Greedy Tree Research Articles

Energy Constrained Depth First Search

AbstractDepth first search is a natural algorithmic technique for constructing a closed route that visits all vertices of a graph. The length of such a route equals, in an edge-weighted tree, twice the total weight of all edges of the tree and this is asymptotically optimal over all exploration strategies. This paper considers a variant of such search strategies where the length of each route is bounded by a positive integer B (e.g. due to limited energy resources of the searcher). The objective is to cover all the edges of a tree T using the minimum number of routes, each starting and ending at the root and each being of length at most B. To this end, we analyze the following natural greedy tree traversal process that is based on decomposing a depth first search traversal into a sequence of limited length routes. Given any arbitrary depth first search traversal R of the tree T, we cover R with routes $$R_1,\ldots ,R_l$$ R 1 , … , R l , each of length at most B such that: $$R_i$$ R i starts at the root, reaches directly the farthest point of R visited by $$R_{i-1}$$ R i - 1 , then $$R_i$$ R i continues along the path R as far as possible, and finally $$R_i$$ R i returns to the root. We call the above algorithm piecemeal-DFS and we prove that it achieves the asymptotically minimal number of routes l, regardless of the choice of R. Our analysis also shows that the total length of the traversal (and thus the traversal time) of piecemeal-DFS is asymptotically minimum over all energy-constrained exploration strategies. The fact that R can be chosen arbitrarily means that the exploration strategy can be constructed in an online fashion when the input tree T is not known in advance. Each route $$R_i$$ R i can be constructed without any knowledge of the yet unvisited part of T. Surprisingly, our results show that depth first search is efficient for energy constrained exploration of trees, even though it is known that the same does not hold for energy constrained exploration of arbitrary graphs.

Energy efficient greedy tree based algorithm for data aggregation in wireless sensor network

Secure data aggregation is essential in wireless sensor networks for lowering the amount of data transmitted and extending the lifetime of the network. The foundation underlying significantly greater industrial internet of things applications is primarily wireless sensor nodes. Sensors that have already been integrated within can be used to sense data in any form of real-time IoT application. In real-time physical surroundings, sensors utilize as little power as feasible to conduct operations including sensing, communicating, and processing data. Many investigations are being carried out to improve sensor node energy efficiency and network lifetime. To save energy, more attention must be paid to the clustering and routing aspects of communication. In this paper, we introduce Energy Efficient Greedy Tree based Data Aggregation (EE–GTDA) algorithm for efficient data aggregation with increased reliability and reduced energy consumption. It is a two-fold homogeneous technique that supervises safe energy-efficient connectivity and data aggregation with the greedy tree based solution that emphasizes multi-objective function. These methodologies are based on minimizing sensor energy consumption to maximize network lifetime simultaneously decreasing communication overhead. A trade-off between energy and safety is accomplished in order to increase efficient energy consumption with a higher packet delivery ratio.

Discrimination of Quartz Genesis Based on Explainable Machine Learning

Quartz is an important mineral in many metal deposits and can provide important indications about the deposit's origin through its chemical composition. However, traditional low-dimensional analysis methods are ineffective in utilizing quartz's chemical composition to reveal the deposit's origin type. In this study, 1140 quartz samples from eight geological environments were collected, and three machine learning (ML) models—random forest, eXtremely Greedy tree Boosting (XGBoost), and light gradient boosting machine (LightGBM) were used to classify quartz deposits. The application of the Shapley Additive Explanation (SHAP) algorithm and Spearman correlation analysis is utilized to interpret the predictive results of the model and analyze feature correlations, aiming to enhance the credibility of the classification results and discover underlying patterns. Finally, a visualization method based on XGBoost and t-SNE was proposed. By calculating SHAP values, the key geochemical indicators that differentiate each type of quartz deposit were determined. Furthermore, the impact of varying concentrations of different trace elements on the identification of quartz deposits was analyzed. This study demonstrated the effectiveness of using machine-learning algorithms based on trace elements to classify quartz and provided new insights into the relationships between trace elements and quartz genesis, as well as the effects of different trace element combinations and concentrations on quartz identification.

An alternative proof of the Sombor index minimizing property of greedy trees

Recently, Gutman defined a new graph invariant which is named the Sombor index SO(G) of a graph G and is computed via the expression SO(G) = ?u~v? qdeg(u)2 + deg(v)2, where deg(u) represents the degree of the vertex u in G and the summing is performed across all the unordered pairs of adjacent vertices u and v. Damnjanovic et al. have implemented an earlier result obtained by Wang in order to show that, among all the trees TD that have a specified degree sequence D, the greedy tree must attain the minimum Sombor index. Here we provide an alternative proof of this same result by constructing an auxiliary graph invariant named the pseudo-Sombor index and without relying on any other earlier results.

