The anti-Ramsey number, ar(G, H) is the minimum integer k such that in any edge colouring of G with k colours there is a rainbow subgraph isomorphic to H, namely, a copy of H with each of its edges assigned a differen...
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The anti-Ramsey number, ar(G, H) is the minimum integer k such that in any edge colouring of G with k colours there is a rainbow subgraph isomorphic to H, namely, a copy of H with each of its edges assigned a different colour. The notion was introduced by Erdos and Simonovits in 1973. Since then the parameter has been studied extensively. The case when H is a star graph was considered by several graph theorists from the combinatorial point of view. Recently this case received the attention of researchers from the algorithm community because of its applications in interface modelling of wireless networks. To the algorithm community, the problem is known as maximum edge q-colouring problem: Find a colouring of the edges of G, maximizing the number of colours satisfying the constraint that each vertex spans at most q colours on its incident edges. It is easy to see that the maximum value of the above optimization problem equals ar(G, K1,q+1) - 1. In this paper, we study the maximum edge 2-colouring problem from the approx-imation algorithm point of view. The case q = 2 is particularly interesting due to its application in real-life problems. algorithmically, this problem is known to be NP-hard for q - 2. For the case of q = 2, it is also known that no polynomial-time algorithm can approximate to a factor less than 3/2 assuming the unique games conjecture. Feng et al. showed a 2-approximation algorithm for this problem. Later Adamaszek and Popa presented a 5/3-approximation algorithm with the additional assumption that the input graph has a perfect matching. Note that the obvious but the only known algorithm issues different colours to the edges of a maximum matching (say M) and different colours to the connected components of G \ M. In this article, we give a new analysis of the aforementioned algorithm to show that for triangle-free graphs with perfect matching the approximation ratio is 8/5. We also show that this algorithm cannot achieve a factor better than 58/37 o
In the literature, many algorithms have been proposed for finding cutnodes on undirected graphs, since cutnodes are crucial to graph connectivity. Here, a cutnode of an undirected graph G is a node of G, whose deletio...
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In the literature, many algorithms have been proposed for finding cutnodes on undirected graphs, since cutnodes are crucial to graph connectivity. Here, a cutnode of an undirected graph G is a node of G, whose deletion will cause a reachable pair of the other nodes in G to be unreachable. Currently, the difficulty of maintaining the entire G in the main memory makes researchers pay attention to compute cutnodes on semi-external memory model. This paper shows that traditional semi-external algorithms are limited by their unbounded time and I/O consumption, making them impractical when G is relatively large or complex. Thus, we propose a linear semi -external cutnode computation algorithm, named SECN. Assuming that G has n nodes and m edges, and B is the disk block size. SECN is the first that can find all the cutnodes of G in O(m thorn n) time and with O(m/B) I/O cost on the semi-external memory model, as far as we know. SECN also has a smaller minimum memory space requirement than traditional algorithms. Our experimental evaluation conducted on both synthetic and real graphs confirms that SECN significantly outperforms existing algorithms (up to 103 times faster). (c) 2022 Elsevier Inc. All rights reserved.
This article presents I/O-efficient algorithms for topologically sorting a directed acyclic graph and for the more general problem identifying and topologically sorting the strongly connected components of a directed ...
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This article presents I/O-efficient algorithms for topologically sorting a directed acyclic graph and for the more general problem identifying and topologically sorting the strongly connected components of a directed graph G = (V, E). Both algorithms are randomized and have I/O-costs O(sort (E) center dot poly(log V)), with high probability, where sort (E) = O(E/B log(M/B) (E/B)) is the I/O cost of sorting an |E|-element array on a machine with size-B blocks and size-M cache/internal memory. These are the first algorithms for these problems that do not incur at least one I/O per vertex, and as such these are the first I/O-efficient algorithms for sparse graphs. By applying the technique of time-forward processing, these algorithms also imply I/O-efficient algorithms for most problems on directed acyclic graphs, such as shortest paths, as well as the single-source reachability problem on arbitrary directed graphs.
Suffix trees are an important data structure at the core of optimal solutions to many fundamental string problems, such as exact pattern matching, longest common substring, matching statistics, and longest repeated su...
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Suffix trees are an important data structure at the core of optimal solutions to many fundamental string problems, such as exact pattern matching, longest common substring, matching statistics, and longest repeated substring. Recent lines of research focused on extending some of these problems to vertex-labeled graphs, either by using efficient ad-hoc approaches which do not generalize to all input graphs, or by indexing difficult graphs and having worst-case exponential complexities. In the absence of an ubiquitous and polynomial tool like the suffix tree for labeled graphs, we introduce the labeled direct product of two graphs as a general tool for obtaining optimal algorithms in the worst case: we obtain conceptually simpler algorithms for the quadratic problems of string matching (SMLG) and longest common substring (LCSP) in labeled graphs. Our algorithms run in time linear in the size of the labeled product graph, which may be smaller than quadratic for some inputs, and their run-time is predictable, because the size of the labeled direct product graph can be precomputed efficiently. We also solve LCSP on graphs containing cycles, which was left as an open problem by Shimohira et al. in 2011. To show the power of the labeled product graph, we also apply it to solve the matching statistics (MSP) and the longest repeated string (LRSP) problems in labeled graphs. Moreover, we show that our (worst-case quadratic) algorithms are also optimal, conditioned on the Orthogonal Vectors Hypothesis. Finally, we complete the complexity picture around LRSP by studying it on undirected graphs.
