Quasi-linear Time Research Articles

Median quartet tree search algorithms using optimal subtree prune and regraft

Gene trees can be different from the species tree due to biological processes and inference errors. One way to obtain a species tree is to find one that maximizes some measure of similarity to a set of gene trees. The number of shared quartets between a potential species tree and gene trees provides a statistically justifiable score; if maximized properly, it could result in a statistically consistent estimator of the species tree under several statistical models of discordance. However, finding the median quartet score tree, one that maximizes this score, is NP-Hard, motivating several existing heuristic algorithms. These heuristics do not follow the hill-climbing paradigm used extensively in phylogenetics. In this paper, we make theoretical contributions that enable an efficient hill-climbing approach. Specifically, we show that a subtree of size m can be placed optimally on a tree of size n in quasi-linear time with respect to n and (almost) independently of m. This result enables us to perform subtree prune and regraft (SPR) rearrangements as part of a hill-climbing search. We show that this approach can slightly improve upon the results of widely-used methods such as ASTRAL in terms of the optimization score but not necessarily accuracy.

Comparison of polynomial and rational function cornea models for effective dimensionality reduction

This study aims to examine geometric models of the corneal surface that can be used to reduce in reasonable time the dimensionality of datasets of normal anterior corneas. Polynomial models (P) like Zernike polynomials (ZP) and spherical harmonic polynomials (SHP) were obvious candidates along with their rational function (R) counterparts, namely Zernike rational functions (ZR) and spherical harmonic rational functions (SHR, new model). Knowing that both SHP and ZR were more accurate than ZP for the modeling of normal and keratoconus corneas, it was expected that both spherical harmonic (SH) models (SHP and SHR) would be more accurate than their Zernike (Z) counterparts (ZP and ZR, respectively), and both rational (R) models (SHR and ZR) more accurate than their polynomial counterparts (SHP and ZP, respectively) for a low dimensional space (coefficient number J < 30). This was the case. The SH factor contributed more to accuracy than the R factor. Considering the corneal processing time as a function of J, P models were processed in quasi-linear time with a quasi-null slope and rational models in polynomial time. Z models were faster than SH models, and increasingly so in their R version. In sum, for corneal dimensionality reduction, SHR is the most accurate model, but its processing time is increasingly prohibitive unless the best coefficient combination is identified beforehand. ZP is the fastest model and is reasonably accurate with normal corneas for exploratory tasks. SHP is the best compromise between accuracy and speed.

Augmenting Deep Residual Surrogates with Fourier Neural Operators for Rapid Two-Phase Flow and Transport Simulations

Summary Accurate numerical modeling of multiphase flow and transport mechanisms is essential to study varied, complex physical phenomena including flow in subsurface oil and gas reservoirs and subsurface aquifers subject to CO2 sequestration. State-of-the-art complete physics-based solvers suffer from many computational challenges. High-fidelity data-driven surrogate models that solve the governing partial differential equations (PDEs) have the potential to optimize the time to solution and increase confidence in critical business and engineering decisions through better quantification of solution statistics. We leverage the recently proposed Fourier neural operators (FNOs) with quasilinear time complexity to capture the spectral information from feature maps to solve the coupled porous flow and transport PDEs. Embedding Fourier layers within the residual blocks results in a highly effective structure that, while achieving competitive accuracy, also enables efficient training of deeper networks with a dramatically reduced number of trainable parameters. The resulting novel deep-learning (DL) architecture is coined as FResNet++. FResNet++ uses squeeze and excitation blocks, atrous spatial pyramid pooling (ASPP), and attention blocks to increase its sensitivity to the relevant features and capture multiscale information, and it is specifically tuned to operate optimally to learn from and predict numerically simulated flow (pressure and saturation) fields. We demonstrate the ability of FResNet++ to generalize over multiple high-dimensional input parameter spaces that describe subsurface permeability and porosity heterogeneity. The resulting DL architecture accurately captures the complex interplay between viscous forces and highly heterogeneous permeability and porosity fields. We investigate two-phase flow in porous media, which is the archetypal problem for reservoir simulation giving rise to a system of nonlinearly coupled PDEs with highly heterogeneous coefficients. We show in blind tests that FResNet++ predicts saturation fields more accurately compared to ResU-Net and original FNO with fully connected linear layers. We additionally investigate the effects of using alternative loss functions and an alternative way of utilizing FResNet++ to increase its effectiveness. For the first time in the literature, we show that the spatiotemporal evolution of pressure and saturation fields can be jointly predicted with good accuracy using a single FResNet++ network over long time horizons in response to previously unseen permeability and porosity fields. After a moderate training investment on graphics processing units (GPUs), FResNet++ yields a speedup of at least four orders of magnitude compared to a conventional numerical PDE solver and operates with notably fewer trainable parameters compared to the original FNO. Our numerical experiments validate that FNOs can be utilized in various convolutional neural network-based architectures and can effectively substitute for repetitive physics-based forward simulations for scenario testing.

