look-up tables, posing significant implementation challenges for ultra-low power hardware implementations. To overcome this, this paper presents an innovative binary tree random number generator (TreeGRNG) allowing the use of ultra-low-cost constant comparators instead of arithmetic units. We further enhance the TreeGRNG proposal with a set of hardware-aware optimizations exploiting the Gaussian properties. The optimized TreeGRNG surpasses the State-of-the-Art (SoTA) in terms of distribution accuracy while achieving a 3.7 × reduction in energy per sample and boosting the throughput per unit area by 5.8×. Moreover, our TreeGRNG proposal possesses a distinct advantage over the current SoTA in terms of flexibility, as it easily enables designers to adjust the shape of the sampled probability distribution, extending beyond the capabilities of traditional GRNGs, opening the horizon towards future probabilistic AI designs. The TreeGRNG design is available open-source in the link<sup>1</sup><sup>1</sup>https://github.com/KULeuven-MICAS/TreeGRNG.
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Guangliang Guo, Yuan Wang, Wei Su et al. · 2012
A novel Dynamic Element Matching (DEM) method is presented to improve the static and dynamic performance of Nyquist-rate current-steering digital to analog converter (DAC). Compared to conventional DEM methods, this approach only increases or decreases the number of selected unit current sources randomly when the input code changes. Thus, signal dependent distortions can be eliminated efficiently at high sampling frequencies. A 6-bit current-steering DAC model adopting the proposed DEM coding method was built up to verify the performance in MATLAB. It could reach a better SFDR (56dB) value and a better DNL (0.26LSB) and INL (0.45LSB) value compared with traditional DEM methods.
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Yamamoto, Ken, Yamazaki, Yoshihiro · arXiv (Cornell University) · 2009
A central limit theorem for binary tree is numerically examined. Two types of central limit theorem for higher-order branches are formulated. A topological structure of a binary tree is expressed by a binary sequence, and the Horton-Strahler indices are calculated by using the sequence. By fitting the Gaussian distribution function to our numerical data, the values of variances are determined and written in simple forms.
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Microprocessors and Microsystems · 1993
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