Recursive Tree Research Articles

Quenched worst-case scenario for root deletion in targeted cutting of random recursive trees

Abstract We propose a method for cutting down a random recursive tree that focuses on its higher-degree vertices. Enumerate the vertices of a random recursive tree of size n according to the decreasing order of their degrees; namely, let $(v^{(i)})_{i=1}^{n}$ be such that $\deg(v^{(1)}) \geq \cdots \geq \deg (v^{(n)})$ . The targeted vertex-cutting process is performed by sequentially removing vertices $v^{(1)}, v^{(2)}, \ldots, v^{(n)}$ and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed. The total number of steps for this procedure, $K_n$ , is upper bounded by $Z_{\geq D}$ , which denotes the number of vertices that have degree at least as large as the degree of the root. We prove that $\ln Z_{\geq D}$ grows as $\ln n$ asymptotically and obtain its limiting behavior in probability. Moreover, we obtain that the kth moment of $\ln Z_{\geq D}$ is proportional to $(\!\ln n)^k$ . As a consequence, we obtain that the first-order growth of $K_n$ is upper bounded by $n^{1-\ln 2}$ , which is substantially smaller than the required number of removals if, instead, the vertices were selected uniformly at random.

Height of weighted recursive trees with sub-polynomially growing total weight

Height of weighted recursive trees with sub-polynomially growing total weight

Predictions of Colloidal Molecular Aggregation Using AI/ML Models.

To facilitate the triage of hits from small molecule screens, we have used various AI/ML techniques and experimentally observed data sets to build models aimed at predicting colloidal aggregation of small organic molecules in aqueous solution. We have found that Naïve Bayesian and deep neural networks outperform logistic regression, recursive partitioning tree, support vector machine, and random forest techniques by having the lowest balanced error rate (BER) for the test set. Derived predictive classification models consistently and successfully discriminated aggregator molecules from nonaggregator hits. An analysis of molecular descriptors in favor of colloidal aggregation confirms previous observations (hydrophobicity, molecular weight, and solubility) in addition to undescribed molecular descriptors such as the fraction of sp3 carbon atoms (Fsp3), and electrotopological state of hydroxyl groups (ES_Sum_sOH). Naïve Bayesian modeling and scaffold tree analysis have revealed chemical features/scaffolds contributing the most to colloidal aggregation and nonaggregation, respectively. These results highlight the importance of scaffolds with high Fsp3 values in promoting nonaggregation. Matched molecular pair analysis (MMPA) has also deciphered context-dependent substitutions, which can be used to design nonaggregator molecules. We found that most matched molecular pairs have a neutral effect on aggregation propensity. We have prospectively applied our predictive models to assist in chemical library triage for optimal plate selection diversity and purchase for high throughput screening (HTS) in drug discovery projects.

Limit Law for Zagreb and Wiener Indices of Random Exponential Recursive Trees

The Wiener index is the sum of distances of all pairs of nodes in a graph; and the Zagreb index is defined as the sum of squares of the degrees of nodes in a rooted tree. In this note, we calculate the first two moments of the Wiener and Zagreb indices of random exponential recursive trees (random ERTs) from two systems of recurrence relations. Then, by an application of the contraction method, we characterize the limit law for a scaled Zagreb index of ERTs. Via the martingale convergence theorem, we also show the almost sure convergence and quadratic mean convergence of an appropriately scaled Wiener index that is indicative of the distance of two randomly chosen nodes.

Depths in random recursive metric spaces

Abstract As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labelled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure, and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. We use martingale theory to prove a law of large numbers and a central limit theorem for the insertion depth, the distance from the master hook to the latch chosen. We also apply our results to further generalizations of random trees, hooking networks, and continuous spaces constructed from line segments.

An extended strategy of “Recursive Tree” for characterization of drug metabolites in vivo and in vitro and actional mechanism study based on network Pharmacology: Formononetin as a study case

