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4. Matrix Multiplication with Less Arithmetic Complexity and IO Complexity

Author

Wu, Pu, Jiang, Huiqing, Shao, Zehui, and Xu, Jin

Subjects
Computer Science - Symbolic Computation, Computer Science - Discrete Mathematics, 68R01, F.2.1, I.1.2
Abstract

After Strassen presented the first sub-cubic matrix multiplication algorithm, many Strassen-like algorithms are presented. Most of them with low asymptotic cost have large hidden leading coefficient which are thus impractical. To reduce the leading coefficient, Cenk and Hasan give a general approach reducing the leading coefficient of $<2,2,2;7>$-algorithm to $5$ but increasing IO complexity. In 2017, Karstadt and Schwartz also reduce the leading coefficient of $<2,2,2;7>$-algorithm to $5$ by the Alternative Basis Matrix Multiplication method. Meanwhile, their method reduces the IO complexity and low-order monomials in arithmetic complexity. In 2019, Beniamini and Schwartz generalize Alternative Basis Matrix Multiplication method reducing leading coefficient in arithmetic complexity but increasing IO complexity. In this paper, we propose a new matrix multiplication algorithm which reduces leading coefficient both in arithmetic complexity and IO complexity. We apply our method to Strassen-like algorithms improving arithmetic complexity and IO complexity (the comparison with previous results are shown in Tables 1 and 2). Surprisingly, our IO complexity of $<3,3,3;23>$-algorithm is $14n^{\log_323}M^{-\frac{1}{2}} + o(n^{\log_323})$ which breaks Ballard's IO complexity low bound ($\Omega(n^{\log_323}M^{1-\frac{\log_323}{2}})$) for recursive Strassen-like algorithms., Comment: 24 pages and one figure

Published
2022

19. Author Correction: Mortality outcomes with hydroxychloroquine and chloroquine in COVID-19 from an international collaborative meta-analysis of randomized trials

Author

Cathrine Axfors, Andreas M. Schmitt, Perrine Janiaud, Janneke van’t Hooft, Sherief Abd-Elsalam, Ehab F. Abdo, Benjamin S. Abella, Javed Akram, Ravi K. Amaravadi, Derek C. Angus, Yaseen M. Arabi, Shehnoor Azhar, Lindsey R. Baden, Arthur W. Baker, Leila Belkhir, Thomas Benfield, Marvin A. H. Berrevoets, Cheng-Pin Chen, Tsung-Chia Chen, Shu-Hsing Cheng, Chien-Yu Cheng, Wei-Sheng Chung, Yehuda Z. Cohen, Lisa N. Cowan, Olav Dalgard, Fernando F. de Almeida e Val, Marcus V. G. de Lacerda, Gisely C. de Melo, Lennie Derde, Vincent Dubee, Anissa Elfakir, Anthony C. Gordon, Carmen M. Hernandez-Cardenas, Thomas Hills, Andy I. M. Hoepelman, Yi-Wen Huang, Bruno Igau, Ronghua Jin, Felipe Jurado-Camacho, Khalid S. Khan, Peter G. Kremsner, Benno Kreuels, Cheng-Yu Kuo, Thuy Le, Yi-Chun Lin, Wu-Pu Lin, Tse-Hung Lin, Magnus Nakrem Lyngbakken, Colin McArthur, Bryan J. McVerry, Patricia Meza-Meneses, Wuelton M. Monteiro, Susan C. Morpeth, Ahmad Mourad, Mark J. Mulligan, Srinivas Murthy, Susanna Naggie, Shanti Narayanasamy, Alistair Nichol, Lewis A. Novack, Sean M. O’Brien, Nwora Lance Okeke, Léna Perez, Rogelio Perez-Padilla, Laurent Perrin, Arantxa Remigio-Luna, Norma E. Rivera-Martinez, Frank W. Rockhold, Sebastian Rodriguez-Llamazares, Robert Rolfe, Rossana Rosa, Helge Røsjø, Vanderson S. Sampaio, Todd B. Seto, Muhammad Shahzad, Shaimaa Soliman, Jason E. Stout, Ireri Thirion-Romero, Andrea B. Troxel, Ting-Yu Tseng, Nicholas A. Turner, Robert J. Ulrich, Stephen R. Walsh, Steve A. Webb, Jesper M. Weehuizen, Maria Velinova, Hon-Lai Wong, Rebekah Wrenn, Fernando G. Zampieri, Wu Zhong, David Moher, Steven N. Goodman, John P. A. Ioannidis, and Lars G. Hemkens

Subjects
Science
Published
2024
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21. Solvent engineering for scalable fabrication of perovskite/silicon tandem solar cells in air

