Ternary Tree Research Articles

Structure of Primitive Pythagorean Triples in Generating Trees

A Pythagorean triple is a triple of positive integers $(a,b,c)$ such that $a^2+b^2=c^2$. If $a,b$ are coprime, then it is called a primitive Pythagorean triple. It is known that every primitive Pythagorean triple can be generated from the triple $(3,4,5)$ using multiplication by unique number and order of three specific $3\times3$ matrices, which yields a ternary tree of triplets. Two such trees were described by Berggren and Price, respectively. A different approach is to view the primitive Pythagorean triples as points in the three-dimensional Euclidean space. In this paper, we prove that the triple of descendants of any primitive Pythagorean triple in Berggren's or Price's tree forms a triangle (and therefore defines a plane), and we present our results related to these triangles (and these planes).

Основанный на генетическом подходе алгоритм внутрикодирования для H.266/VVC

This paper presents a genetic approach for optimizing intra coding in H.266/VVC. The proposed algorithm efficiently selects coding tools and Multi-Type Tree (MTT) partitions to achieve a balance between encoding time and video quality. The fitness evaluation function, which combines perceptual metrics and coding efficiency metrics, is used to assess the quality of each candidate solution. The results demonstrate a significant reduction in encoding time without compromising video quality. The proposed algorithm selects coding tools from a set of available tools in H.266/VVC. These tools include intra prediction modes, transform units, quantization parameters, and entropy coding modes. The MTT partitioning scheme includes four types of partitions: quadtree, binary tree, ternary tree, and quad-binary tree. Perceptual metrics are used to evaluate the visual quality of the encoded video. Coding efficiency metrics are used to evaluate the coding efficiency of the encoded video. The fitness evaluation function combines perceptual metrics and coding efficiency metrics to assess the quality of each candidate solution.

Cutoff estimation and construction of their confidence intervals for continuous biomarkers under ternary umbrella and tree stochastic ordering settings.

Tuberculosis (TB) studies often involve four different states under consideration, namely, "healthy," "latent infection," "pulmonary active disease," and "extra-pulmonary active disease." While highly accurate clinical diagnosis tests do exist, they are expensive and generally not accessible in regions where they are most needed; thus, there is an interest in assessing the accuracy of new and easily obtainable biomarkers. For some such biomarkers, the typical stochastic ordering assumption might not be justified for all disease classes under study, and usual ROC methodologies that involve ROC surfaces and hypersurfaces are inadequate. Different types of orderings may be appropriate depending on the setting, and these may involve a number of ambiguously ordered groups that stochastically exhibit larger (or lower) marker scores than the remaining groups. Recently, there has been scientific interest on ROC methods that can accommodate these so-called "tree" or "umbrella" orderings. However, there is limited work discussing the estimation of cutoffs in such settings. In this article, we discuss the estimation and inference around optimized cutoffs when accounting for such configurations. We explore different cutoff alternatives and provide parametric, flexible parametric, and non-parametric kernel-based approaches for estimation and inference. We evaluate our approaches using simulations and illustrate them through a real data set that involves TB patients.

Polo: Adaptive Trie-Based Log Parser for Anomaly Detection

Automated log parsing is essential for many log-mining applications, as logs provide a vast range of information on events and variations within an operating system or software at runtime. Over the years, various methods have been proposed for log parsing. With improved log-parsing methods, log-mining applications can gain deeper insights into system behaviors and identify anomalies or failures promptly. However, current log parsers still face limitations, such as insufficient parsing of log templates and a lack of parallelism, as well as inaccurate log template parsing. To overcome these limitations, we have designed Polo, a parser that leverages a prefix forest composed of ternary search trees to mine templates from logs. We then conducted extensive experiments to evaluate the accuracy of Polo on nine representative system logs, achieving an average accuracy of 0.987. It is 9.93% to 40.95% faster than the state-of-the-art parsing methods. Furthermore, we evaluated our approach on a downstream log analysis task, specifically anomaly detection. The experimental results demonstrated that, in terms of F1-score, our parser outperformed Deeplog, LogAnomaly, CNN, and LogRobust by 11.5%, 4%, 1%, and 19.1%, respectively, exhibiting a promising recall score of 0.971. These results indicate the effectiveness of Polo for anomaly detection.

