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"abstract": "State-of-the-art optical see-through head-mounted displays (OST-HMD) for augmented reality applications lack the ability to correctly render light blocking behavior between digital and physical objects, known as mutual occlusion capability. In this article, we present a novel optical architecture for enabling a high performance, occlusion-capable optical see-through head-mounted display (OCOST-HMD). The design utilizes a single-layer, double-pass architecture, creating a compact OCOST-HMD that is capable of rendering per-pixel mutual occlusion, correctly pupil-matched viewing perspective between virtual and real scenes, and a wide see-through field of view (FOV). Based on this architecture, we present a design embodiment and a compact prototype implementation. The prototype demonstrates a virtual display with an FOV of 34° by 22°, an angular resolution of 1.06 arc minutes per pixel, and an average image contrast greater than 40 percent at the Nyquist frequency of 53 cycles/mm. Furthermore, the device achieves a see-through FOV of 90° by 50°, within which about 40° diagonally is occlusion-enabled, and has an angular resolution of 1.0 arc minutes (comparable to a 20/20 vision) and a dynamic range greater than 100:1. We conclude the paper with a quantitative comparison of the key optical performance such as modulation transfer function, image contrast, and color rendering accuracy of our OCOST-HMD system with and without occlusion enabled for various lighting environments.",
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"normalizedAbstract": "From rate to temporal encoding, spiking information processing has demonstrated advantages across diverse neuromorphic applications. In the aspects of data capacity and robustness, multiplexing encoding outperforms alternative encoding schemes. In this work, we aim to implement a new class of multiplexing temporal encoders, patterning stimuli in multiple timescales to improve the information processing capability, and robustness of systems deployed in noisy environments. Benefitted by the internal reference frame using subthreshold membrane oscillation (SMO), the encoded spike patterns are less sensitive to the input noise, increasing the encoder’s robustness. Our design results in a tremendous saving on power consumption and silicon area compared with the power-hungry analog-to-digital converters. Furthermore, a working prototype of the multiplexing temporal encoder built based on an interspike interval (ISI) encoding scheme is implemented on a silicon chip using the standard 180-nm CMOS process. To the best of our knowledge, our introduced encoder demonstrates the first integrated circuit (IC) implementation of neural encoding with multiplexing topology. Finally, the accuracy and efficiency of our design are evaluated through standard machine learning benchmarks, including Modified National Institute of Standards and Technology (MNIST), Canadian Institute For Advanced Research (CIFAR)-10, Street View House Number (SVHN), and spectrum sensing in high-speed communication networks. While our multiplexing temporal encoder demonstrates a higher classification accuracy across all the benchmarks, the power consumption and dissipated energy per spike reach merely - and 95 fJ/spike, respectively, with an effective frame rate of 300 MHz. Compared with alternative encoding schemes, our multiplexing temporal encoder achieves at most 100% higher data capacity, 11.4% more accurate in classification, and 25% more robust against noise. Compared with the state-of-the-art designs, our work achieves up to - power efficiency without significantly increasing the silicon area.",
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"abstract": "We present an efficient locomotion technique that can reduce cybersickness through aligning the visual and vestibular induced self-motion illusion. Our locomotion technique stimulates proprioception consistent with the visual sense by intentional head motion, which includes both the head's translational movement and yaw rotation. A locomotion event is triggered by the hand-held controller together with an intended physical head motion simultaneously. Based on our method, we further explore the connections between the level of cybersickness and the velocity of self motion through a series of experiments. We first conduct Experiment 1 to investigate the cybersickness induced by different translation velocities using our method and then conduct Experiment 2 to investigate the cybersickness induced by different angular velocities. Our user studies from these two experiments reveal a new finding on the correlation between translation/angular velocities and the level of cybersickness. The cybersickness is greatest at the lowest velocity using our method, and the statistical analysis also indicates a possible U-shaped relation between the translation/angular velocity and cybersickness degree. Finally, we conduct Experiment 3 to evaluate the performances of our method and other commonly-used locomotion approaches, i.e., joystick-based steering and teleportation. The results show that our method can significantly reduce cybersickness compared with the joystick-based steering and obtain a higher presence compared with the teleportation. These advantages demonstrate that our method can be an optional locomotion solution for immersive VR applications using commercially available HMD suites only.",
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"abstract": "We present an interactive wave-based sound propagation system that generates accurate, realistic sound in virtual environments for dynamic (moving) sources and listeners. We propose a novel algorithm to accurately solve the wave equation for dynamic sources and listeners using a combination of precomputation techniques and GPU-based runtime evaluation. Our system can handle large environments typically used in VR applications, compute spatial sound corresponding to listener's motion (including head tracking) and handle both omnidirectional and directional sources, all at interactive rates. As compared to prior wave-based techniques applied to large scenes with moving sources, we observe significant improvement in runtime memory. The overall sound-propagation and rendering system has been integrated with the Half-Life 2 game engine, Oculus-Rift head-mounted display, and the Xbox game controller to enable users to experience high-quality acoustic effects (e.g., amplification, diffraction low-passing, high-order scattering) and spatial audio, based on their interactions in the VR application. We provide the results of preliminary user evaluations, conducted to study the impact of wave-based acoustic effects and spatial audio on users' navigation performance in virtual environments.",
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"abstract": "This paper presents the transformation method of the system of algebraic equations describing the symmetric cipher into the QUBO problem. After transformation of given equations <inline-formula><tex-math notation=\"LaTeX\">Z_$f_0, f_1, \\ldots, f_{n-1}$_Z</tex-math></inline-formula> to equations over integers <inline-formula><tex-math notation=\"LaTeX\">Z_$f^{\\prime }_0, f^{\\prime }_1, \\ldots, f^{\\prime }_{n-1}$_Z</tex-math></inline-formula>, one can linearize each, obtaining <inline-formula><tex-math notation=\"LaTeX\">Z_$f^{\\prime }_{lin_i}=lin(f^{\\prime }_i)$_Z</tex-math></inline-formula>, for <inline-formula><tex-math notation=\"LaTeX\">Z_$i=\\overline{0, n-1}$_Z</tex-math></inline-formula>, where <inline-formula><tex-math notation=\"LaTeX\">Z_$lin$_Z</tex-math></inline-formula> denotes linearization operation. Finally, one can obtain problem in the QUBO form as <inline-formula><tex-math notation=\"LaTeX\">Z_$(f^{\\prime }_{lin_0} )^2+\\cdots +(f^{\\prime }_{lin_{n-1}} )^2+Pen-C$_Z</tex-math></inline-formula>, where <inline-formula><tex-math notation=\"LaTeX\">Z_$Pen$_Z</tex-math></inline-formula> denotes penalties obtained during linearization of equations, <inline-formula><tex-math notation=\"LaTeX\">Z_$n$_Z</tex-math></inline-formula> is the number of equations and <inline-formula><tex-math notation=\"LaTeX\">Z_$C$_Z</tex-math></inline-formula> is constant appearing in the polynomial <inline-formula><tex-math notation=\"LaTeX\">Z_$(f^{\\prime }_{lin_0} )^2+\\cdots +(f^{\\prime }_{lin_{n-1}} )^2+Pen$_Z</tex-math></inline-formula>. This paper presents the transformation method of SPN block ciphers to the QUBO problem. What is more, we present the results of the transformation of the complete AES-128 cipher to the QUBO problem, where the number of variables of the equivalent QUBO problem equals approximately 30,026. It is worth noting that AES-128 is much easier to solve using quantum annealing than the factorization problem and the discrete logarithm problem of a similar level of security. For example, factorizing a 3072 bit long RSA integer using quantum annealing requires a QUBO problem of about 2,360,000 variables.",