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Gaussian processes are widely used as priors for unknown functions in statistics and machine learning. To achieve computationally feasible inference for large datasets, a popular approach is the Vecchia approximation, which is an ordered conditional approximation of the data vector that implies a sparse Cholesky factor of the precision matrix. The ordering and sparsity pattern are typically determined based on Euclidean distance of the inputs or locations corresponding to the data points. Here, we propose instead to use a correlation-based distance metric, which implicitly applies the Vecchia approximation in a suitable transformed input space. The correlation-based algorithm can be carried out in quasilinear time in the size of the dataset, and so it can be applied even for iterative inference on unknown parameters in the correlation structure. The correlation-based approach has two advantages for complex settings: It can result in more accurate approximations, and it offers a simple, automatic strategy that can be applied to any covariance, even when Euclidean distance is not applicable. We demonstrate these advantages in several settings, including anisotropic, nonstationary, multivariate, and spatio-temporal processes. We also illustrate our method on multivariate spatio-temporal temperature fields produced by a regional climate model.

Tractable Orders for Direct Access to Ranked Answers of Conjunctive Queries

We study the question of when we can provide direct access to the k-th answer to a Conjunctive Query (CQ) according to a specified order over the answers in time logarithmic in the size of the database, following a preprocessing step that constructs a data structure in time quasilinear in database size. Specifically, we embark on the challenge of identifying the tractable answer orderings , that is, those orders that allow for such complexity guarantees. To better understand the computational challenge at hand, we also investigate the more modest task of providing access to only a single answer (i.e., finding the answer at a given position), a task that we refer to as the selection problem , and ask when it can be performed in quasilinear time. We also explore the question of when selection is indeed easier than ranked direct access. We begin with lexicographic orders . For each of the two problems, we give a decidable characterization (under conventional complexity assumptions) of the class of tractable lexicographic orders for every CQ without self-joins. We then continue to the more general orders by the sum of attribute weights and establish the corresponding decidable characterizations, for each of the two problems, of the tractable CQs without self-joins. Finally, we explore the question of when the satisfaction of Functional Dependencies (FDs) can be utilized for tractability and establish the corresponding generalizations of our characterizations for every set of unary FDs.

Certified Newton schemes for the evaluation of low-genus theta functions

Theta functions and theta constants in low genus, especially genus 1 and 2, can be evaluated at any given point in quasi-linear time in the required precision using Newton schemes based on Borchardt sequences. Our goal in this paper is to provide the necessary tools to implement these algorithms in a provably correct way. In particular, we obtain uniform and explicit convergence results in the case of theta constants in genus 1 and 2, and theta functions in genus 1: the associated Newton schemes will converge starting from approximations to N bits of precision for N = 60, 300, and 1600 respectively, for all suitably reduced arguments. We also describe a uniform quasi-linear time algorithm to evaluate genus 2 theta constants on the Siegel fundamental domain. Our main tool is a detailed study of Borchardt means as multivariate analytic functions.

Conditional regression for single-index models

The single-index model is a statistical model for intrinsic regression where responses are assumed to depend on a single yet unknown linear combination of the predictors, allowing to express the regression function as E[Y|X]=f(⟨v,X⟩) for some unknown index vector v and link function f. Conditional methods provide a simple and effective approach to estimate v by averaging moments of X conditioned on Y, but depend on parameters whose optimal choice is unknown and do not provide generalization bounds on f. In this paper we propose a new conditional method converging at n rate under an explicit parameter characterization. Moreover, we prove that polynomial partitioning estimates achieve the 1-dimensional min-max rate for regression of Hölder functions when combined to any n-convergent index estimator. Overall this yields an estimator for dimension reduction and regression of single-index models that attains statistical optimality in quasilinear time.