In this paper, we address the problem of finding a representation of a subtree distance, which is an extension of a tree metric. We show that a minimal representation is uniquely determined by a given subtree distance...
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In this paper, we address the problem of finding a representation of a subtree distance, which is an extension of a tree metric. We show that a minimal representation is uniquely determined by a given subtree distance, and give an O(n(2)) time algorithm that finds such a representation, where n is the size of the ground set. Since a lower bound of the problem is Omega(n(2)), our algorithm achieves the optimal time complexity.
Background: COVID-19 is an infectious disease caused by SARS-CoV-2. The symptoms of COVID-19 vary from mild-to-moderate respiratory illnesses, and it sometimes requires urgent medication. Therefore, it is crucial to d...
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Background: COVID-19 is an infectious disease caused by SARS-CoV-2. The symptoms of COVID-19 vary from mild-to-moderate respiratory illnesses, and it sometimes requires urgent medication. Therefore, it is crucial to detect COVID-19 at an early stage through specific clinical tests, testing kits, and medical devices. However, these tests are not always available during the time of the pandemic. Therefore, this study developed an automatic, intelligent, rapid, and real-time diagnostic model for the early detection of COVID-19 based on its ***: The COVID-19 knowledge graph (KG) constructed based on literature from heterogeneous data is imported to understand the COVID-19 different relations. We added human disease ontology to the COVID-19 KG and applied a node-embedding graph algorithm called fast random projection to extract an extra feature from the COVID-19 dataset. Subsequently, experiments were conducted using two machine learning (ML) pipelines to predict COVID-19 infection from its symptoms. Additionally, automatic tuning of the model hyperparameters was ***: We compared two graph-based ML models, logistic regression (LR) and random forest (RF) models. The proposed graph-based RF model achieved a small error rate = 0.0064 and the best scores on all performance metrics, including specificity = 98.71%, accuracy = 99.36%, precision = 99.65%, recall = 99.53%, and F1-score = 99.59%. Furthermore, the Matthews correlation coefficient achieved by the RF model was higher than that of the LR model. Comparative analysis with other ML algorithms and with studies from the literature showed that the proposed RF model exhibited the best detection ***: The graph-based RF model registered high performance in classifying the symptoms of COVID-19 infection, thereby indicating that the graph data science, in conjunction with ML techniques, helps improve performance and accelerate innovations.
An essential step towards gaining a deeper insight into intricate mechanisms underlying the formation and functioning of complex networks is extracting and understanding their building blocks encoded in the clustering...
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An essential step towards gaining a deeper insight into intricate mechanisms underlying the formation and functioning of complex networks is extracting and understanding their building blocks encoded in the clustering structure. At its core, the problem of partitioning vertices into clusters may be regarded as a dual problem to vertex colouring and, as such, permitted us to leverage the Petford-Welsh colouring algorithm to devise a highly scalable decentralised heuristic approach to cluster detection. As long as the graph under scrutiny admits a fairly well-defined clustering structure per se, the modified Petford-Welsh algorithm tends to perform on a par with or even surpasses existing techniques.
Given a directed graph G and a pair of nodes s and t, an s-t bridge of G is an edge whose removal breaks all s-t paths of G (and thus appears in all s-t paths). Computing all s-t bridges of G is a basic graph problem,...
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Given a directed graph G and a pair of nodes s and t, an s-t bridge of G is an edge whose removal breaks all s-t paths of G (and thus appears in all s-t paths). Computing all s-t bridges of G is a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of "safety" from bioinformatics. We say that a walk W is safe with respect to a set W of s-t walks, if W is a subwalk of all walks in W. We start by considering the maximal safe walks when W consists of: all s-t paths, all s-t trails, or all s-t walks of G. We show that the solutions for the first two problems immediately follow from finding all s-t bridges after incorporating simple characterisations. However, solving the third problem requires non-trivial techniques for incorporating its characterisation. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. Our solutions also directly extend to multigraphs, except for the second problem, which requires a more involved approach. We further generalise these problems, by assuming that safety is defined only with respect to a subset of visible edges. Here we prove a dichotomy between the s-t paths and s-t trails cases, and the s-t walks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations of s-t articulation points (nodes appearing in all s-t paths). We thus obtain the best possible results for natural "safety"-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice.
Despite the long history of genome assembly research, there remains a large gap between the theoretical and practical work. There is practical software with little theoretical underpinning of accuracy on one hand and ...
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A connected feedback vertex set of a graph is a connected subgraph of the graph whose removal makes the graph cycle free. In this paper, we give an approximation algorithm that computes a connected feedback vertex set...
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ISBN:
(纸本)9783031343469;9783031343476
A connected feedback vertex set of a graph is a connected subgraph of the graph whose removal makes the graph cycle free. In this paper, we give an approximation algorithm that computes a connected feedback vertex set of size (1.9091OPT + 6) on 2-connected AT-free graphs with running time O(n(8)m(2)). Also, we give another approximation algorithm that computes a connected feedback vertex set of size (2.9091OPT + 6) on the same graph class with more efficient running time O(min{m(log(n)), n(2)}).
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