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"content": "This paper presents the transformation method of the system of algebraic equations describing the symmetric cipher into the QUBO problem. After transformation of given equations <inline-formula><tex-math notation=\"LaTeX\">$f_0, f_1, \\ldots, f_{n-1}$</tex-math><alternatives><mml:math><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>...</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href=\"wronski-ieq1-3143152.gif\"/></alternatives></inline-formula> to equations over integers <inline-formula><tex-math notation=\"LaTeX\">$f^{\\prime }_0, f^{\\prime }_1, \\ldots, f^{\\prime }_{n-1}$</tex-math><alternatives><mml:math><mml:mrow><mml:msubsup><mml:mi>f</mml:mi><mml:mn>0</mml:mn><mml:mo>'</mml:mo></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mn>1</mml:mn><mml:mo>'</mml:mo></mml:msubsup><mml:mo>,</mml:mo><mml:mo>...</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>'</mml:mo></mml:msubsup></mml:mrow></mml:math><inline-graphic xlink:href=\"wronski-ieq2-3143152.gif\"/></alternatives></inline-formula>, one can linearize each, obtaining <inline-formula><tex-math notation=\"LaTeX\">$f^{\\prime }_{lin_i}=lin(f^{\\prime }_i)$</tex-math><alternatives><mml:math><mml:mrow><mml:msubsup><mml:mi>f</mml:mi><mml:mrow><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>'</mml:mo></mml:msubsup><mml:mo>=</mml:mo><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mi>i</mml:mi><mml:mo>'</mml:mo></mml:msubsup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href=\"wronski-ieq3-3143152.gif\"/></alternatives></inline-formula>, for <inline-formula><tex-math notation=\"LaTeX\">$i=\\overline{0, n-1}$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math><inline-graphic xlink:href=\"wronski-ieq4-3143152.gif\"/></alternatives></inline-formula>, where <inline-formula><tex-math notation=\"LaTeX\">$lin$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"wronski-ieq5-3143152.gif\"/></alternatives></inline-formula> denotes linearization operation. Finally, one can obtain problem in the QUBO form as <inline-formula><tex-math notation=\"LaTeX\">$(f^{\\prime }_{lin_0} )^2+\\cdots +(f^{\\prime }_{lin_{n-1}} )^2+Pen-C$</tex-math><alternatives><mml:math><mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mrow><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>'</mml:mo></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mrow><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>'</mml:mo></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mi>e</mml:mi><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"wronski-ieq6-3143152.gif\"/></alternatives></inline-formula>, where <inline-formula><tex-math notation=\"LaTeX\">$Pen$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>P</mml:mi><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"wronski-ieq7-3143152.gif\"/></alternatives></inline-formula> denotes penalties obtained during linearization of equations, <inline-formula><tex-math notation=\"LaTeX\">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href=\"wronski-ieq8-3143152.gif\"/></alternatives></inline-formula> is the number of equations and <inline-formula><tex-math notation=\"LaTeX\">$C$</tex-math><alternatives><mml:math><mml:mi>C</mml:mi></mml:math><inline-graphic xlink:href=\"wronski-ieq9-3143152.gif\"/></alternatives></inline-formula> is constant appearing in the polynomial <inline-formula><tex-math notation=\"LaTeX\">$(f^{\\prime }_{lin_0} )^2+\\cdots +(f^{\\prime }_{lin_{n-1}} )^2+Pen$</tex-math><alternatives><mml:math><mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mrow><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>'</mml:mo></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mrow><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>'</mml:mo></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"wronski-ieq10-3143152.gif\"/></alternatives></inline-formula>. This paper presents the transformation method of SPN block ciphers to the QUBO problem. What is more, we present the results of the transformation of the complete AES-128 cipher to the QUBO problem, where the number of variables of the equivalent QUBO problem equals approximately 30,026. It is worth noting that AES-128 is much easier to solve using quantum annealing than the factorization problem and the discrete logarithm problem of a similar level of security. For example, factorizing a 3072 bit long RSA integer using quantum annealing requires a QUBO problem of about 2,360,000 variables.",
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"affiliation": "Department of Cybernetics, Military University of Technology, Warsaw, Poland",
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"abstract": "In recent decades, unsupervised feature selection methods have become increasingly popular. Nevertheless, most of the existing unsupervised feature selection methods suffer from two major problems that lead to suboptimal solutions. Many methods impose a hard linear projection constraint on original data, which is overly strict in nature and not suitable for dealing with data sampled from nonlinear manifolds. Second, most existing methods use <inline-formula><tex-math notation=\"LaTeX\">Z_$\\ell _{2,p}$_Z</tex-math></inline-formula>-norm (<inline-formula><tex-math notation=\"LaTeX\">Z_$0< p\\leq 1$_Z</tex-math></inline-formula>) regularization on projection matrix to obtain row sparse matrix and then calculate scores of each feature, which would introduce extra parameter with a slight possibility to directly obtain indexes of discriminative features. To solve the above problems, we propose two novel unsupervised feature selection methods called SF<inline-formula><tex-math notation=\"LaTeX\">Z_$^{2}$_Z</tex-math></inline-formula>S and SF<inline-formula><tex-math notation=\"LaTeX\">Z_$^{2}$_Z</tex-math></inline-formula>SOG, which can simultaneously learn optimal flexible projections and obtain an orthogonal sparse projection to directly select discriminative features by applying <inline-formula><tex-math notation=\"LaTeX\">Z_$\\ell _{2,0}$_Z</tex-math></inline-formula>-norm constraint. Moreover, we propose to explore the local structure of flexible embedding through preserving the manifold structure of original data and adaptively constructing an optimal graph in subspace. Third, the novel iterative optimization algorithms are presented to solve objective functions guaranteeing convergence theoretically. Various evaluation experiments on synthetic and real-world datasets demonstrate the effectiveness and superiority of our proposed methods.",
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"normalizedAbstract": "In recent decades, unsupervised feature selection methods have become increasingly popular. Nevertheless, most of the existing unsupervised feature selection methods suffer from two major problems that lead to suboptimal solutions. Many methods impose a hard linear projection constraint on original data, which is overly strict in nature and not suitable for dealing with data sampled from nonlinear manifolds. Second, most existing methods use --norm (-) regularization on projection matrix to obtain row sparse matrix and then calculate scores of each feature, which would introduce extra parameter with a slight possibility to directly obtain indexes of discriminative features. To solve the above problems, we propose two novel unsupervised feature selection methods called SF-S and SF-SOG, which can simultaneously learn optimal flexible projections and obtain an orthogonal sparse projection to directly select discriminative features by applying --norm constraint. Moreover, we propose to explore the local structure of flexible embedding through preserving the manifold structure of original data and adaptively constructing an optimal graph in subspace. Third, the novel iterative optimization algorithms are presented to solve objective functions guaranteeing convergence theoretically. Various evaluation experiments on synthetic and real-world datasets demonstrate the effectiveness and superiority of our proposed methods.",
"title": "Sparse and Flexible Projections for Unsupervised Feature Selection",