A Note on Extremal Sombor Indices of Trees with a Given Degree Sequence

We note here that the problem of determining extremal values of Sombor index for trees with a given degree sequence fits within the framework of results by Hua Wang from [Cent. Eur. J. Math. 12 (2014) 1656-1663], implying that the greedy tree has the minimum Sombor index, while an alternating greedy tree has the maximum Sombor index.

Discriminating Tectonic Setting of Igneous Rocks Using Biotite Major Element chemistry−A Machine Learning Approach

AbstractThe composition of igneous biotite is a potential indicator of the geologic environment of its host rock. Here, we apply two machine learning models−eXtreemly Greedy tree Boosting (XGBoost) and light gradient boosting machine (LightGBM) to classify biotite in igneous rocks from five tectonic settings−oceanic intraplate, continental intraplate, continental arc, island arc, and rift, using their major element chemistry compiled from a global dataset. The two models are successfully able to discriminate among biotite from the five tectonic settings. The classifiers quantitatively search for unique geochemical signatures in the training dataset, mapping each biotite analysis to its corresponding tectonic setting. Both models yielded good classification accuracy (∼90% on average) on an unseen dataset, suggesting that biotite major element chemistry can successfully discriminate the tectonic setting of many igneous rocks. The Shapley Additive exPlanations algorithm, which measures the impact of every element, indicates that Na, Mn, Ba, Mg, Al, Cr, and O2− constitute important geochemical discriminators. According to both models, high Na, low Na, low Ba, high Mn, and low Mn of biotite have the strongest influence on the prediction of continental arc, continental intraplate, island arc, oceanic intraplate, and rift settings, respectively. The classifier models have been applied to investigate the Neoproterozoic geodynamics of the Aravalli‐Delhi Belt, northwestern India. The models show that igneous activity related to the Erinpura granites (∼835 Ma) and the Malani Igneous Suite (∼750 Ma) can be ascribed to continental intraplate setting related to extensional tectonics prior to the break‐up of the Rodinia supercontinent.

G-CAS: Greedy Algorithm-Based Security Event Correlation System for Critical Infrastructure Network

The attacks on the critical infrastructure network have increased sharply, and the strict management measures of the critical infrastructure network have caused its correlation analysis technology for security events to be relatively backward; this makes the critical infrastructure network’s security situation more severe. Currently, there is no common correlation analysis technology for the critical infrastructure network, and most technologies focus on expanding the dimension of data analysis, but with less attention to the optimization of analysis performance. The analysis performance does not meet the practical environment, and real-time analysis is even more impossible; as a result, the efficiency of security threat detection is greatly declined. To solve this issue, we propose the greedy tree algorithm, a correlation analysis approach based on the greedy algorithm, which optimizes event analysis steps and significantly improves the performance, so the real-time correlation analysis can be realized. We first verify the performance of the algorithm through formalization, and then the G-CAS (Greedy Correlation Analysis System) is implemented based on this algorithm and is applied in a real critical infrastructure network, which outperformed the current mainstream products.

Uncovering feature interdependencies in high-noise environments with stepwise lookahead decision forests

Conventionally, random forests are built from “greedy” decision trees which each consider only one split at a time during their construction. The sub-optimality of greedy implementation has been well-known, yet mainstream adoption of more sophisticated tree building algorithms has been lacking. We examine under what circumstances an implementation of less greedy decision trees actually yields outperformance. To this end, a “stepwise lookahead” variation of the random forest algorithm is presented for its ability to better uncover binary feature interdependencies. In contrast to the greedy approach, the decision trees included in this random forest algorithm, each simultaneously consider three split nodes in tiers of depth two. It is demonstrated on synthetic data and financial price time series that the lookahead version significantly outperforms the greedy one when (a) certain non-linear relationships between feature-pairs are present and (b) if the signal-to-noise ratio is particularly low. A long-short trading strategy for copper futures is then backtested by training both greedy and stepwise lookahead random forests to predict the signs of daily price returns. The resulting superior performance of the lookahead algorithm is at least partially explained by the presence of “XOR-like” relationships between long-term and short-term technical indicators. More generally, across all examined datasets, when no such relationships between features are present, performance across random forests is similar. Given its enhanced ability to understand the feature-interdependencies present in complex systems, this lookahead variation is a useful extension to the toolkit of data scientists, in particular for financial machine learning, where conditions (a) and (b) are typically met.

Extremal Trees with Fixed Degree Sequence

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.

