Recursive Tree Research Articles

A tree search algorithm towards solving Ising formulated combinatorial optimization problems

Simulated annealing (SA) attracts more attention among classical heuristic algorithms because many combinatorial optimization problems can be easily recast as the ground state search problem of the Ising Hamiltonian. However, for practical implementation, the annealing process cannot be arbitrarily slow and hence, it may deviate from the expected stationary Boltzmann distribution and become trapped in a local energy minimum. To overcome this problem, this paper proposes a heuristic search algorithm by expanding search space from a Markov chain to a recursive depth limited tree based on SA, where the parent and child nodes represent the current and future spin states. At each iteration, the algorithm selects the best near-optimal solution within the feasible search space by exploring along the tree in the sense of “look ahead”. Furthermore, motivated by the coherent Ising machine (CIM), the discrete representation of spin states is relaxed to a continuous representation with a regularization term, which enables the use of the reduced dynamics of the oscillators to explore the surrounding neighborhood of the selected tree nodes. We tested our algorithm on a representative NP-hard problem (MAX-CUT) to illustrate the effectiveness of the proposed algorithm compared to semi-definite programming (SDP), SA, and simulated CIM. Our results show that with the primal heuristics SA and CIM, our high-level tree search strategy is able to provide solutions within fewer epochs for Ising formulated combinatorial optimization problems.

Commutativity of entropy for nonautonomous systems on trees

In this paper, we show that the commutativity of the entropy for nonautonomous Markov systems holds on an aperiodic recursive tree. Then the similar result is extended to p-periodic and anisotropic systems. Finally, we give a characterization for a Markov tree to be a complete recursive tree in terms of the properties of the associated adjacency matrix.

High-performance GPU-accelerated evaluation of electron repulsion integrals

ABSTRACT A novel methodology for the evaluation of two electron integrals up to f functions using Graphics Processing Units (GPUs) is presented. The Head-Gordon-Pople recursion relationships are solved via a simple heuristic methodology to minimize the number of evaluated intermediates in the recursion trees. Automatic code generation is used to generate highly optimized CUDA kernels. A novel approach for f functions is presented in which integral classes are split into smaller subclasses to minimize register pressure and exploit additional parallelism at the cost of recomputing a small number of intermediates. Alongside optimized kernels, the ERI evaluation scheme works in conjunction with an efficient work distribution scheme which guarantees load-balancing during computation. The new HGP scheme shows excellent speedups of 2 × to above 60 × against existing GPU code. Additionally, when coupled with digestion into the Fock matrix, the scaling is excellent on up to 7 GPUs with an 85% parallel efficiency for the 6-31G(d) basis set.

Recursive tree grammar autoencoders

Machine learning on trees has been mostly focused on trees as input. Much less research has investigated trees as output, which has many applications, such as molecule optimization for drug discovery, or hint generation for intelligent tutoring systems. In this work, we propose a novel autoencoder approach, called recursive tree grammar autoencoder (RTG-AE), which encodes trees via a bottom-up parser and decodes trees via a tree grammar, both learned via recursive neural networks that minimize the variational autoencoder loss. The resulting encoder and decoder can then be utilized in subsequent tasks, such as optimization and time series prediction. RTG-AEs are the first model to combine three features: recursive processing, grammatical knowledge, and deep learning. Our key message is that this unique combination of all three features outperforms models which combine any two of the three. Experimentally, we show that RTG-AE improves the autoencoding error, training time, and optimization score on synthetic as well as real datasets compared to four baselines. We further prove that RTG-AEs parse and generate trees in linear time and are expressive enough to handle all regular tree grammars.

Distinct Fringe Subtrees in Random Trees

A fringe subtree of a rooted tree is a subtree induced by one of the vertices and all its descendants. We consider the problem of estimating the number of distinct fringe subtrees in random trees under a generalized notion of distinctness, which allows for many different interpretations of what “distinct” trees are. The random tree models considered are simply generated trees and families of increasing trees (recursive trees, d-ary increasing trees and generalized plane-oriented recursive trees). We prove that the order of magnitude of the number of distinct fringe subtrees (under rather mild assumptions on what ‘distinct’ means) in random trees with n vertices is n/sqrt{log n} for simply generated trees and n/log n for increasing trees.

Limit laws for two distance-based indices in random recursive tree models

Abstract In this paper, we derive several results related to total path length and Sackin index in two classes of random recursive trees. A limiting distribution of the normalized version of the Sackin index is given by the contraction method in random recursive trees. Also, we show the normalized total path length converges in L2 and almost surely to a limiting random variable in plane-oriented recursive trees via martingales.