"normalizedTitle": "Sparse and Flexible Projections for Unsupervised Feature Selection",
"fno": "09759987",
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"keywords": [
"Feature Extraction",
"Sparse Matrices",
"Optimization",
"Manifolds",
"Data Models",
"Convergence",
"Optics",
"Feature Selection",
"Unsupervised Learning",
"Flexible Projection",
"<named-content xmlns:xlink=\"http://www.w3.org/1999/xlink\" xmlns:ali=\"http://www.niso.org/schemas/ali/1.0/\" xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:xsi=\"http://www.w3.org/2001/XMLSchema-instance\" content-type=\"math\" xlink:type=\"simple\"> <inline-formula> <tex-math notation=\"LaTeX\">Z_$\\ell _{2,0}$_Z</tex-math> </inline-formula> </named-content>-norm",
"Optimal Graph"
],
"authors": [
{
"givenName": "Rong",
"surname": "Wang",
"fullName": "Rong Wang",
"affiliation": "Key Laboratory of Intelligent Interaction and Applications (Ministry of Industry and Information Technology), School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an, China",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Canyu",
"surname": "Zhang",
"fullName": "Canyu Zhang",
"affiliation": "Key Laboratory of Intelligent Interaction and Applications (Ministry of Industry and Information Technology), School of Computer Science and School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an, China",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Jintang",
"surname": "Bian",
"fullName": "Jintang Bian",
"affiliation": "Key Laboratory of Intelligent Interaction and Applications (Ministry of Industry and Information Technology), School of Computer Science and School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an, China",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Zheng",
"surname": "Wang",
"fullName": "Zheng Wang",
"affiliation": "Key Laboratory of Intelligent Interaction and Applications (Ministry of Industry and Information Technology), School of Computer Science and School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an, China",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Feiping",
"surname": "Nie",
"fullName": "Feiping Nie",
"affiliation": "Key Laboratory of Intelligent Interaction and Applications (Ministry of Industry and Information Technology), School of Computer Science and School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an, China",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Xuelong",
"surname": "Li",
"fullName": "Xuelong Li",
"affiliation": "Key Laboratory of Intelligent Interaction and Applications (Ministry of Industry and Information Technology), School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an, China",
"__typename": "ArticleAuthorType"
}
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"issueNum": "06",
"pubDate": "2023-06-01 00:00:00",
"pubType": "trans",
"pages": "6362-6375",
"year": "2023",
"issn": "1041-4347",
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"issue": {
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"title": "Oct.",
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"doi": "10.1109/TC.2020.3023120",
"abstract": "The attackers in a network may have a tendency of targeting on a group of clustered nodes, and they hope to avoid the existence of significant large communication groups in the remaining network, such as botnet attack, DDoS attack, and Local Area Network Denial attack. Current various kinds of connectivity do not well reflect the fault tolerance of a network under these attacks. This observation inspires a new measure for network reliability to resist the block attack by taking into account of the dispersity of the remaining nodes. Let <inline-formula><tex-math notation=\"LaTeX\">Z_$G$_Z</tex-math></inline-formula> be a network, <inline-formula><tex-math notation=\"LaTeX\">Z_$C\\subset V(G)$_Z</tex-math></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">Z_$G[C]$_Z</tex-math></inline-formula> be a <italic>connected subgraph</italic>. Then <inline-formula><tex-math notation=\"LaTeX\">Z_$C$_Z</tex-math></inline-formula> is called an <italic><inline-formula><tex-math notation=\"LaTeX\">Z_$h$_Z</tex-math></inline-formula>-faulty-block</italic> of <inline-formula><tex-math notation=\"LaTeX\">Z_$G$_Z</tex-math></inline-formula> if <inline-formula><tex-math notation=\"LaTeX\">Z_$G-C$_Z</tex-math></inline-formula> is disconnected, and every component of <inline-formula><tex-math notation=\"LaTeX\">Z_$G-C$_Z</tex-math></inline-formula> has at least <inline-formula><tex-math notation=\"LaTeX\">Z_$h+1$_Z</tex-math></inline-formula> nodes. The minimum cardinality over all <inline-formula><tex-math notation=\"LaTeX\">Z_$h$_Z</tex-math></inline-formula>-faulty-blocks of <inline-formula><tex-math notation=\"LaTeX\">Z_$G$_Z</tex-math></inline-formula> is called <italic><inline-formula><tex-math notation=\"LaTeX\">Z_$h$_Z</tex-math></inline-formula>-faulty-block connectivity</italic> of <inline-formula><tex-math notation=\"LaTeX\">Z_$G$_Z</tex-math></inline-formula>, denoted by <inline-formula><tex-math notation=\"LaTeX\">Z_${FB}\\kappa _h(G)$_Z</tex-math></inline-formula>. In this article, we determine <inline-formula><tex-math notation=\"LaTeX\">Z_${FB}\\kappa _h(Q_n)$_Z</tex-math></inline-formula> for <inline-formula><tex-math notation=\"LaTeX\">Z_$n$_Z</tex-math></inline-formula>-dimensional hypercube <inline-formula><tex-math notation=\"LaTeX\">Z_$Q_n$_Z</tex-math></inline-formula> (<inline-formula><tex-math notation=\"LaTeX\">Z_$n\\geq 4$_Z</tex-math></inline-formula>), a classic interconnection network. We establish that <inline-formula><tex-math notation=\"LaTeX\">Z_${FB}\\kappa _h(Q_n)=(h+2)n-3h-1$_Z</tex-math></inline-formula> for <inline-formula><tex-math notation=\"LaTeX\">Z_$0\\leq h\\leq 1$_Z</tex-math></inline-formula>, and <inline-formula><tex-math notation=\"LaTeX\">Z_${FB}\\kappa _h(Q_n)=(h+2)n-4h+1$_Z</tex-math></inline-formula> for <inline-formula><tex-math notation=\"LaTeX\">Z_$2\\leq h\\leq n-2$_Z</tex-math></inline-formula>, respectively. Larger <inline-formula><tex-math notation=\"LaTeX\">Z_$h$_Z</tex-math></inline-formula>-faulty-block connectivity implies that an attacker will have to stage an attack to a bigger block of connected nodes, so that each remaining components will not be too small, which will in turn limit the size of large components. In other words, there will not be great disparity in sizes between any two remaining components, and hence there will less likely be a significantly large remaining communication group. The larger the <inline-formula><tex-math notation=\"LaTeX\">Z_$h$_Z</tex-math></inline-formula>-faulty-block, the more difficult for an attacker to achieve that goal. As a consequence, the resistance of the network against the attacker will increase. Our experiments also show that as <inline-formula><tex-math notation=\"LaTeX\">Z_$h$_Z</tex-math></inline-formula> increases, the <inline-formula><tex-math notation=\"LaTeX\">Z_$h$_Z</tex-math></inline-formula>-faulty-block gets larger, and the size disparity between any two remaining components decreases. In turn, as expected, the size of the largest remaining communication group becomes smaller.",
"abstracts": [
{
"abstractType": "Regular",
"content": "The attackers in a network may have a tendency of targeting on a group of clustered nodes, and they hope to avoid the existence of significant large communication groups in the remaining network, such as botnet attack, DDoS attack, and Local Area Network Denial attack. Current various kinds of connectivity do not well reflect the fault tolerance of a network under these attacks. This observation inspires a new measure for network reliability to resist the block attack by taking into account of the dispersity of the remaining nodes. Let <inline-formula><tex-math notation=\"LaTeX\">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq1-3023120.gif\"/></alternatives></inline-formula> be a network, <inline-formula><tex-math notation=\"LaTeX\">$C\\subset V(G)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>C</mml:mi><mml:mo>⊂</mml:mo><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq2-3023120.gif\"/></alternatives></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">$G[C]$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>G</mml:mi><mml:mo>[</mml:mo><mml:mi>C</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq3-3023120.gif\"/></alternatives></inline-formula> be a <italic>connected subgraph</italic>. Then <inline-formula><tex-math notation=\"LaTeX\">$C$</tex-math><alternatives><mml:math><mml:mi>C</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq4-3023120.gif\"/></alternatives></inline-formula> is