The screening and identification of drug metabolites in biological matrices is challenging, and ultra-high performance liquid chromatography-Q-Exactive Orbitrap mass spectrometry (UHPLC-Q-Exactive Orbitrap MS) has become a powerful technological tool for drug metabolites analysis due to its high sensitivity. However, the spectral information contained in existing chemical standards and databases is very limited, and the UHPLC-Q-Exactive Orbitrap MS technique alone cannot satisfy the identification of complex and diverse metabolites. Therefore, there is an urgent need for a new strategy to achieve comprehensive drug metabolic profile. Based on this, we have innovatively constructed a “recursive tree” analysis strategy and bridged it with network pharmacology for elucidating the pharmacological mechanisms of drugs. In this paper, we investigated the overall metabolic profile of formononetin as an example and utilized the primary branching metabolites of formononetin as effective ingredients for the study of the anti-NAFLD mechanism. The results showed that a total of 131 metabolites (prototype drug included) were detected and identified. Among them, 106 metabolites were found in rats and 31 metabolites were found in liver microsomes. Glucose conjugation, demethylation, sulfation, glucuronidation, and their complex reactions were the major processes of formononetin biotransformation. Network pharmacology results screened 104 potential targets and 20 major signaling pathways. Their mechanisms may be additive and/or synergistic effects. In addition, the therapeutic effects of formononetin against NAFLD were investigated based on palmitic acid / oleic acid-induced HepG2 cells. In summary, the recursive tree analysis strategy provides a convenient method for the identification of metabolites, and its seamless integration with network pharmacology lays the foundation for studying the pharmacological activities of natural products.

Assessment of functional diversities in patients with Asthma, COPD, Asthma-COPD overlap, and Cystic Fibrosis (CF).

The objectives of the present study were to evaluate the discriminating power of spirometric and plethysmographic lung function parameters to differenciate the diagnosis of asthma, ACO, COPD, and to define functional characteristics for more precise classification of obstructive lung diseases. From the databases of 4 centers, a total of 756 lung function tests (194 healthy subjects, 175 with asthma, 71 with ACO, 78 with COPD and 238 with CF) were collected, and gradients among combinations of target parameters from spirometry (forced expiratory volume one second: FEV1; FEV1/forced vital capacity: FEV1/FVC; forced expiratory flow between 25-75% FVC: FEF25-75), and plethysmography (effective, resistive airway resistance: sReff; aerodynamic work of breathing at rest: sWOB), separately for in- and expiration (sReffIN, sReffEX, sWOBin, sWOBex) as well as static lung volumes (total lung capacity: TLC; functional residual capacity: FRCpleth; residual volume: RV), the control of breathing (mouth occlusion pressure: P0.1; mean inspiratory flow: VT/TI; the inspiratory to total time ratio: TI/Ttot) and the inspiratory impedance (Zinpleth = P0.1/VT/TI) were explored. Linear discriminant analyses (LDA) were applied to identify discriminant functions and classification rules using recursive partitioning decision trees. LDA showed a high classification accuracy (sensitivity and specificity > 90%) for healthy subjects, COPD and CF. The accuracy dropped for asthma (~70%) and even more for ACO (~60%). The decision tree revealed that P0.1, sRtot, and VT/TI differentiate most between healthy and asthma (68.9%), COPD (82.1%), and CF (60.6%). Moreover, using sWOBex and Zinpleth ACO can be discriminated from asthma and COPD (60%). Thus, the functional complexity of obstructive lung diseases can be understood, if specific spirometric and plethysmographic parameters are used. Moreover, the newly described parameters of airway dynamics and the central control of breathing including Zinpleth may well serve as promising functional marker in the field of precision medicine.

The relationship between positive airway pressure tolerance and adherence: defining a new metric.

Positive airway pressure (PAP) therapy adherence rates range from 30% to 60%, yet adherent patients may still express dissatisfaction with treatment. The identification of factors affecting PAP tolerance could provide insight into its impact on adherence. Patients with obstructive sleep apnea presenting for first follow-up visit after newly initiating PAP therapy were given a 10-question PAP tolerance survey encompassing domains of psychosocial perception, practical issues, and side effects, utilizing 10-point visual analog scales. Relationships between adherence data, tolerance scores, and patient variables (demographics, sleep-related factors, comorbidities, usage data) were explored via 2-tailed t tests, multivariable regression analysis, and recursive partitioning regression trees with a significance level of P ≤ .05. For 105 patients, tolerance scores were higher in patients considered adherent to therapy (P = .033), as were scores for individual survey questions addressing the ability to fall asleep (P = .013) and sleep through the night (P = .020). Depression positively (P = .006) and insomnia medication use negatively (P = .010) predicted tolerance score. Data-driven tolerance score cutoffs were identified to correlate with PAP adherence, with higher tolerance scores correlating with greater adherence rates. PAP tolerance may play an important role in therapy adherence. Tolerance can be statistically defined and categorized based on prior adherence data. Its utility as a predictive tool in assessing future adherence is warranted. Tekumalla S, Plawecki A, Kaffenberger T, etal. The relationship between positive airway pressure tolerance and adherence: defining a new metric. J Clin Sleep Med. 2024;20(7):1033-1038.