Correction terms for the height of weighted recursive trees

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of weights, the first order for the height of such trees has been recently established by one of the authors. In this paper, we obtain the second and third orders in the asymptotic expansion of the height of weighted recursive trees, under similar assumptions. Our methods are inspired from those used to prove similar results for branching random walks. Our results also apply to a related model of growing trees, called the preferential attachment tree with additive fitnesses.

Dynamical models for random simplicial complexes

We study a general model of random dynamical simplicial complexes and derive a formula for the asymptotic degree distribution. This asymptotic formula generalises results for a number of existing models, including random Apollonian networks and the weighted random recursive tree. It also confirms results on the scale-free nature of complex quantum network manifolds in dimensions d>2, and special types of network geometry with Flavour models studied in the physics literature by Bianconi and Rahmede [Sci. Rep. 5 (2015) 13979 and Phys. Rev. E 93 (2016) 032315].

Pricing Total Return Swaps

Total return swap (TRS) involves a pricing dilemma: LIBOR discounting of its premium leg forces upfront payment of future funding premium, and yet replacing LIBOR with a firm’s own funding rate falls into the well-known FVA debate trap. We consider TRS hedge financing from a repo market perspective and apply postcrisis derivatives valuation with collateralization and funding to TRS. We find that the financing cost of the TRS hedge should be reflected on the security leg, and the funding premium can only be discounted in conjunction with the TRS as a whole, depending on margining schemes. An easy to implement, recursive tree model is developed to value TRS with repo-style margining or defaultable underlying, together with any value adjustments.

Proof of the satisfiability conjecture for large $k$

We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_{\rm sat}(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha\lt \alpha_{\rm sat}$, and unsatisfiable for $\alpha>\alpha_{\rm sat}$. We show that the threshold $\alpha_{\rm sat}(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the $1$-RSB universality class.

Stark: Fast and Scalable Strassen’s Matrix Multiplication Using Apache Spark

This article presents a new fast, highly scalable distributed matrix multiplication algorithm on Apache Spark, called <i>Stark</i> , based on Strassen’s matrix multiplication algorithm. Stark preserves Strassen’s seven multiplications scheme in a distributed environment and thus achieves asymptotically faster execution time. It creates a distributed recursion tree of computation where each level of the tree corresponds to division and combination of distributed matrix blocks stored in the form of Resilient Distributed Datasets (RDDs). It processes each divide and combine step in parallel and memorises the sub-matrices by intelligently tagging matrix blocks in it. To the best of our knowledge, Stark is the first implementation of a distribute Strassen’s algorithm on Spark platform. We also report a detailed complexity analysis for the proposed algorithm, taking into account computation and communication costs. Experimental results suggest that Stark outperforms existing distributed matrix multiplication implementations on Spark – <i>Marlin</i> and <i>MLLib</i> , for high matrix sizes ( <inline-formula><tex-math notation="LaTeX">$\geq 16384\times 16384$</tex-math></inline-formula> ). Our experiments reveal optimal block sizes for each matrix size, which is also shown from theoretical analysis. We also show that the experimental and theoretical running times for Stark match closely. It has also been shown experimentally that Stark exhibits strong scalability with increasing number of executors.

Caregiver Reports of Infant Distress and Injury in Abused Infants

To determine whether in infants evaluated for physical abuse, medical encounters for infant distress are correlated with physical abuse or a history of sentinel injuries. This retrospective, case-control analysis of infants aged <12months evaluated for physical abuse identified demographic characteristics, prior injuries, and medical encounters for infant distress. Variables were compared between abused infants and nonabused infants with and without sentinel injuries. A nonparametric recursive classification tree analysis assessed interactions between variables. Infant distress was associated with abuse (67.9% vs 44.7%; P=.008; OR, 2.6; 95% CI, 1.3-5.2). Infants with sentinel injuries had higher rates of infant distress (74.1% vs 42.4%; P≤.001) and crying (81.5% vs 62.7%; P=.012). Previous falls (32.6% vs 18.1%; P=.03) and nonsentinel injuries (18.2% vs 5.4%; P=.002) also were associated with abuse, although sentinel injuries were the most important predictor of abuse, followed by infant distress. Infants with medical encounters for distress and injury may be at higher risk for abuse and may benefit from intensive educational and support services for their caregivers. Additional research evaluating the most effective interventions for caregivers of fussy infants is needed.