called an <italic><inline-formula><tex-math notation=\"LaTeX\">$h$</tex-math><alternatives><mml:math><mml:mi>h</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq5-3023120.gif\"/></alternatives></inline-formula>-faulty-block</italic> of <inline-formula><tex-math notation=\"LaTeX\">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq6-3023120.gif\"/></alternatives></inline-formula> if <inline-formula><tex-math notation=\"LaTeX\">$G-C$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>G</mml:mi><mml:mo>-</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq7-3023120.gif\"/></alternatives></inline-formula> is disconnected, and every component of <inline-formula><tex-math notation=\"LaTeX\">$G-C$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>G</mml:mi><mml:mo>-</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq8-3023120.gif\"/></alternatives></inline-formula> has at least <inline-formula><tex-math notation=\"LaTeX\">$h+1$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>h</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq9-3023120.gif\"/></alternatives></inline-formula> nodes. The minimum cardinality over all <inline-formula><tex-math notation=\"LaTeX\">$h$</tex-math><alternatives><mml:math><mml:mi>h</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq10-3023120.gif\"/></alternatives></inline-formula>-faulty-blocks of <inline-formula><tex-math notation=\"LaTeX\">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq11-3023120.gif\"/></alternatives></inline-formula> is called <italic><inline-formula><tex-math notation=\"LaTeX\">$h$</tex-math><alternatives><mml:math><mml:mi>h</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq12-3023120.gif\"/></alternatives></inline-formula>-faulty-block connectivity</italic> of <inline-formula><tex-math notation=\"LaTeX\">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq13-3023120.gif\"/></alternatives></inline-formula>, denoted by <inline-formula><tex-math notation=\"LaTeX\">${FB}\\kappa _h(G)$</tex-math><alternatives><mml:math><mml:mrow><mml:mrow><mml:mi>F</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq14-3023120.gif\"/></alternatives></inline-formula>. In this article, we determine <inline-formula><tex-math notation=\"LaTeX\">${FB}\\kappa _h(Q_n)$</tex-math><alternatives><mml:math><mml:mrow><mml:mrow><mml:mi>F</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq15-3023120.gif\"/></alternatives></inline-formula> for <inline-formula><tex-math notation=\"LaTeX\">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq16-3023120.gif\"/></alternatives></inline-formula>-dimensional hypercube <inline-formula><tex-math notation=\"LaTeX\">$Q_n$</tex-math><alternatives><mml:math><mml:msub><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math><inline-graphic xlink:href=\"lin-ieq17-3023120.gif\"/></alternatives></inline-formula> (<inline-formula><tex-math notation=\"LaTeX\">$n\\geq 4$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq18-3023120.gif\"/></alternatives></inline-formula>), a classic interconnection network. We establish that <inline-formula><tex-math notation=\"LaTeX\">${FB}\\kappa _h(Q_n)=(h+2)n-3h-1$</tex-math><alternatives><mml:math><mml:mrow><mml:mrow><mml:mi>F</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>h</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mi>h</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq19-3023120.gif\"/></alternatives></inline-formula> for <inline-formula><tex-math notation=\"LaTeX\">$0\\leq h\\leq 1$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>h</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq20-3023120.gif\"/></alternatives></inline-formula>, and <inline-formula><tex-math notation=\"LaTeX\">${FB}\\kappa _h(Q_n)=(h+2)n-4h+1$</tex-math><alternatives><mml:math><mml:mrow><mml:mrow><mml:mi>F</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>h</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mi>h</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq21-3023120.gif\"/></alternatives></inline-formula> for <inline-formula><tex-math notation=\"LaTeX\">$2\\leq h\\leq n-2$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>h</mml:mi><mml:mo>≤</mml:mo><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"lin-ieq22-3023120.gif\"/></alternatives></inline-formula>, respectively. Larger <inline-formula><tex-math notation=\"LaTeX\">$h$</tex-math><alternatives><mml:math><mml:mi>h</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq23-3023120.gif\"/></alternatives></inline-formula>-faulty-block connectivity implies that an attacker will have to stage an attack to a bigger block of connected nodes, so that each remaining components will not be too small, which will in turn limit the size of large components. In other words, there will not be great disparity in sizes between any two remaining components, and hence there will less likely be a significantly large remaining communication group. The larger the <inline-formula><tex-math notation=\"LaTeX\">$h$</tex-math><alternatives><mml:math><mml:mi>h</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq24-3023120.gif\"/></alternatives></inline-formula>-faulty-block, the more difficult for an attacker to achieve that goal. As a consequence, the resistance of the network against the attacker will increase. Our experiments also show that as <inline-formula><tex-math notation=\"LaTeX\">$h$</tex-math><alternatives><mml:math><mml:mi>h</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq25-3023120.gif\"/></alternatives></inline-formula> increases, the <inline-formula><tex-math notation=\"LaTeX\">$h$</tex-math><alternatives><mml:math><mml:mi>h</mml:mi></mml:math><inline-graphic xlink:href=\"lin-ieq26-3023120.gif\"/></alternatives></inline-formula>-faulty-block gets larger, and the size disparity between any two remaining components decreases. In turn, as expected, the size of the largest remaining communication group becomes smaller.",
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"normalizedAbstract": "The attackers in a network may have a tendency of targeting on a group of clustered nodes, and they hope to avoid the existence of significant large communication groups in the remaining network, such as botnet attack, DDoS attack, and Local Area Network Denial attack. Current various kinds of connectivity do not well reflect the fault tolerance of a network under these attacks. This observation inspires a new measure for network reliability to resist the block attack by taking into account of the dispersity of the remaining nodes. Let - be a network, - and - be a connected subgraph. Then - is called an --faulty-block of - if - is disconnected, and every component of - has at least - nodes. The minimum cardinality over all --faulty-blocks of - is called --faulty-block connectivity of -, denoted by -. In this article, we determine - for --dimensional hypercube - (-), a classic interconnection network. We establish that - for -, and - for -, respectively. Larger --faulty-block connectivity implies that an attacker will have to stage an attack to a bigger block of connected nodes, so that each remaining components will not be too small, which will in turn limit the size of large components. In other words, there will not be great disparity in sizes between any two remaining components, and hence there will less likely be a significantly large remaining communication group. The larger the --faulty-block, the more difficult for an attacker to achieve that goal. As a consequence, the resistance of the network against the attacker will increase. Our experiments also show that as - increases, the --faulty-block gets larger, and the size disparity between any two remaining components decreases. In turn, as expected, the size of the largest remaining communication group becomes smaller.",
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"surname": "Huang",
"fullName": "Yanze Huang",
"affiliation": "School of Mathematics and Physics, Fujian University of Technology, Fuzhou, Fujian, P.R. China",
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"surname": "Wang",
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"affiliation": "Department of Computer Science, Montclair State University, Montclair, NJ, USA",
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"fullName": "Sun-Yuan Hsieh",
"affiliation": "Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan, Taiwan",
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"givenName": "Li",
"surname": "Xu",
"fullName": "Li Xu",
"affiliation": "College of Mathematics and Informatics, Key Laboratory of Network Security and Cryptology, Center for Applied Mathematics of Fujian Province, Fujian Normal University, Fuzhou, Fujian, P.R. China",