Machine Learning Based Efficient QT-MTT Partitioning Scheme for VVC Intra Encoders

The next-generation Versatile Video Coding (VVC) standard introduces a new Multi-Type Tree (MTT) block partitioning structure that supports Binary-Tree (BT) and Ternary-Tree (TT) splits in both vertical and horizontal directions. This new approach leads to five possible splits at each block depth. It thereby improves the coding efficiency of VVC over that of the preceding High Efficiency Video Coding (HEVC) standard, which only supports Quad-Tree (QT) partitioning with a single split per block depth. However, MTT also has brought a considerable impact on encoder computational complexity. This paper proposes a two-stage learning-based technique to tackle the complexity overhead of MTT in VVC intra encoders. In our scheme, the input block is first processed by a Convolutional Neural Network (CNN) to predict its spatial features through a vector of probabilities describing the partition at each 4×4 edge. Subsequently, a Decision Tree (DT) model leverages this vector of spatial features to predict the most likely splits at each block. Finally, based on this prediction, only the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> most likely splits are processed by the Rate-Distortion (RD) process of the encoder. In order to train our CNN and DT models on a wide range of image contents, we also propose a public VVC frame partitioning dataset based on existing image dataset encoded with the VVC reference software encoder. Our solution relying on the top-3 configuration reaches 47.4% complexity reduction for a negligible bitrate increase of 0.79%. A top-2 configuration enables a higher complexity reduction of 70.4% for 2.49% bitrate loss. These results emphasize a better trade-off between VTM intra-coding efficiency and complexity reduction compared to the state-of-the-art solutions. The source code of the proposed method and the training dataset are made publicly available at GitHub.

Visual Perception Based Intra Coding Algorithm for H.266/VVC

The latest international video coding standard, H.266/Versatile Video Coding (VVC), supports high-definition videos, with resolutions from 4 K to 8 K or even larger. It offers a higher compression ratio than its predecessor, H.265/High Efficiency Video Coding (HEVC). In addition to the quadtree partition structure of H.265/HEVC, the nested multi-type tree (MTT) structure of H.266/VVC provides more diverse splits through binary and ternary trees. It also includes many new coding tools, which tremendously increases the encoding complexity. This paper proposes a fast intra coding algorithm for H.266/VVC based on visual perception analysis. The algorithm applies the factor of average background luminance for just-noticeable-distortion to identify the visually distinguishable (VD) pixels within a coding unit (CU). We propose calculating the variances of the numbers of VD pixels in various MTT splits of a CU. Intra sub-partitions and matrix weighted intra prediction are turned off conditionally based on the variance of the four variances for MTT splits and a thresholding criterion. The fast horizontal/vertical splitting decisions for binary and ternary trees are proposed by utilizing random forest classifiers of machine learning techniques, which use the information of VD pixels and the quantization parameter. Experimental results show that the proposed algorithm achieves around 47.26% encoding time reduction with a Bjøntegaard Delta Bitrate (BDBR) of 1.535% on average under the All Intra configuration. Overall, this algorithm can significantly speed up H.266/VVC intra coding and outperform previous studies.

Good Things Come in Threes: A Ternary Tree for Triangular Triples

Summary A common saying is that good things come in threes. In 1934, B. Berggren may have agreed when unearthing three matrices which generate a ternary tree of all primitive Pythagorean triples from the initial triple . We show that a similar structure exists when replacing primitive Pythagorean triples with triangular triples. The construction of both ternary trees are intertwined, involving Berggren’s three matrices and a third ternary tree. Perhaps good things really do come in threes!

Restricted rotation distance between k-ary trees

We study restricted rotation distance between ternary and higher-valence trees using approaches based upon generalizations of Thompson's group $F$. We obtain bounds and a method for computing these distances exactly in linear time, as well as a linear-time algorithm for computing rotations needed to realize these distances. Unlike the binary case, the higher-valence notions of rotation distance do not give Tamari lattices, so there are fewer tools for analysis in the higher-valence settings. Higher-valence trees arise in a range of database and filesystem applications where balance is important for efficient performance.

Clifford Algebras, Spin Groups and Qubit Trees

Representations of Spin groups and Clifford algebras derived from the structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deletion of superfluous branches. The usual Jordan–Wigner construction also may be formally obtained in this approach by bringing the process up to trivial qubit chain (trunk). The methods can also be used for effective simulation of some quantum circuits corresponding to the binary tree structure. The modeling of more general qubit trees, as well as the relationship with the mapping used in the Bravyi–Kitaev transformation, are also briefly discussed.Quanta 2022; 11: 97–114.