Sign choices in the AGM for genus two theta constants

Existing algorithms to compute genus 2 theta constants in quasi-linear time use Borchardt sequences, an analogue of the arithmetic-geometric mean for four complex numbers. In this paper, we show that these Borchardt sequences are given by good choices of square roots only, as in the genus 1 case. This removes the sign indeterminacies in the algorithm without relying on numerical integration.

Eccentricity queries and beyond using hub labels

Hub labeling schemes are popular methods for computing distances on road networks and other large complex networks, often answering to a query within a few microseconds for graphs with millions of edges. In this work, we study their algorithmic applications beyond distance queries. Indeed, several implementations of hub labels were reported to have good practical performances, both in terms of pre-processing time and maximum label size. There are also a few relevant graph classes for applications for which we know how to compute hub labelings with sublinear labels in quasi linear time. These positive results raise the question of what can be computed efficiently given a graph and a hub labeling of small maximum label size as input. We focus on eccentricity queries and distance-sum queries, for several versions of these problems on directed weighted graphs, that is in part motivated by their importance in facility location problems. On the negative side, we show conditional lower bounds for these above problems on unweighted undirected sparse graphs, via standard constructions from “Fine-grained” complexity. Specifically, under the Strong Exponential-Time Hypothesis (SETH), answering to these above queries requires Ω(|V|2−o(1)) pre-processing time or Ω(|V|1−o(1)) query time, even if we are given as input a hub labeling of maximum label size in ω(log⁡|V|). However, things take a different turn when the hub labels have a sublogarithmic size. Indeed, for every ε>0 there exists a δε>0 such that the following hold, being given a hub labeling of maximum label size ≤k: after pre-processing the labels in total O(2δε⋅k⋅|V|1+ε) time, we can compute both the eccentricity and the distance-sum of any vertex in O(2δε⋅k⋅|V|ε) time. Our data structure is a novel application of the orthogonal range query framework of Cabello and Knauer (2009) [17]. It can also be applied to the fast global computation of some topological indices. Finally, as a by-product of our approach, on any fixed class of unweighted graphs with bounded expansion, we can decide whether the diameter of an n-vertex graph in the class is at most k in fε(k)⋅n1+ε time for any ε>0, for some “explicit” function fε that depends on ε and the graph class considered. This result is further motivated by the empirical evidence that many classes of complex networks have bounded expansion (Demaine et al., 2019 [24]), and that some of these networks, such as the online social networks, have a relatively small diameter.

Logspace and compressed-word computations in nilpotent groups

For finitely generated nilpotent groups, we employ Mal'cev coordinates to solve several classical algorithmic problems efficiently. Computation of normal forms, the membership problem, the conjugacy problem, and computation of presentations for subgroups are solved using only logarithmic space and quasilinear time. Logarithmic space presentation-uniform versions of these algorithms are provided. Compressed-word versions of the same problems, in which each input word is provided as a straight-line program, are solved in polynomial time.

Linear Aggregation in Tree-Based Estimators

Regression trees and their ensemble methods are popular methods for nonparametric regression: they combine strong predictive performance with interpretable estimators. To improve their utility for locally smooth response surfaces, we study regression trees and random forests with linear aggregation functions. We introduce a new algorithm that finds the best axis-aligned split to fit linear aggregation functions on the corresponding nodes, and we offer a quasilinear time implementation. We demonstrate the algorithm’s favorable performance on real-world benchmarks and in an extensive simulation study, and we demonstrate its improved interpretability using a large get-out-the-vote experiment. We provide an open-source software package that implements several tree-based estimators with linear aggregation functions. Supplementary materials for this article are available online.

Completing gene trees without species trees in sub-quadratic time.

As genome-wide reconstruction of phylogenetic trees becomes more widespread, limitations of available data are being appreciated more than ever before. One issue is that phylogenomic datasets are riddled with missing data, and gene trees, in particular, almost always lack representatives from some species otherwise available in the dataset. Since many downstream applications of gene trees require or can benefit from access to complete gene trees, it will be beneficial to algorithmically complete gene trees. Also, gene trees are often unrooted, and rooting them is useful for downstream applications. While completing and rooting a gene tree with respect to a given species tree has been studied, those problems are not studied in depth when we lack such a reference species tree. We study completion of gene trees without a need for a reference species tree. We formulate an optimization problem to complete the gene trees while minimizing their quartet distance to the given set of gene trees. We extend a seminal algorithm by Brodal et al. to solve this problem in quasi-linear time. In simulated studies and on a large empirical data, we show that completion of gene trees using other gene trees is relatively accurate and, unlike the case where a species tree is available, is unbiased. Our method, tripVote, is available at https://github.com/uym2/tripVote. Supplementary data are available at Bioinformatics online.