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"abstract": "In the context of wireless sensor networks (WSNs), the problem of virtual backbones (VBs) for undertaking routing tasks to alleviate broadcast storms has been extensively studied. In practical applications, different nodes in a WSN may have different transmission ranges because of differences in power control or functionality, among other reasons. In such a situation, a disk graph (DG) can be used as a mathematical model of the WSN, and a strongly connected dominating and absorbent set (SCDAS) in the DG can be treated as a VB in the corresponding WSN. In a WSN with faulty nodes, a fault-tolerant VB is superior to a traditional one. Thus, it is desirable to construct a <inline-formula><tex-math notation=\"LaTeX\">Z_$ k$_Z</tex-math></inline-formula>-strongly connected <inline-formula><tex-math notation=\"LaTeX\">Z_$ m$_Z</tex-math></inline-formula>-dominating and absorbent set (<inline-formula><tex-math notation=\"LaTeX\">Z_$ (k,m)$_Z</tex-math></inline-formula>-SCDAS) in the DG to serve as a fault-tolerant VB in the corresponding WSN. In this article, to enable the construction of a high-quality <inline-formula><tex-math notation=\"LaTeX\">Z_$ (k,m)$_Z</tex-math></inline-formula>-SCDAS in a DG, a constant approximation algorithm with a performance ratio of <inline-formula><tex-math notation=\"LaTeX\">Z_$(2k(5^{k-1}-1)+1)(R+m+4(\\frac{R}{m}+1))$_Z</tex-math></inline-formula> is proposed, where <inline-formula><tex-math notation=\"LaTeX\">Z_$ k$_Z</tex-math></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">Z_$ m$_Z</tex-math></inline-formula> are constants (<inline-formula><tex-math notation=\"LaTeX\">Z_$ 2 \\;\\leqslant\\; k \\;\\leqslant\\; m$_Z</tex-math></inline-formula>) and <inline-formula><tex-math notation=\"LaTeX\">Z_$ R$_Z</tex-math></inline-formula> is the maximum number of independent nodes within the transmission range of a node in the DG. A theoretical analysis and simulation results show that our work is superior to previous approaches.",
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"content": "In the context of wireless sensor networks (WSNs), the problem of virtual backbones (VBs) for undertaking routing tasks to alleviate broadcast storms has been extensively studied. In practical applications, different nodes in a WSN may have different transmission ranges because of differences in power control or functionality, among other reasons. In such a situation, a disk graph (DG) can be used as a mathematical model of the WSN, and a strongly connected dominating and absorbent set (SCDAS) in the DG can be treated as a VB in the corresponding WSN. In a WSN with faulty nodes, a fault-tolerant VB is superior to a traditional one. Thus, it is desirable to construct a <inline-formula><tex-math notation=\"LaTeX\">$ k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href=\"zhang-ieq1-3048960.gif\"/></alternatives></inline-formula>-strongly connected <inline-formula><tex-math notation=\"LaTeX\">$ m$</tex-math><alternatives><mml:math><mml:mi>m</mml:mi></mml:math><inline-graphic xlink:href=\"zhang-ieq2-3048960.gif\"/></alternatives></inline-formula>-dominating and absorbent set (<inline-formula><tex-math notation=\"LaTeX\">$ (k,m)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"zhang-ieq3-3048960.gif\"/></alternatives></inline-formula>-SCDAS) in the DG to serve as a fault-tolerant VB in the corresponding WSN. In this article, to enable the construction of a high-quality <inline-formula><tex-math notation=\"LaTeX\">$ (k,m)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"zhang-ieq4-3048960.gif\"/></alternatives></inline-formula>-SCDAS in a DG, a constant approximation algorithm with a performance ratio of <inline-formula><tex-math notation=\"LaTeX\">$(2k(5^{k-1}-1)+1)(R+m+4(\\frac{R}{m}+1))$</tex-math><alternatives><mml:math><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mn>5</mml:mn><mml:mrow><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>R</mml:mi><mml:mo>+</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>R</mml:mi><mml:mi>m</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href=\"zhang-ieq5-3048960.gif\"/></alternatives></inline-formula> is proposed, where <inline-formula><tex-math notation=\"LaTeX\">$ k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href=\"zhang-ieq6-3048960.gif\"/></alternatives></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">$ m$</tex-math><alternatives><mml:math><mml:mi>m</mml:mi></mml:math><inline-graphic xlink:href=\"zhang-ieq7-3048960.gif\"/></alternatives></inline-formula> are constants (<inline-formula><tex-math notation=\"LaTeX\">$ 2 \\;\\leqslant\\; k \\;\\leqslant\\; m$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>2</mml:mn><mml:mspace width=\"0.166667em\"/><mml:mi>⩽</mml:mi><mml:mspace width=\"0.166667em\"/><mml:mi>k</mml:mi><mml:mspace width=\"0.166667em\"/><mml:mi>⩽</mml:mi><mml:mspace width=\"0.166667em\"/><mml:mi>m</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"zhang-ieq8-3048960.gif\"/></alternatives></inline-formula>) and <inline-formula><tex-math notation=\"LaTeX\">$ R$</tex-math><alternatives><mml:math><mml:mi>R</mml:mi></mml:math><inline-graphic xlink:href=\"zhang-ieq9-3048960.gif\"/></alternatives></inline-formula> is the maximum number of independent nodes within the transmission range of a node in the DG. A theoretical analysis and simulation results show that our work is superior to previous approaches.",
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"title": "On the Computation of Virtual Backbones With Fault Tolerance in Heterogeneous Wireless Sensor Networks",
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"Wireless Sensor Network",
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{
"givenName": "Weiguang",
"surname": "Zhang",
"fullName": "Weiguang Zhang",
"affiliation": "School of Computer, Electronics and Information and Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Nanning, P. R. China",
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"givenName": "Jiarong",
"surname": "Liang",
"fullName": "Jiarong Liang",
"affiliation": "School of Computer, Electronics and Information and Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Nanning, P. R. China",
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{
"givenName": "Xinyu",
"surname": "Liang",
"fullName": "Xinyu Liang",
"affiliation": "School of Electrical Engineering, Guangxi University, Nanning, China",
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"abstract": "The Chinese remainder theorem (CRT)-based multiplier is a new type of hybrid bit-parallel multiplier, which can achieve nearly the same time complexity compared with the fastest multiplier known to date with reduced space complexity. However, the current CRT-based multipliers are only applicable to trinomials. In this article, we propose an efficient CRT-based bit-parallel multiplier for a special type of pentanomial <inline-formula><tex-math notation=\"LaTeX\">Z_$x^m+x^{m-k}+x^{m-2k}+x^{m-3k}+1, 5k+1<m\\leq 11k$_Z</tex-math></inline-formula>. Through transforming the non-constant part <inline-formula><tex-math notation=\"LaTeX\">Z_$x^m+x^{m-k}+x^{m-2k}+x^{m-3k}$_Z</tex-math></inline-formula> into a binomial, we can obtain relatively simpler quotient and remainder computations, which lead to faster implementation with reduced space complexity compared with classic quadratic multipliers for the same pentanomials. Moreover, for some <inline-formula><tex-math notation=\"LaTeX\">Z_$m$_Z</tex-math></inline-formula>, our proposal can match the fastest multipliers for irreducible Type I, Type II, and Type C.1 pentanomials of the same degree, but space complexities are roughly reduced by 8 percent.",