Enumeration of Some Classes of Pattern Avoiding Matchings, with a Glimpse into the Matching Pattern Poset

A matching of the set [2n]={1,2,ldots ,2n} is a partition of [2n] into blocks with two elements, i.e. a graph on [2n], such that every vertex has degree one. Given two matchings sigma and tau , we say that sigma is a pattern of tau when sigma can be obtained from tau by deleting some of its edges and consistently relabelling the remaining vertices. This is a partial order relation turning the set of all matchings into a poset, which will be called the matching pattern poset. In this paper, we continue the study of classes of pattern avoiding matchings (see below for previous work on this subject). In particular, we work out explicit formulas to enumerate the class of matchings avoiding two new patterns, obtained by juxtaposition of smaller patterns, and we describe a recursive formula for the generating function of the class of matchings avoiding the lifting of a pattern and two additional patterns. Moreover, we introduce the notion of unlabeled pattern, as a combinatorial way to collect patterns, and we provide enumerative formulas for two classes of matchings avoiding an unlabeled pattern of order three. In one case, the enumeration follows from an interesting bijection between the matchings of the class and ternary trees. The last part of the paper initiates the study of the matching pattern poset, by providing some preliminary results about its Möbius functions and the structure of some simple intervals.

Object-Cooperated Ternary Tree Partitioning Decision Method for Versatile Video Coding.

In this paper, we propose an object-cooperated decision method for efficient ternary tree (TT) partitioning that reduces the encoding complexity of versatile video coding (VVC). In most previous studies, the VVC complexity was reduced using decision schemes based on the encoding context, which do not apply object detecion models. We assume that high-level objects are important for deciding whether complex TT partitioning is required because they can provide hints on the characteristics of a video. Herein, we apply an object detection model that discovers and extracts the high-level object features—the number and ratio of objects from frames in a video sequence. Using the extracted features, we propose machine learning (ML)-based classifiers for each TT-split direction to efficiently reduce the encoding complexity of VVC and decide whether the TT-split process can be skipped in the vertical or horizontal direction. The TT-split decision of classifiers is formulated as a binary classification problem. Experimental results show that the proposed method more effectively decreases the encoding complexity of VVC than a state-of-the-art model based on ML.

CMash: fast, multi-resolution estimation of k-mer-based Jaccard and containment indices.

Motivation K-mer-based methods are used ubiquitously in the field of computational biology. However, determining the optimal value of k for a specific application often remains heuristic. Simply reconstructing a new k-mer set with another k-mer size is computationally expensive, especially in metagenomic analysis where datasets are large. Here, we introduce a hashing-based technique that leverages a kind of bottom-m sketch as well as a k-mer ternary search tree (KTST) to obtain k-mer-based similarity estimates for a range of k values. By truncating k-mers stored in a pre-built KTST with a large value, we can simultaneously obtain k-mer-based estimates for all k values up to kmax. This truncation approach circumvents the reconstruction of new k-mer sets when changing k values, making analysis more time and space-efficient.ResultsWe derived the theoretical expression of the bias factor due to truncation. And we showed that the biases are negligible in practice: when using a KTST to estimate the containment index between a RefSeq-based microbial reference database and simulated metagenome data for 10 values of k, the running time was close to 10× faster compared to a classic MinHash approach while using less than one-fifth the space to store the data structure.Availability and implementationA python implementation of this method, CMash, is available at https://github.com/dkoslicki/CMash. The reproduction of all experiments presented herein can be accessed via https://github.com/KoslickiLab/CMASH-reproducibles.Supplementary information Supplementary data are available at Bioinformatics online.

A differential evolution algorithm with ternary search tree for solving the three-dimensional packing problem

The three-dimensional packing problem, which is a strongly NP-hard problem and extremely difficult to solve in practice, is a typical issue in container loading. The container loading problem (CLP) is a combinatorial optimization problem with complex constraints. Its solution process is complicated, and the amount of computation is large. Although some studies used exact algorithms, they could not obtain the optimal solution in a reasonable time. Therefore, heuristic algorithms and intelligent optimization algorithms have become the preferred choice for solving the three-dimensional container loading problem (3D-CLP). Differential evolution algorithm (DE) is an efficient intelligent optimization algorithm, which has remarkable effect in dealing with optimization problems. In this paper, we propose a novel technique that combines a differential evolution algorithm with a ternary search tree model (TSTDE) to solve the 3D-CLP. TSTDE utilizes the ternary tree model to generate a set of suboptimal solutions as the initial population of DE. The improved differential evolution algorithm is used to search the feasible solution of the problem. In this paper, we used the Bischoff and Ratcliff instances to test the TSTDE algorithm, and the results solving the 3D-CLP were compared with those of other advanced algorithms. The experimental results verified the feasibility and superiority of TSTDE in handling the 3D-CLP.