LEADER–FOLLOWER COHERENCE OF THE WEIGHTED RECURSIVE TREE NETWORKS

Recently, the consensus problem has attracted extensive attention of researchers from different scientific communities. In this paper, we study the leader–follower network coherence (i.e. the mean steady-state variance of the deviation from the static value of the leader nodes) in the weighted recursive tree networks with assigning the leaders orderly in the initial state. For the weighted recursive tree networks model, we study the characteristic polynomial of the Laplacian submatrix and obtain the relationship for the eigenvalues in two successive generations. The analytical formula of the leader–follower network coherence related by the sum of the reciprocal of all eigenvalues of this submatrix is obtained. The result shows that more number of leaders and the greater weight factor lead to better consensus.

Inclusive Trial Designs in Acute Spinal Cord Injuries: Prediction–Based Stratification of Clinical Walking Outcome and Projected Enrolment Frequencies

Background New therapeutic approaches in neurological disorders are progressing into clinical development. Past failures in translational research have underlined the critical importance of selecting appropriate inclusion criteria and primary outcomes. Narrow inclusion criteria provide sensitivity, but increase trial duration and cost to the point of infeasibility, while broader requirements amplify confounding, increasing the risk of trial failure. This dilemma is perhaps most pronounced in spinal cord injury (SCI), but applies to all neurological disorders with low frequency and/or heterogeneous clinical manifestations. Objective Stratification of homogeneous patient cohorts to enable the design of clinical trials with broad inclusion criteria. Methods Prospectively–gathered data from patients with acute cervical SCI were analysed using an unbiased recursive partitioning conditional inference tree (URP–CTREE) approach. Performance in the 6-minute walk test at 6 months after injury was classified based on standardized neurological assessments within the first 15 days of injury. Functional and neurological outcomes were tracked throughout rehabilitation up to 6 months after injury. Results URP–CTREE identified homogeneous outcome cohorts in a study group of 309 SCI patients. These cohorts were validated by an internal, yet independent, validation group of 172 patients. The study group cohorts identified demonstrated distinct recovery profiles throughout rehabilitation. The baseline characteristics of the analysed groups were compared to a reference group of 477 patients. Conclusion URP–CTREE enables inclusive trial design by revealing the distribution of outcome cohorts, discerning distinct recovery profiles and projecting potential patient enrolment by providing estimates of the relative frequencies of cohorts to improve the design of clinical trials in SCI and beyond.

A new approach to Pólya urn schemes and its infinite color generalization

In this work, we introduce a generalization of the classical Pólya urn scheme (Ann. Inst. Henri Poincaré 1 (1930) 117–161) with colors indexed by a Polish space, say, S. The urns are defined as finite measures on S endowed with the Borel σ-algebra, say, S. The generalization is an extension of a model introduced earlier by Blackwell and MacQueen (Ann. Statist. 1 (1973) 353–355). We present a novel approach of representing the observed sequence of colors from such a scheme in terms an associated branching Markov chain on the random recursive tree. The work presents fairly general asymptotic results for this new generalized urn models. As special cases, we show that the results on classical urns, as well as, some of the results proved recently for infinite color urn models in (Bernoulli 23 (2017) 3243–3267; Statist. Probab. Lett. 92 (2014) 232–240), can easily be derived using the general asymptotic. We also demonstrate some newer results for infinite color urns.

Broadcasting on random recursive trees

We study the broadcasting problem when the underlying tree is a random recursive tree. The root of the tree has a random bit value assigned. Every other vertex has the same bit value as its parent with probability 1−q and the opposite value with probability q, where q∈[0,1]. The broadcasting problem consists in estimating the value of the root bit upon observing the unlabeled tree, together with the bit value associated with every vertex. In a more difficult version of the problem, the unlabeled tree is observed but only the bit values of the leaves are observed. When the underlying tree is a uniform random recursive tree, in both variants of the problem we characterize the values of q for which the optimal reconstruction method has a probability of error bounded away from 1/2. We also show that the probability of error is bounded by a constant times q. Two simple reconstruction rules are analyzed in detail. One of them is the simple majority vote, the other is the bit value of the centroid of the tree. Most results are extended to linear preferential attachment trees as well.

Motor Recovery Depends on Timing of Surgery in Patients With Lumbar Disk Herniation.