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"content": "The Chinese remainder theorem (CRT)-based multiplier is a new type of hybrid bit-parallel multiplier, which can achieve nearly the same time complexity compared with the fastest multiplier known to date with reduced space complexity. However, the current CRT-based multipliers are only applicable to trinomials. In this article, we propose an efficient CRT-based bit-parallel multiplier for a special type of pentanomial <inline-formula><tex-math notation=\"LaTeX\">$x^m+x^{m-k}+x^{m-2k}+x^{m-3k}+1, 5k+1<m\\leq 11k$</tex-math><alternatives><mml:math><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mi>m</mml:mi><mml:mo>≤</mml:mo><mml:mn>11</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"li-ieq1-3058346.gif\"/></alternatives></inline-formula>. Through transforming the non-constant part <inline-formula><tex-math notation=\"LaTeX\">$x^m+x^{m-k}+x^{m-2k}+x^{m-3k}$</tex-math><alternatives><mml:math><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href=\"li-ieq2-3058346.gif\"/></alternatives></inline-formula> into a binomial, we can obtain relatively simpler quotient and remainder computations, which lead to faster implementation with reduced space complexity compared with classic quadratic multipliers for the same pentanomials. Moreover, for some <inline-formula><tex-math notation=\"LaTeX\">$m$</tex-math><alternatives><mml:math><mml:mi>m</mml:mi></mml:math><inline-graphic xlink:href=\"li-ieq3-3058346.gif\"/></alternatives></inline-formula>, our proposal can match the fastest multipliers for irreducible Type I, Type II, and Type C.1 pentanomials of the same degree, but space complexities are roughly reduced by 8 percent.",
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"title": "An Efficient CRT-Based Bit-Parallel Multiplier for Special Pentanomials",
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"abstract": "Despite the success of stochastic variance-reduced gradient (SVRG) algorithms in solving large-scale problems, their stochastic gradient complexity often scales linearly with data size and is expensive for huge data. Accordingly, we propose a hybrid stochastic-deterministic minibatch proximal gradient (<sc>HSDMPG</sc>) algorithm for strongly convex problems with linear prediction structure, e.g., least squares and logistic/softmax regression. <sc>HSDMPG</sc> enjoys improved computational complexity that is data-size-independent for large-scale problems. It iteratively samples an evolving minibatch of individual losses to estimate the original problem, and can efficiently minimize the sampled subproblems. For strongly convex loss of <inline-formula><tex-math notation=\"LaTeX\">Z_$n$_Z</tex-math></inline-formula> components, <sc>HSDMPG</sc> attains an <inline-formula><tex-math notation=\"LaTeX\">Z_$\\epsilon$_Z</tex-math></inline-formula>-optimization-error within <inline-formula><tex-math notation=\"LaTeX\">Z_$\\mathcal {O} \\left(\\kappa \\log ^{\\zeta +1}\\left(\\frac{1}{\\epsilon }\\right)\\frac{1}{\\epsilon }\\bigwedge n\\log ^{\\zeta }\\left(\\frac{1}{\\epsilon }\\right)\\right)$_Z</tex-math></inline-formula> stochastic gradient evaluations, where <inline-formula><tex-math notation=\"LaTeX\">Z_$\\kappa$_Z</tex-math></inline-formula> is condition number, <inline-formula><tex-math notation=\"LaTeX\">Z_$\\zeta =1$_Z</tex-math></inline-formula> for quadratic loss and <inline-formula><tex-math notation=\"LaTeX\">Z_$\\zeta =2$_Z</tex-math></inline-formula> for generic loss. For large-scale problems, our complexity outperforms those of SVRG-type algorithms with/without dependence on data size. Particularly, when <inline-formula><tex-math notation=\"LaTeX\">Z_$\\epsilon =\\mathcal {O}(1/\\sqrt{n})$_Z</tex-math></inline-formula> which matches the intrinsic excess error of a learning model and is sufficient for generalization, our complexity for quadratic and generic losses is respectively <inline-formula><tex-math notation=\"LaTeX\">Z_$\\mathcal {O} (n^{0.5}\\log ^{2}(n))$_Z</tex-math></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">Z_$\\mathcal {O} (n^{0.5}\\log ^{3}(n))$_Z</tex-math></inline-formula>, which for the first time achieves optimal generalization in less than a single pass over data. Besides, we extend <sc>HSDMPG</sc> to online strongly convex problems and prove its higher efficiency over the prior algorithms. Numerical results demonstrate the computational advantages of <sc>HSDMPG</sc>.",
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For large-scale problems, our complexity outperforms those of SVRG-type algorithms with/without dependence on data size. Particularly, when <inline-formula><tex-math notation=\"LaTeX\">$\\epsilon =\\mathcal {O}(1/\\sqrt{n})$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>ε</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"script\">O</mml:mi><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"zhou-ieq7-3087328.gif\"/></alternatives></inline-formula> which matches the intrinsic excess error of a learning model and is sufficient for generalization, our complexity for quadratic and generic losses is respectively <inline-formula><tex-math notation=\"LaTeX\">$\\mathcal {O} (n^{0.5}\\log ^{2}(n))$</tex-math><alternatives><mml:math><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mo form=\"prefix\">log</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"zhou-ieq8-3087328.gif\"/></alternatives></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">$\\mathcal {O} (n^{0.5}\\log ^{3}(n))$</tex-math><alternatives><mml:math><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mo form=\"prefix\">log</mml:mo><mml:mn>3</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"zhou-ieq9-3087328.gif\"/></alternatives></inline-formula>, which for the first time achieves optimal generalization in less than a single pass over data. Besides, we extend <sc>HSDMPG</sc> to online strongly convex problems and prove its higher efficiency over the prior algorithms. Numerical results demonstrate the computational advantages of <sc>HSDMPG</sc>.",
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"abstract": "Geodesics measure the shortest distance (either locally or globally) between two points on a curved surface and serve as a fundamental tool in digital geometry processing. Suppose that we have a parameterized path <inline-formula><tex-math notation=\"LaTeX\">Z_$\\gamma (t)=\\mathbf {x}(u(t),v(t))$_Z</tex-math></inline-formula> on a surface <inline-formula><tex-math notation=\"LaTeX\">Z_$\\mathbf {x}=\\mathbf {x}(u,v)$_Z</tex-math></inline-formula> with <inline-formula><tex-math notation=\"LaTeX\">Z_$\\gamma (0)=p$_Z</tex-math></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">Z_$\\gamma (1)=q$_Z</tex-math></inline-formula>. We formulate the two-point geodesic problem into a minimization problem <inline-formula><tex-math notation=\"LaTeX\">Z_$\\int _0^1 H(\\Vert \\mathbf {x}_uu^{\\prime }(t)+\\mathbf {x}_vv^{\\prime }(t)\\Vert)\\text{d}t$_Z</tex-math></inline-formula>, where <inline-formula><tex-math notation=\"LaTeX\">Z_$H(s)$_Z</tex-math></inline-formula> satisfies <inline-formula><tex-math notation=\"LaTeX\">Z_$H(0)=0,H^{\\prime }(s)>0$_Z</tex-math></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">Z_$H^{\\prime \\prime }(s)\\geq 0$_Z</tex-math></inline-formula> for <inline-formula><tex-math notation=\"LaTeX\">Z_$s>0$_Z</tex-math></inline-formula>. In our implementation, we choose <inline-formula><tex-math notation=\"LaTeX\">Z_$H(s)=e^{s^2}-1$_Z</tex-math></inline-formula> and show that it has several unique advantages over other choices such as <inline-formula><tex-math notation=\"LaTeX\">Z_$H(s)=s^2$_Z</tex-math></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">Z_$H(s)=s$_Z</tex-math></inline-formula>. It is also a minimizer of the traditional geodesic length variational and able to guarantee the uniqueness and regularity in terms of curve parameterization. In the discrete setting, we construct the initial path by a sequence of moveable points <inline-formula><tex-math notation=\"LaTeX\">Z_$\\lbrace x_i\\rbrace _{i=1}^n$_Z</tex-math></inline-formula> and minimize <inline-formula><tex-math notation=\"LaTeX\">Z_$\\sum _{i=1}^{n} H(\\Vert x_i - x_{i+1}\\Vert)$_Z</tex-math></inline-formula>. The resulting points are evenly spaced along the path. It’s obvious that our algorithm can deal with parametric surfaces. Considering that meshes, point clouds and implicit surfaces can be transformed into a signed distance function (SDF), we also discuss its implementation on a general SDF. Finally, we show that our method can be extended to solve a general least-cost path problem. We validate the proposed algorithm in terms of accuracy, performance and scalability, and demonstrate the advantages by extensive comparisons.",