Mc2g: An Efficient Algorithm for Matrix Completion With Social and Item Similarity Graphs

In this paper, we design and analyze MC2G (Matrix Completion with 2 Graphs), an algorithm that performs matrix completion in the presence of social and item similarity graphs. MC2G runs in quasilinear time and is parameter free. It is based on spectral clustering and local refinement steps. For the matrix completion problem which possesses additional block structures in its rows and columns, we derive the expected number of sampled entries required for MC2G to succeed, and further show that it matches an information-theoretic lower bound up to a constant factor for a wide range of parameters. We perform extensive experiments on both synthetic datasets and a semi-real dataset inspired by real graphs. The experimental results show that MC2G outperforms other state-of-the-art matrix completion algorithms.

Fast Diameter Computation within Split Graphs

When can we compute the diameter of a graph in quasi linear time? We address this question for the class of {\em split graphs}, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either $2$ or $3$, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of an $n$-vertex $m$-edge split graph in less than quadratic time -- in the size $n+m$ of the input. Therefore it is worth to study the complexity of diameter computation on {\em subclasses} of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded {\em clique-interval number} and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the {\em VC-dimension} and the {\em stabbing number} of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs: - For the $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in quasi linear time if $k=o(\log{n})$ and in addition a corresponding ordering of the vertices in the clique is given. However, under SETH this cannot be done in truly subquadratic time for any $k = \omega(\log{n})$. - For the {\em complements} of $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in time ${\cal O}(km)$ if a corresponding ordering of the vertices in the stable set is given. Again this latter result is optimal under SETH up to polylogarithmic factors. Our findings raise the question whether a $k$-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for $k=1$ and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs -- including the ones mentioned above -- have a bounded clique-interval number.

Approximate inversion of discrete Fourier integral operators

This paper introduces a factorization for the inverse of discrete Fourier integral operators of size N×N that can be applied in quasi-linear time. The factorization starts by approximating the operator with the butterfly factorization. Next, a hierarchical matrix representation is constructed for the hermitian matrix arising from composing the Fourier integral operator with its adjoint. This representation is inverted efficiently with a new algorithm based on the hierarchical interpolative factorization. By combining these two factorizations, an approximate inverse factorization for the Fourier integral operator is obtained as a product of O(log⁡N) sparse matrices with O(N) entries. The resulting approximate inverse factorization can be used as a direct solver or as a preconditioner. Numerical examples on 1D and 2D Fourier integral operators, including a generalized Radon transform, demonstrate the performance of this new approach.

Fast computation of elliptic curve isogenies in characteristic two

We propose an algorithm that calculates isogenies between elliptic curves defined over an extension $K$ of $\mathbb{Q}_2$. It consists in efficiently solving with a logarithmic loss of $2$-adic precision the first order differential equation satisfied by the isogeny. We give some applications, especially computing over finite fields of characteristic 2 isogenies of elliptic curves and irreducible polynomials, both in quasi-linear time in the degree.