Although approximately half of the patients undergoing lumbar disk surgery present with motor deficits, timing of surgery for radicular weakness is largely unclear. To evaluate the impact of surgical timing on motor recovery in patients with lumbar disk herniation (LDH) and to identify an ideal time window for intervention. In a single-center observational trial, 390 patients with LDH-associated motor deficits were prospectively followed for a minimum of 12 months after nonelective microscopic disk surgery. The duration of motor deficit before surgery was documented. Motor function was graded according to the Medical Research Council (MRC) scale. Statistical analysis of motor recovery applied unbiased recursive partitioning conditional inference tree to determine cutoff times for optimal surgical intervention. The slope of recovery calculated as the change of the MRC grade over time served as the primary outcome. A preoperative motor deficit of MRC ≤2/5 and the duration of paresis were identified as the most important predictors of recovery ( P < .001). Surgery within 3 days was associated with a better recovery for both severe and moderate/mild deficits ( P = .017 for MRC ≤ 2/5; P < .001 for MRC > 2/5; number needed to treat [NNT] <2). A sensitivity analysis in mild motor deficits indicated a cutoff of 8 days. Timing of surgery is crucial for motor recovery in LDH-associated deficits. Immediate diagnosis, imaging, and referral should be aimed for to allow disk surgery within 3 days in patients with severe and moderate radicular weakness. If functionally disabling, even mild deficits may warrant decompression within a week.

The containment profile of hyper-recursive trees

AbstractWe investigate vertex levels of containment in a random hypergraph grown in the spirit of a recursive tree. We consider a local profile tracking the evolution of the containment of a particular vertex over time, and a global profile concerned with counts of the number of vertices of a particular containment level.For the local containment profile, we obtain the exact mean, variance, and probability distribution in terms of standard combinatorial quantities such as generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: the early vertices have an asymptotically normal distribution, intermediate vertices have a Poisson distribution, and late vertices have a degenerate distribution.As for the global containment profile, we establish an asymptotically normal distribution for the number of vertices at the smallest containment level as well as their covariances with the number of vertices at the second smallest containment level and the variances of these numbers. The orders in the variance–covariance matrix establish concentration laws.

Depth of vertices with high degree in random recursive trees

Let $T_n$ be a random recursive tree with $n$ nodes. List vertices of $T_n$ in decreasing order of degree as $v^1,\ldots,v^n$, and write $d^i$ and $h^i$ for the degree of $v^i$ and the distance of $v^i$ from the root, respectively. We prove that, as $n \to \infty$ along suitable subsequences, \[ \bigg(d^i - \lfloor \log_2 n \rfloor, \frac{h^i - \mu\ln n}{\sqrt{\sigma^2\ln n}}\bigg) \to ((P_i,i \ge 1),(N_i,i \ge 1))\, , \] where $\mu=1-(\log_2 e)/2$, $\sigma^2=1-(\log_2 e)/4$, $(P_i,i \ge 1)$ is a Poisson point process on $\mathbb{Z}$ and $(N_i,i \ge 1)$ is a vector of independent standard Gaussians. We additionally establish joint normality for the depths of uniformly random vertices in $T_n$, which extends results for the case of a single random vertex. The joint limit holds even if the random vertices are conditioned to have large degree, provided the normalizing constants are adjusted accordingly; however, both the mean and variance of the conditinal depths remain of orden $\ln n$. Our results are based on a simple relationship between random recursive trees and Kingman's $n$-coalescent; a utility that seems to have been largely overlooked.

Growing random graphs with a preferential attachment structure

The aim of this paper is to develop a method for proving almost sure convergence in Gromov-Hausodorff-Prokhorov topology for a class of models of growing random graphs that generalises Remy's algorithm for binary trees. We describe the obtained limits using some iterative gluing construction that generalises the famous line-breaking construction of Aldous' Brownian tree. In order to do that, we develop a framework in which a metric space is constructed by gluing smaller metric spaces, called \emph{blocks}, along the structure of a (possibly infinite) discrete tree. Our growing random graphs seen as metric spaces can be understood in this framework, that is, as evolving blocks glued along a growing discrete tree structure. Their scaling limit convergence can then be obtained by separately proving the almost sure convergence of every block and verifying some relative compactness property for the whole structure. For the particular models that we study, the discrete tree structure behind the construction has the distribution of an affine preferential attachment tree or a weighted recursive tree. We strongly rely on results concerning those two models of random trees and their connection, obtained in a companion paper.