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"content": "Geodesics measure the shortest distance (either locally or globally) between two points on a curved surface and serve as a fundamental tool in digital geometry processing. Suppose that we have a parameterized path <inline-formula><tex-math notation=\"LaTeX\">$\\gamma (t)=\\mathbf {x}(u(t),v(t))$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>γ</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold\">x</mml:mi><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq1-3135021.gif\"/></alternatives></inline-formula> on a surface <inline-formula><tex-math notation=\"LaTeX\">$\\mathbf {x}=\\mathbf {x}(u,v)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi mathvariant=\"bold\">x</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold\">x</mml:mi><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq2-3135021.gif\"/></alternatives></inline-formula> with <inline-formula><tex-math notation=\"LaTeX\">$\\gamma (0)=p$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>γ</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq3-3135021.gif\"/></alternatives></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">$\\gamma (1)=q$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>γ</mml:mi><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq4-3135021.gif\"/></alternatives></inline-formula>. We formulate the two-point geodesic problem into a minimization problem <inline-formula><tex-math notation=\"LaTeX\">$\\int _0^1 H(\\Vert \\mathbf {x}_uu^{\\prime }(t)+\\mathbf {x}_vv^{\\prime }(t)\\Vert)\\text{d}t$</tex-math><alternatives><mml:math><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mo>∥</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">x</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:msup><mml:mi>u</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">x</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msup><mml:mi>v</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>∥</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mtext>d</mml:mtext><mml:mi>t</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq5-3135021.gif\"/></alternatives></inline-formula>, where <inline-formula><tex-math notation=\"LaTeX\">$H(s)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq6-3135021.gif\"/></alternatives></inline-formula> satisfies <inline-formula><tex-math notation=\"LaTeX\">$H(0)=0,H^{\\prime }(s)>0$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>'</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq7-3135021.gif\"/></alternatives></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">$H^{\\prime \\prime }(s)\\geq 0$</tex-math><alternatives><mml:math><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mo>'</mml:mo><mml:mo>'</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq8-3135021.gif\"/></alternatives></inline-formula> for <inline-formula><tex-math notation=\"LaTeX\">$s>0$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>s</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq9-3135021.gif\"/></alternatives></inline-formula>. In our implementation, we choose <inline-formula><tex-math notation=\"LaTeX\">$H(s)=e^{s^2}-1$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:msup><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq10-3135021.gif\"/></alternatives></inline-formula> and show that it has several unique advantages over other choices such as <inline-formula><tex-math notation=\"LaTeX\">$H(s)=s^2$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq11-3135021.gif\"/></alternatives></inline-formula> and <inline-formula><tex-math notation=\"LaTeX\">$H(s)=s$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq12-3135021.gif\"/></alternatives></inline-formula>. It is also a minimizer of the traditional geodesic length variational and able to guarantee the uniqueness and regularity in terms of curve parameterization. In the discrete setting, we construct the initial path by a sequence of moveable points <inline-formula><tex-math notation=\"LaTeX\">$\\lbrace x_i\\rbrace _{i=1}^n$</tex-math><alternatives><mml:math><mml:msubsup><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>}</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup></mml:math><inline-graphic xlink:href=\"xin-ieq13-3135021.gif\"/></alternatives></inline-formula> and minimize <inline-formula><tex-math notation=\"LaTeX\">$\\sum _{i=1}^{n} H(\\Vert x_i - x_{i+1}\\Vert)$</tex-math><alternatives><mml:math><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mo>∥</mml:mo></mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>∥</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href=\"xin-ieq14-3135021.gif\"/></alternatives></inline-formula>. The resulting points are evenly spaced along the path. It’s obvious that our algorithm can deal with parametric surfaces. Considering that meshes, point clouds and implicit surfaces can be transformed into a signed distance function (SDF), we also discuss its implementation on a general SDF. Finally, we show that our method can be extended to solve a general least-cost path problem. We validate the proposed algorithm in terms of accuracy, performance and scalability, and demonstrate the advantages by extensive comparisons.",
"__typename": "ArticleAbstractType"
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"normalizedAbstract": "Geodesics measure the shortest distance (either locally or globally) between two points on a curved surface and serve as a fundamental tool in digital geometry processing. Suppose that we have a parameterized path - on a surface - with - and -. We formulate the two-point geodesic problem into a minimization problem -, where - satisfies - and - for -. In our implementation, we choose - and show that it has several unique advantages over other choices such as - and -. It is also a minimizer of the traditional geodesic length variational and able to guarantee the uniqueness and regularity in terms of curve parameterization. In the discrete setting, we construct the initial path by a sequence of moveable points - and minimize -. The resulting points are evenly spaced along the path. It’s obvious that our algorithm can deal with parametric surfaces. Considering that meshes, point clouds and implicit surfaces can be transformed into a signed distance function (SDF), we also discuss its implementation on a general SDF. Finally, we show that our method can be extended to solve a general least-cost path problem. We validate the proposed algorithm in terms of accuracy, performance and scalability, and demonstrate the advantages by extensive comparisons.",
"title": "A Variational Framework for Curve Shortening in Various Geometric Domains",
"normalizedTitle": "A Variational Framework for Curve Shortening in Various Geometric Domains",
"fno": "09650532",
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"keywords": [
"Computational Complexity",
"Computational Geometry",
"Differential Geometry",
"Geometry",
"Minimisation",
"E Lt Sup",
"Parameterized Path Lt I Gt X 03 B 3",
"Two Point Geodesic Problem",
"Heating Systems",
"Three Dimensional Displays",
"Point Cloud Compression",
"Shortest Path Problem",
"Costs",
"Approximation Algorithms",
"Surface Waves",
"Variational Method",
"Shortest Paths",
"Geodesics",
"Least Cost Paths"
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"authors": [
{
"givenName": "Na",
"surname": "Yuan",
"fullName": "Na Yuan",
"affiliation": "School of Computer Science and Technology, Shandong University, Ji Nan Shi, Shan Dong Sheng, China",
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{
"givenName": "Peihui",
"surname": "Wang",
"fullName": "Peihui Wang",
"affiliation": "ArcSoft, Hangzhou, Zhejiang, China",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Wenlong",
"surname": "Meng",
"fullName": "Wenlong Meng",
"affiliation": "School of Computer Science and Technology, Shandong University, Ji Nan Shi, Shan Dong Sheng, China",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Shuangmin",
"surname": "Chen",
"fullName": "Shuangmin Chen",
"affiliation": "School of Information and Technology, Qingdao University of Science and Technology, Qingdao, Shandong, China",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Jian",
"surname": "Xu",
"fullName": "Jian Xu",
"affiliation": "Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Beijing, China",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Shiqing",
"surname": "Xin",
"fullName": "Shiqing Xin",
"affiliation": "School of Computer Science and Technology, Shandong University, Ji Nan Shi, Shan Dong Sheng, China",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Ying",
"surname": "He",
"fullName": "Ying He",
"affiliation": "School of Computer Science and Engineering, Nanyang Technological University, Singapore",
"__typename": "ArticleAuthorType"
},
{
"givenName": "Wenping",
"surname": "Wang",
"fullName": "Wenping Wang",
"affiliation": "Department of Computer Science, The University of Hong Kong, Hong Kong",
"__typename": "ArticleAuthorType"
}
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"issueNum": "04",
"pubDate": "2023-04-01 00:00:00",
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"pages": "1951-1963",
"year": "2023",
"issn": "1077-2626",