The location of high-degree vertices in weighted recursive graphs with bounded random weights

AbstractWe study the asymptotic growth rate of the labels of high-degree vertices in weighted recursive graphs (WRGs) when the weights are independent, identically distributed, almost surely bounded random variables, and as a result confirm a conjecture by Lodewijks and Ortgiese (‘The maximal degree in random recursive graphs with random weights’, preprint, 2020). WRGs are a generalisation of the random recursive tree and directed acyclic graph models, in which vertices are assigned vertex-weights and where new vertices attach to $m\in\mathbb{N}$ predecessors, each selected independently with a probability proportional to the vertex-weight of the predecessor. Prior work established the asymptotic growth rate of the maximum degree of the WRG model, and here we show that there exists a critical exponent $\mu_m$ such that the typical label size of the maximum-degree vertex equals $n^{\mu_m(1+o(1))}$ almost surely as n, the size of the graph, tends to infinity. These results extend results on the asymptotic behaviour of the location of the maximum degree, formerly only known for the random recursive tree model, to the more general weighted multigraph case of the WRG model. Moreover, for the weighted recursive tree model, that is, the WRG model with $m=1$ , we prove the joint convergence of the rescaled degree and label of high-degree vertices under additional assumptions on the vertex-weight distribution, and also extend results on the growth rate of the maximum degree obtained by Eslava, Lodewijks, and Ortgiese (Stoch. Process. Appl.158, 2023).

On the protected nodes in exponential recursive trees

The exponential recursive trees model several kinds of networks. At each step of growing of these trees, each node independently attracts a new node with probability p, or fails to do with probability 1 − p. Here, we investigate the number of protected nodes, total path length of protected nodes, and a mean study of the protected node profile of such trees.

On finding a satisfactory partition in an undirected graph: algorithm design and results

<p>A satisfactory partition is a partition of undirected-graph vertices such that the partition has only two nonempty parts, and every vertex has at least as many adjacent vertices in its part as it has in the other part. Generally, the problem of determining whether a given undirected graph has a satisfactory partition is known to be NP-complete. In this paper, we show that for a given undirected graph with $ n $ vertices, a satisfactory partition (if any exists) can be computed recursively with a recursion tree of depth of $ \mathcal{O}(\ln n) $ in expectation. Subsequently, we show that a satisfactory partition for those undirected graphs with recursion tree depth meeting the expectation can be computed in time $ \mathcal{O}(n^{3} 2^{\mathcal{O}(\ln n)}) $.</p>

On the time complexity of achieving optimal throughput in time division multiple access communication networks

<abstract><p>The fundamental problem of finding transmission schedules for achieving optimal throughput in time division multiple access (TDMA) communication networks is known to be NP-hard. Let $ \mathcal{N} $ be a scheduled $ k $-time slot TDMA network with $ n $ stations and $ m $ links. We showed that an optimal link schedule for $ \mathcal{N} $ can be computed recursively with a recursion tree of logarithmic depth $ \mathcal{O}(\ln m) $ in expectation. Additionally, we showed that optimal link schedules for those TDMA networks, with recursion trees of depth meeting the expectation, can be found in time $ \mathcal{O}(m^{2+\ln k}) $. Likewise, we discuss analogous results for computing optimal station schedules of TDMA networks.</p></abstract>

Counterbalancing steps at random in a random walk

A random walk with counterbalanced steps is a process of partial sums \check S(n)=\check X_{1}+ \cdots + \check X_{n} whose steps \check X_{n} are given recursively as follows. For each n\geq 2 , with a fixed probability p , \check X_{n} is a new independent sample from some fixed law \mu , and with complementary probability 1-p , \check X_{n}= -\check X_{v(n)} counterbalances a previous step, with v(n) a uniform random pick from \{1, \ldots, n-\nobreak 1\} . We determine the asymptotic behavior of \check S(n) in terms of p and the first two moments of \mu . Our approach relies on a coupling with a reinforcement algorithm due to H. A. Simon, and on properties of random recursive trees and Eulerian numbers, which may be of independent interest. The method can be adapted to the situation where the step distribution \mu belongs to the domain of attraction of a stable law.

A growth-fragmentation-isolation process on random recursive trees and contact tracing

A growth-fragmentation-isolation process on random recursive trees and contact tracing

Number-enhanced representation with hierarchical recursive tree decoding for math word problem solving

Automatic solving math word problems (MWPs) is a number-intensive application in natural language processing (NLP). However, these existing methods are far from achieving acceptable levels of numeracy learning. As a result, the performance of these models is limited for mathematical reasoning. In addition, the mainstream tree decoder suffers from early-stage information loss, resulting in an unsatisfactory performance for complex problems with more operators. In this paper, we propose NERHRT (Number-Enhanced Representation with Hierarchical Recursive Tree Decoding), a simple yet effective number embedding method that produces the numerical reasoning via the decimal notation-based embedding and a dual-direction graph attention network. In addition, a hierarchical recursive tree-structured decoder is introduced to aggregate information from all ancestor nodes. Experiments show that our approach obtains the best performance on four popular benchmark datasets, and beats the state-of-the-art models with a large margin.