The least dimonoid congruences on relatively free trioids

When Loday and Ronco studied ternary planar trees, they introduced types of algebras,called trioids and trialgebras. A trioid is a nonempty set equipped with three binary associativeoperations satisfying additional eight axioms relating these operations, while a trialgebra is justa linear analog of a trioid. If all operations of a trioid (trialgebra) coincide, we obtain the notionof a semigroup (associative algebra), and if two concrete operations of a trioid (trialgebra)coincide, we obtain the notion of a dimonoid (dialgebra) and so, trioids (trialgebras) are ageneralization of semigroups (associative algebras) and dimonoids (dialgebras). Trioids andtrialgebras have close relationships with the Hopf algebras, the Leibniz 3-algebras, the Rota-Baxter operators, and the post-Jordan algebras. Originally, these structures arose in algebraictopology. One of the most useful concepts in algebra is the free object. Every variety containsfree algebras and free objects in any variety of algebras are important in the study of thatvariety. Loday and Ronco constructed the free trioid of rank 1 and the free trialgebra. Recently,the free trioid of an arbitrary rank, the free commutative trioid, the free n-nilpotent trioid, thefree rectangular triband, the free left n-trinilpotent trioid and the free abelian trioid wereconstructed and the least dimonoid congruences as well as the least semigroup congruence onthe first four free algebras were characterized. However, just mentioned congruences on freeleft (right) n-trinilpotent trioids and free abelian trioids were not considered. In this paper, wecharacterize the least dimonoid congruences and the least semigroup congruence on free left(right) n-trinilpotent trioids and free abelian trioids.

Activity-Based Block Partitioning Decision Method for Versatile Video Coding

Versatile Video Coding (VVC), the latest international video coding standard, has more than twice the compression performance of High-Efficiency Video Coding (HEVC) through adopting various coding techniques. The multi-type tree (MTT) block structure offers more advanced flexible block partitioning by allowing the binary tree (BT) and ternary tree (TT) structures, as well as the quadtree (QT) structure. Because VVC selects the optimal block partition by performing encoding on all possible CU partitions, the encoding complexity increases enormously. In this paper, we observe the relationship between block partitions and activity that indicates block texture complexity. Based on experimental observations, we propose an activity-based fast block partitioning decision method to reduce the encoding complexity. The proposed method uses only information of the current block without using the information of neighboring or upper blocks, and also minimizes the dependency on QP. For these reasons, the proposed algorithm is simple and parallelizable. In addition, by utilizing the gradient calculation used in VVC’s ALF, a VVC-friendly fast algorithm was designed. The proposed method consists of two-step decision-making processes. The first step terminates the block partitioning early based on observed posterior probability through the relationship between the block size and activity per sample. Next, the sub-activities of the current block are used to determine the type and direction of partitioning. The experimental results show that in the all-intra configuration, the proposed method can reduce the encoding time of the VVC test model (VTM) by up to 45.15% with 2.80% BD-rate loss.

A Weighted Error Distance Metrics (WEDM) for Performance Evaluation on Multiple Change-Point (MCP) Detection in Synthetic Time Series.

Change-point detection (CPD) is to find abrupt changes in time-series data. Various computational algorithms have been developed for CPD applications. To compare the different CPD models, many performance metrics have been introduced to evaluate the algorithms. Each of the previous evaluation methods measures the different aspects of the methods. Based on the existing weighted error distance (WED) method on single change-point (CP) detection, a novel WED metrics (WEDM) was proposed to evaluate the overall performance of a CPD model across not only repetitive tests on single CP detection, but also successive tests on multiple change-point (MCP) detection on synthetic time series under the random slide window (RSW) and fixed slide window (FSW) frameworks. In the proposed WEDM method, a concept of normalized error distance was introduced that allows comparisons of the distance between the estimated change-point (eCP) position and the target change point (tCP) in the synthetic time series. In the successive MCPs detection, the proposed WEDM method first divides the original time-series sample into a series of data segments in terms of the assigned tCPs set and then calculates a normalized error distance (NED) value for each segment. Next, our WEDM presents the frequency and WED distribution of the resultant eCPs from all data segments in the normalized positive-error distance (NPED) and the normalized negative-error distance (NNED) intervals in the same coordinates. Last, the mean WED (MWED) and MWTD (1-MWED) were obtained and then dealt with as important performance evaluation indexes. Based on the synthetic datasets in the Matlab platform, repetitive tests on single CP detection were executed by using different CPD models, including ternary search tree (TST), binary search tree (BST), Kolmogorov–Smirnov (KS) tests, t-tests (T), and singular spectrum analysis (SSA) algorithms. Meanwhile, successive tests on MCPs detection were implemented under the fixed slide window (FSW) and random slide window (RSW) frameworks. These CPD models mentioned above were evaluated in terms of our WED metrics, together with supplementary indexes for evaluating the convergence of different CPD models, including rates of hit, miss, error, and computing time, respectively. The experimental results showed the value of this WEDM method.