Dismantling Networks by Skeleton Extraction and Greedy Tree Breaking

Network dismantling is one of the important NP-hard problems in the field of social network analysis. It aims to break down networks into many small components of limited size by only removing a small group of nodes. One feasible way is to decycle (eliminating all the cycles) the network first and then break the acyclic graph. However, existing decycling-based algorithms mainly concentrate on the decycling step, ignoring the importance of the tree breaking process. Besides, none of the algorithms try to pre-process the network, which may bring improvement in both effectiveness and efficiency. In this paper, we fill these two gaps by proposing a novel network dismantling algorithm that combines skeleton extraction and greedy tree breaking (SEGTB). Network skeleton supports the whole network structure, whose removal would leave a much looser structure and serves as an effective pre-processing for the dismantling problem. The presented tree breaking method is provided with theoretical proofs on its lower bound. Experiments on ten real-world datasets show that our proposed SEGTB algorithm is both effective and efficient, outperforming the state-of-the-art.

Deep Reinforcement Learning Designed Shinnar-Le Roux RF Pulse Using Root-Flipping: DeepRFSLR.

A novel approach of applying deep reinforcement learning to an RF pulse design is introduced. This method, which is referred to as DeepRFSLR, is designed to minimize the peak amplitude or, equivalently, minimize the pulse duration of a multiband refocusing pulse generated by the Shinar Le-Roux (SLR) algorithm. In the method, the root pattern of SLR polynomial, which determines the RF pulse shape, is optimized by iterative applications of deep reinforcement learning and greedy tree search. When tested for the designs of the multiband pulses with three and seven slices, DeepRFSLR demonstrated improved performance compared to conventional methods, generating shorter duration RF pulses in shorter computational time. In the experiments, the RF pulse from DeepRFSLR produced a slice profile similar to the minimum-phase SLR RF pulse and the profiles matched to that of the computer simulation. Our approach suggests a new way of designing an RF by applying a machine learning algorithm, demonstrating a "machine-designed" MRI sequence.

LP Relaxation and Tree Packing for Minimum $k$-Cut

Karger used spanning tree packings [D. R. Karger, J. ACM, 47 (2000), pp. 46--76] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [D. R. Karger, Random Sampling in Graph Optimization Problems, Ph.D. thesis, Stanford University, Stanford, CA, 1995, D. R. Karger and C. Stein, J. ACM, 43 (1996), pp. 601--640]. Thorup developed a fast deterministic algorithm for the minimum $k$-cut problem via greedy recursive tree packings [M. Thorup, Minimum $k$-way cuts via deterministic greedy tree packing, in Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 159--166]. In this paper we revisit properties of an LP relaxation for cͅut proposed by Naor and Rabani [Tree packing and approximating $k$-cuts, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Vol. 103, SIAM, Philadelphia, 2001, pp. 26--27], and analyzed in [C. Chekuri, S. Guha, and J. Naor, SIAM J. Discrete Math., 20 (2006), pp. 261--271]. We show that the dual of the LP yields a tree packing that, when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the $k$-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of $n$. We also improve the bound on the number of $\alpha$-approximate $k$-cuts. Second, we give a simple proof that the integrality gap of the LP is $2(1-1/n)$. Third, we show that an optimum solution to the LP relaxation, for all values of $k$, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Oper. Res. Lett., 26 (2000), pp. 99--105] and Ravi and Sinha [European J. Oper. Res., 186 (2008), pp. 77--90]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP.

The Aα-spectral radius of trees and unicyclic graphs with given degree sequence

For any real α ∈ [0, 1], Aα(G)=αD(G)+(1−α)A(G) is the Aα-matrix of a graph G, where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the degrees of G. This paper presents some extremal results about the spectral radius λ1(Aα(G)) of Aα(G) that generalize previous results about λ1(A0(G)) and λ1(A12(G)). In this paper, we study the behavior of the Aα-spectral radius under some graph transformations for α ∈ [0, 1). As applications, we show that the greedy tree has the maximum Aα-spectral radius in GD when D is a tree degree sequence firstly. Furthermore, we determine that the greedy unicyclic graph has the largest Aα-spectral radius in GD when D is a unicyclic graphic sequence, where GD={G∣GisconnectedwithDasitsdegreesequence}.

On structural properties of ABC-minimal chemical trees

The atom-bond connectivity (ABC) index of a connected graph G is defined as ABC(G)=∑uv∈E(G)d(u)+d(v)−2d(u)d(v), where d(w) is the degree of vertex w in G and E(G) is the set of edges of G. Given greedy trees perturbed by different degree sequences, we investigate the behaviors on their ABC index in this paper. Moreover, we also give the minimum ABC index and its structural properties of the chemical trees with n vertices and p pendant vertices for n≥3p−2.