ТЕОРЕТИЧНА МОДЕЛЬ ФОРМУВАННЯ ДВОВИМІРНИХ НАНОСТРУКТУР ВЕРТИКАЛЬНОГО ГРАФЕНУ ПІД ДІЄЮ ПЛАЗМИ

Vertically oriented graphene nanostructures have been grown for more than decade, but the mechanisms of their formation are still unclear. A multifactor model is proposed, which is verified by comparison with experimental data and describes the processes of growth of the structure of vertical graphene in plasma. The role of chemical and physical processes that cannot be directly characterized by available experimental methods, such as surface diffusion of adatoms and radicals under the action of ions, has been studied. Ion bombardment is a key factor that significantly accelerates the growth rate through the formation of surface defects and, consequently, increases the energy of surface adsorption. Hydrocarbon radicals formed on the substrate under the bombardment diffuse to the graphene nanosheets and serve as the main source of the construction material. Thus, the leading role in the formation of vertical graphene belongs to surface diffusion, rather than direct deposition from the gas phase. The temperature of the sample is also an important parameter, which affects the growth process according to the following mechanism: at low temperatures the adsorption from the gas phase is more intense, but the diffusion processes are slowed down; elevated temperatures have the opposite effect. The surface density of graphene nanosheets, which can be controlled at the stage of nucleation, strongly affects the height of the structure due to the redistribution of ion fluxes during the growth: as the nanosheets grow, the ion current density decreases to the side edge of the sheet and increases to the upper edge. This process leads to a decrease in the ion current density at the side edge of the nanosheet, and, as a consequence, to a change in the dependence of the graphene sheet length on time: from a saturated curve or a quasilinear time dependence to a parabolic dependence. The assumption of surface diffusion of hydrocarbon radicals as the dominant growth mechanism is consistent with existing experimental data; these results confirm the physical model, and also bring a deeper understanding of the physics of growth of vertical graphene.

Network community detection via iterative edge removal in a flocking-like system

We present a network community-detection technique based on properties that emerge from a nature-inspired flocking system. Our algorithm comprises two alternating mechanisms: first, we control the particles alignment in higher dimensional space and, second, we present an iterative process of edge removal. These mechanisms together can potentially reduce accidental alignment among particles from different communities and, consequently, the model can generate robust community-detection results. In the proposed model, a random-direction unit vector is assigned to each vertex initially. A nonlinear dynamic law is established, so that neighboring vertices try to become aligned with each other. After some time, the system stops and edges that connect the least-aligned pairs of vertices are removed. Then, the evolution starts over without the removed edges, and after enough number of removal rounds, each community becomes a connected component. The proposed approach is evaluated using widely accepted benchmarks and real-world networks. Experimental results reveal that the method is robust and excels on a wide variety of networks. For large sparse networks, the edge-removal process runs in quasilinear time, which enables application in large-scale networks. Moreover, the distributed nature of the process eases the parallel implementation of the model.

Feature Matching Based on Minimum Relative Motion Entropy for Image Registration

Accurate point matching is widely used, and it is a critical and challenging process in feature-based image registration. To improve feature matching accuracy on putative matches with heavy outliers and similar local structures, an accurate and robust feature point matching algorithm based on minimum relative motion entropy (MRME) is proposed, in which the relative motion between the putative matches and their K-nearest neighbors is formulated. Based on the relative motion clustering result, the relative motion entropy is defined to find the coincident relative motions. According to relative motions with MRME, the outliers are removed in a two-stage feature match strategy. With quasi-linear time complexity, outliers with random or irregular relative motion are removed efficiently and accurately, while inliers with coincident relative motion are retained. Three data sets with repetitive patterns, viewpoint changes, low overlapping areas, and local deformations are used to demonstrate the performance of the proposed algorithm. MRME is shown to be more robust and accurate than ten state-of-the-art feature matching algorithms.

Improved Thermal Area Law and Quasilinear Time Algorithm for Quantum Gibbs States

One of the most fundamental problems in quantum many-body physics is the characterization of correlations among thermal states. Of particular relevance is the thermal area law, which justifies the tensor network approximations to thermal states with a bond dimension growing polynomially with the system size. In the regime of sufficiently low temperatures, which is particularly important for practical applications, the existing techniques do not yield optimal bounds. Here, we propose a new thermal area law that holds for generic many-body systems on lattices. We improve the temperature dependence from the original $\mathcal{O}(\beta)$ to $\tilde{\mathcal{O}}(\beta^{2/3})$, thereby suggesting diffusive propagation of entanglement by imaginary time evolution. This qualitatively differs from the real-time evolution which usually induces linear growth of entanglement. We also prove analogous bounds for the R\'enyi entanglement of purification and the entanglement of formation. Our analysis is based on a polynomial approximation to the exponential function which provides a relationship between the imaginary-time evolution and random walks. Moreover, for one-dimensional (1D) systems with $n$ spins, we prove that the Gibbs state is well-approximated by a matrix product operator with a sublinear bond dimension of $e^{\sqrt{\tilde{\mathcal{O}}(\beta \log(n))}}$. This proof allows us to rigorously establish, for the first time, a quasi-linear time classical algorithm for constructing an MPS representation of 1D quantum Gibbs states at arbitrary temperatures of $\beta = o(\log(n))$. Our new technical ingredient is a block decomposition of the Gibbs state, that bears resemblance to the decomposition of real-time evolution given by Haah et al., FOCS'18.