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"abstract": "Federated learning (FL) provides a high efficient decentralized machine learning framework, where the training data remains distributed at remote clients in a network. Though FL enables a privacy-preserving mobile edge computing framework using IoT devices, recent studies have shown that this approach is susceptible to poisoning attacks from the side of remote clients. To address the poisoning attacks on FL, we provide a <italic>two-phase</italic> defense algorithm called <inline-formula><tex-math notation=\"LaTeX\">Z_${\\underline{Lo}cal\\ \\underline{Ma}licious\\ Facto\\underline{r}}$_Z</tex-math></inline-formula> (LoMar). In phase I, LoMar scores model updates from each remote client by measuring the relative distribution over their neighbors using a kernel density estimation method. In phase II, an optimal threshold is approximated to distinguish malicious and clean updates from a statistical perspective. Comprehensive experiments on four real-world datasets have been conducted, and the experimental results show that our defense strategy can effectively protect the FL system. Specifically, the defense performance on Amazon dataset under a label-flipping attack indicates that, compared with FG+Krum, LoMar increases the target label testing accuracy from <inline-formula><tex-math notation=\"LaTeX\">Z_$96.0\\%$_Z</tex-math></inline-formula> to <inline-formula><tex-math notation=\"LaTeX\">Z_$98.8\\%$_Z</tex-math></inline-formula>, and the overall averaged testing accuracy from <inline-formula><tex-math notation=\"LaTeX\">Z_$90.1\\%$_Z</tex-math></inline-formula> to <inline-formula><tex-math notation=\"LaTeX\">Z_$97.0\\%$_Z</tex-math></inline-formula>.",
"abstracts": [
{
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"content": "Federated learning (FL) provides a high efficient decentralized machine learning framework, where the training data remains distributed at remote clients in a network. Though FL enables a privacy-preserving mobile edge computing framework using IoT devices, recent studies have shown that this approach is susceptible to poisoning attacks from the side of remote clients. To address the poisoning attacks on FL, we provide a <italic>two-phase</italic> defense algorithm called <inline-formula><tex-math notation=\"LaTeX\">${\\underline{Lo}cal\\ \\underline{Ma}licious\\ Facto\\underline{r}}$</tex-math><alternatives><mml:math><mml:mrow><mml:munder><mml:mrow><mml:mi>L</mml:mi><mml:mi>o</mml:mi></mml:mrow><mml:mo>̲</mml:mo></mml:munder><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>l</mml:mi><mml:mspace width=\"4pt\"/><mml:munder><mml:mrow><mml:mi>M</mml:mi><mml:mi>a</mml:mi></mml:mrow><mml:mo>̲</mml:mo></mml:munder><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mi>c</mml:mi><mml:mi>i</mml:mi><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>s</mml:mi><mml:mspace width=\"4pt\"/><mml:mi>F</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:munder><mml:mi>r</mml:mi><mml:mo>̲</mml:mo></mml:munder></mml:mrow></mml:math><inline-graphic xlink:href=\"tang-ieq1-3135422.gif\"/></alternatives></inline-formula> (LoMar). In phase I, LoMar scores model updates from each remote client by measuring the relative distribution over their neighbors using a kernel density estimation method. In phase II, an optimal threshold is approximated to distinguish malicious and clean updates from a statistical perspective. Comprehensive experiments on four real-world datasets have been conducted, and the experimental results show that our defense strategy can effectively protect the FL system. Specifically, the defense performance on Amazon dataset under a label-flipping attack indicates that, compared with FG+Krum, LoMar increases the target label testing accuracy from <inline-formula><tex-math notation=\"LaTeX\">$96.0\\%$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>96</mml:mn><mml:mo>.</mml:mo><mml:mn>0</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"tang-ieq2-3135422.gif\"/></alternatives></inline-formula> to <inline-formula><tex-math notation=\"LaTeX\">$98.8\\%$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>98</mml:mn><mml:mo>.</mml:mo><mml:mn>8</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"tang-ieq3-3135422.gif\"/></alternatives></inline-formula>, and the overall averaged testing accuracy from <inline-formula><tex-math notation=\"LaTeX\">$90.1\\%$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>90</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"tang-ieq4-3135422.gif\"/></alternatives></inline-formula> to <inline-formula><tex-math notation=\"LaTeX\">$97.0\\%$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>97</mml:mn><mml:mo>.</mml:mo><mml:mn>0</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"tang-ieq5-3135422.gif\"/></alternatives></inline-formula>.",
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"abstract": "This paper presents a simple, three stage method to simulate the mechanics of wetting of porous solid objects, like sponges and cloth, when they interact with a fluid. In the first stage, we model the absorption of fluid by the object when it comes in contact with the fluid. In the second stage, we model the transport of absorbed fluid inside the object, due to diffusion, as a flow in a deforming, unstructured mesh. The fluid diffuses within the object depending on saturation of its various parts and other body forces. Finally, in the third stage, oversaturated parts of the object shed extra fluid by dripping. The simulation model is motivated by the physics of imbibition of fluids into porous solids in the presence of gravity. It is phenomenologically capable of simulating wicking and imbibition, dripping, surface flows over wet media, material weakening, and volume expansion due to wetting. The model is inherently mass conserving and works for both thin 2D objects like cloth and for 3D volumetric objects like sponges. It is also designed to be computationally efficient and can be easily added to existing cloth, soft body, and fluid simulation pipelines.",
"abstracts": [
{
"abstractType": "Regular",
"content": "This paper presents a simple, three stage method to simulate the mechanics of wetting of porous solid objects, like sponges and cloth, when they interact with a fluid. In the first stage, we model the absorption of fluid by the object when it comes in contact with the fluid. In the second stage, we model the transport of absorbed fluid inside the object, due to diffusion, as a flow in a deforming, unstructured mesh. The fluid diffuses within the object depending on saturation of its various parts and other body forces. Finally, in the third stage, oversaturated parts of the object shed extra fluid by dripping. The simulation model is motivated by the physics of imbibition of fluids into porous solids in the presence of gravity. It is phenomenologically capable of simulating wicking and imbibition, dripping, surface flows over wet media, material weakening, and volume expansion due to wetting. The model is inherently mass conserving and works for both thin 2D objects like cloth and for 3D volumetric objects like sponges. It is also designed to be computationally efficient and can be easily added to existing cloth, soft body, and fluid simulation pipelines.",
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"normalizedAbstract": "This paper presents a simple, three stage method to simulate the mechanics of wetting of porous solid objects, like sponges and cloth, when they interact with a fluid. In the first stage, we model the absorption of fluid by the object when it comes in contact with the fluid. In the second stage, we model the transport of absorbed fluid inside the object, due to diffusion, as a flow in a deforming, unstructured mesh. The fluid diffuses within the object depending on saturation of its various parts and other body forces. Finally, in the third stage, oversaturated parts of the object shed extra fluid by dripping. The simulation model is motivated by the physics of imbibition of fluids into porous solids in the presence of gravity. It is phenomenologically capable of simulating wicking and imbibition, dripping, surface flows over wet media, material weakening, and volume expansion due to wetting. The model is inherently mass conserving and works for both thin 2D objects like cloth and for 3D volumetric objects like sponges. It is also designed to be computationally efficient and can be easily added to existing cloth, soft body, and fluid simulation pipelines.",
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"Mechanics Of Wetting"
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