On the distribution of eigenvalues of increasing trees

We prove that the multiplicity of a fixed eigenvalue α in a random recursive tree on n vertices satisfies a central limit theorem with mean and variance asymptotically equal to μαn and σα2n respectively. It is also shown that μα and σα2 are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue 0, the constants μ0 and σ02 can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.

A neural builder for spatial subdivision hierarchies

Spatial data structures, such as k-d trees and bounding volume hierarchies, are extensively used in computer graphics for the acceleration of spatial queries in ray tracing, nearest neighbour searches and other tasks. Typically, the splitting strategy employed during the construction of such structures is based on the greedy evaluation of a predefined objective function, resulting in a less than optimal subdivision scheme. In this work, for the first time, we propose the use of unsupervised deep learning to infer the structure of a fixed-depth k-d tree from a constant, subsampled set of the input primitives, based on the recursive evaluation of the cost function at hand. This results in high-quality upper spatial hierarchy, inferred in constant time and without paying the intractable price of a fully recursive tree optimisation. The resulting fixed-depth tree can then be further expanded, in parallel, into either a full k-d tree or transformed into a bounding volume hierarchy, with any known conventional tree builder. The approach is generic enough to accommodate different cost functions, such as the popular surface area and volume heuristics. We experimentally validate that the resulting hierarchies have competitive traversal performance with respect to established tree builders, while maintaining minimal overhead in construction times.

Unbiased recursive decision tree for supervised functional data classification with applying on electrocardiogram signals

Unbiased recursive decision tree for supervised functional data classification with applying on electrocardiogram signals

Development, Evaluation, and Multisite Deployment of a Machine Learning Decision Tree Algorithm To Optimize Urinalysis Parameters for Predicting Urine Culture Positivity.

PittUDT, a recursive partitioning decision tree algorithm for predicting urine culture (UC) positivity based on macroscopic and microscopic urinalysis (UA) parameters, was developed in support of a broader system-wide diagnostic stewardship initiative to increase appropriateness of UC testing. Reflex algorithm training utilized results from 19,511 paired UA and UC cases (26.8% UC positive); the average patient age was 57.4 years, and 70% of samples were from female patients. Receiver operating characteristic (ROC) analysis identified urine white blood cells (WBCs), leukocyte esterase, and bacteria as the best predictors of UC positivity, with areas under the ROC curve of 0.79, 0.78, and 0.77, respectively. Using the held-out test data set (9,773 cases; 26.3% UC positive), the PittUDT algorithm met the prespecified target of a negative predictive value above 90% and resulted in a 30 to 60% total negative proportion (true-negative plus false-negative predictions). These data show that a supervised rule-based machine learning algorithm trained on paired UA and UC data has adequate predictive ability for triaging urine specimens by identifying low-risk urine specimens, which are unlikely to grow pathogenic organisms, with a false-negative proportion under 5%. The decision tree approach also generates human-readable rules that can be easily implemented across multiple hospital sites and settings. Our work demonstrates how a data-driven approach can be used to optimize UA parameters for predicting UC positivity in a reflex protocol, with the intent of improving antimicrobial stewardship and UC utilization, a potential avenue for cost savings.

RD-FCA: A resilient distributed framework for formal concept analysis

Formal Concept Analysis (FCA) is a data analysis technique with applications in data mining, artificial intelligence, software engineering, etc. The algorithms for FCA are computationally expensive, and their recursion tree is highly irregular and dynamic in nature. Several distributed FCA algorithms have been proposed to exploit parallelism within and across machines. However, none of the distributed approaches are able to recover from failures in the system. We propose RD-FCA, the first resilient distributed framework for FCA that uses novel load-balancing strategy, handles fail-stop failures and provides at-least-once semantics for concept discovery. Our asynchronous snapshot mechanism with incremental updates reduces the snapshot overhead and minimizes recalculation of concepts during recovery. RD-FCA also supports dynamic addition of workers to accelerate performance. Compared to MapReduce based approaches, RD-FCA performs an order of magnitude faster. We show through extensive evaluation that RD-FCA recovers efficiently from single, multiple, independent and cascading failures.