TAID-LCA: Segmentation Algorithm Based on Ternary Trees

In this work, a statistical method for the segmentation of samples and/or populations is presented, which is based on a ternary tree structure. This approach overcomes known limitations of other segmentation methods such as CHAID, concerning the multivariate response and the non-symmetric relationship between explanatory and response variables. The multivariate response segmentation problem is handled through latent class models, while the factorial decomposition of the explanatory capability of variables is based on the Non-Symmetrical Correspondence Analysis. Stop criteria based on the CATANOVA index and impurity measures are proposed. A Simulated Annealing based post-pruning strategy is considered to avoid over-fitting relative to the training set and guarantee a better generalization capability for the method.

Counting ternary trees according to the number of middle edges and factorizing into (3/2)-ary trees

The sequence A120986 in the Encyclopedia of Integer Sequences counts ternary trees according to the number of edges (equivalently nodes) and the number of middle edges. Using a certain substitution, the underlying cubic equation can be factored. This leads to an extension of the concept of [Formula: see text]-ary trees, introduced by Knuth in his christmas lecture from 2014.

High Efficiency Intra CU Partition and Mode Decision Method for VVC

Versatile video coding (VVC/H.266) is the newest video compression standard, which is developed by the Joint Video Experts Team. Compared with previous encoding schemes, VVC achieves higher compression efficiency by introducing a new partition structure and additional intra prediction modes but results in high computational complexity. To efficiently solve this problem of redundant processing in quad-tree with nested multi-type tree structures and intra mode prediction, we propose a texture analysis&#x2013;based ternary tree (TT) and binary tree (BT) partition strategy, and a gradient-based intra mode decision method to accelerate TT and BT partition and intra mode prediction, separately. The texture complexity and prediction direction of coding unit (CU) is calculated by texture detection method. A texture analysis&#x2013;based TT and BT partition strategy is established by using the regression method based on analyzing the texture complexity of the CU. Then, a texture analysis&#x2013;based TT and BT partition strategy is applied to reduce the redundant partition for each CU. By using the prediction direction of CU, a gradient-based intra mode decision method is established for skipping the impossible modes for each CU. Experimental results revealed that the proposed method could save 49.49% in encoding time and increase the Bjontegaard delta bit rate (BDBR) by only 0.56%. It confirms that the proposed method achieved high efficiency and a good balance between the BDBR and time saving.

Efficient Partition Decision Based on Visual Perception and Machine Learning for H.266/Versatile Video Coding

H.266/Versatile Video Coding (VVC) is the latest international video coding standard to encode ultra-high-definition video effectively. The quadtree with nested multi-type tree (QT-MTT) structure provides various sizes of coding tree partitioning and allows the nested binary tree (BT) split and ternary tree (TT) split at each QT level. Furthermore, numerous advanced coding tools are equipped in the H.266/VVC encoder. However, the encoding time increases tremendously. Previous researches regarding the fast coding algorithm of H.266/VVC seldom mention perceptual redundancy. This paper utilizes the human vision model of just noticeable difference to extract the visually distinguishable pixels that may affect the visual perception. We observe that the distributions acquired by the horizontal and vertical projections of visually distinguishable pixels within the coding unit are related to their corresponding MTT splitting modes. Therefore, the distributions representing the perceptual information of human vision are used to be the input features of machine learning. Fast MTT decision determined by the random forest models of machine learning is proposed to quickly select the partition for intra coding. Experimental results demonstrate that the proposed method can effectively accelerate intra coding process while maintaining good bitrate and video quality based on the properties of the visual perception. The proposed algorithm provides better performance than the previous work.