A greedy constructing tree algorithm for shortest path in perpetual wireless recharging wireless sensor network

Energy recharging has received much attention now because of the development of wireless recharging technology. This paper proposes a greedy constructing tree algorithm for shortest path in perpetual wireless recharging wireless sensor networks (GCTA) which enables the recharger to work while moving through the shortest path. The algorithm can construct the shorter path which passes all the sensors only once and guarantees that each sensor will be fully recharged. It computes the path in advance with lower complexity. And it also uses the graph theory, the tree theory, the Markov chain theorem and the powerful computing function of a computer. The contributions of the paper are itemized as follows: guaranteeing all the sensors to be fully recharged only once and reducing the length of the recharging path through constructing the greedy tree. In the experiment, this algorithm can construct the shorter path. So it is novel, general and can be applied widely.

RAxML-NG: a fast, scalable and user-friendly tool for maximum likelihood phylogenetic inference.

MotivationPhylogenies are important for fundamental biological research, but also have numerous applications in biotechnology, agriculture and medicine. Finding the optimal tree under the popular maximum likelihood (ML) criterion is known to be NP-hard. Thus, highly optimized and scalable codes are needed to analyze constantly growing empirical datasets.ResultsWe present RAxML-NG, a from-scratch re-implementation of the established greedy tree search algorithm of RAxML/ExaML. RAxML-NG offers improved accuracy, flexibility, speed, scalability, and usability compared with RAxML/ExaML. On taxon-rich datasets, RAxML-NG typically finds higher-scoring trees than IQTree, an increasingly popular recent tool for ML-based phylogenetic inference (although IQ-Tree shows better stability). Finally, RAxML-NG introduces several new features, such as the detection of terraces in tree space and the recently introduced transfer bootstrap support metric.Availability and implementationThe code is available under GNU GPL at https://github.com/amkozlov/raxml-ng. RAxML-NG web service (maintained by Vital-IT) is available at https://raxml-ng.vital-it.ch/.Supplementary information Supplementary data are available at Bioinformatics online.

A note on extremal trees with degree conditions

A fundamental question in the study of graph invariants asks for the extremal structures under certain constraints that maximize or minimize a graph invariant. In this note, we summarize some recent work on the extremal trees of distance-based and degree-based graph invariants under various degree conditions. We note that many of such extremal structures turned out to be identical for different but similar invariants. Such common extremal structures are investigated through the greedy trees and majorization between degree sequences. We show that many of the known extremal results can be obtained through this line of arguments. We also introduce some new extremal results as immediate consequences.

On majorization of closed walk vectors of trees with given degree sequences

Let Cv(k; T) be the number of closed walks of length k starting at vertex v in a tree T. We prove that for any tree T with a given degree sequence π, the vector C(k; T) ≡ (Cv(k; T), v ∈ V(T)) is weakly majorized by the vector C(k;Tπ*)≡(Cv(k;Tπ*),v∈V(Tπ*)), where Tπ* is the greedy tree with the degree sequence π. In addition, for two trees degree sequences π and π′, if π is majorized by π′, then C(k;Tπ*) is weakly majorized by C(k;Tπ′*).

A low overhead flow-holding algorithm in software-defined networks

Software-Defined Networking (SDN) allows flexible and efficient management of networks. However, the limited capacity of flow tables in SDN switches hinders the deployment of SDN. In this paper, we propose a novel routing scheme to improve the efficiency of flow tables in SDNs. To efficiently use the routing scheme, we formulate an optimization problem with the objective to maximize the number of flows in the network, constrained by the limited flow table space in SDN switches. The problem is NP-hard, and we propose the K Similar Greedy Tree (KSGT) algorithm to solve it. We evaluate the performance of KSGT against “traditional” SDN solutions with real-world topologies and traffic. The results show that, compared to the existing solutions, KSGT can reduce about 60% of flow entries when processing the same amount of flows, and improve about 25% of the successful installation and forwarding flows under the same flow table space.

Improving Resiliency in SDN using Routing Tree Algorithms

The ability to recover the control logic after a failure is detected in specific time window is called resiliency. The Software Defined Network (SDN) is an emerged and powerful architecture which allow to separate the control plane from forwarding. This decoupling architecture brings new difficulties to the network resiliency because link failure between switch and controller could defunct the forwarding plane. It has been identified that the resiliency of the network can be improved by choosing the correct place for the controller and by choosing proper routing tree once the controller location is known. In this work, we have analysed the performance of various Routing Tree algorithms on different network topology generated by Bernoulli Random Graph model and found that Greedy Routing Tree (GRT) provides the maximum resiliency. The Closeness Centrality Theorem has proposed to find the best controller position and later analysed the performance of various single controller placement algorithms on GRT for finding the overall improvement of the resiliency of the network.