Author

Zheng, Xuntian, Kong, Wenchi, Wen, Jin, Hong, Jiajia, Luo, Haowen, Xia, Rui, Huang, Zilong, Luo, Xin, Liu, Zhou, Li, Hongjiang, Sun, Hongfei, Wang, Yurui, Liu, Chenshuaiyu, Wu, Pu, Gao, Han, Li, Manya, Bui, Anh Dinh, Mo, Yi, Zhang, Xueling, Yang, Guangtao, Chen, Yifeng, Feng, Zhiqiang, Nguyen, Hieu T., Lin, Renxing, Li, Ludong, Gao, Jifan, and Tan, Hairen

Published
2024
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24. Author Correction: Mortality outcomes with hydroxychloroquine and chloroquine in COVID-19 from an international collaborative meta-analysis of randomized trials

Author

Axfors, Cathrine, Schmitt, Andreas M., Janiaud, Perrine, van’t Hooft, Janneke, Abd-Elsalam, Sherief, Abdo, Ehab F., Abella, Benjamin S., Akram, Javed, Amaravadi, Ravi K., Angus, Derek C., Arabi, Yaseen M., Azhar, Shehnoor, Baden, Lindsey R., Baker, Arthur W., Belkhir, Leila, Benfield, Thomas, Berrevoets, Marvin A. H., Chen, Cheng-Pin, Chen, Tsung-Chia, Cheng, Shu-Hsing, Cheng, Chien-Yu, Chung, Wei-Sheng, Cohen, Yehuda Z., Cowan, Lisa N., Dalgard, Olav, de Almeida e Val, Fernando F., de Lacerda, Marcus V. G., de Melo, Gisely C., Derde, Lennie, Dubee, Vincent, Elfakir, Anissa, Gordon, Anthony C., Hernandez-Cardenas, Carmen M., Hills, Thomas, Hoepelman, Andy I. M., Huang, Yi-Wen, Igau, Bruno, Jin, Ronghua, Jurado-Camacho, Felipe, Khan, Khalid S., Kremsner, Peter G., Kreuels, Benno, Kuo, Cheng-Yu, Le, Thuy, Lin, Yi-Chun, Lin, Wu-Pu, Lin, Tse-Hung, Lyngbakken, Magnus Nakrem, McArthur, Colin, McVerry, Bryan J., Meza-Meneses, Patricia, Monteiro, Wuelton M., Morpeth, Susan C., Mourad, Ahmad, Mulligan, Mark J., Murthy, Srinivas, Naggie, Susanna, Narayanasamy, Shanti, Nichol, Alistair, Novack, Lewis A., O’Brien, Sean M., Okeke, Nwora Lance, Perez, Léna, Perez-Padilla, Rogelio, Perrin, Laurent, Remigio-Luna, Arantxa, Rivera-Martinez, Norma E., Rockhold, Frank W., Rodriguez-Llamazares, Sebastian, Rolfe, Robert, Rosa, Rossana, Røsjø, Helge, Sampaio, Vanderson S., Seto, Todd B., Shahzad, Muhammad, Soliman, Shaimaa, Stout, Jason E., Thirion-Romero, Ireri, Troxel, Andrea B., Tseng, Ting-Yu, Turner, Nicholas A., Ulrich, Robert J., Walsh, Stephen R., Webb, Steve A., Weehuizen, Jesper M., Velinova, Maria, Wong, Hon-Lai, Wrenn, Rebekah, Zampieri, Fernando G., Zhong, Wu, Moher, David, Goodman, Steven N., Ioannidis, John P. A., and Hemkens, Lars G.

Published
2024
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28. Changing and unchanging 2-rainbow independent domination

Author

Wu, Pu, Shao, Zehui, Samodivkin, Vladimir, Sheikholeslami, S. M., Soroudi, M., and Wang, Shaohui

Subjects
Mathematics - Combinatorics
Abstract

For a function $f : V(G ) \rightarrow \{0, 1, 2\}$ we denote by $V_i$ the set of vertices to which the value $i$ is assigned by $f$, i.e. $V_i = \{ x \in V (G ) : f(x ) = i \}$. If a function $f: V(G) \rightarrow \{0,1,2\}$ satisfying the condition that $V_i$ is independent for $i \in \{1,2\}$ and every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = i$ for each $i \in \{1,2\}$, then $f$ is called a 2-rainbow independent dominating function (2RiDF). The weight $w(f)$ of a 2RiDF $f$ is the value $w(f) = |V_1|+|V_2|$. The minimum weight of a 2RiDF on a graph $G$ is called the \emph{2-rainbow independent domination number} of $G$. A graph $G$ is 2-rainbow independent domination stable if the 2-rainbow independent domination number of $G$ remains unchanged under removal of any vertex. In this paper, we characterize 2-rainbow independent domination stable trees and we study the effect of edge removal on 2-rainbow independent domination number in trees.

Published
2018

38. The characterization of perfect Roman domination stable trees

Author

Li, Zepeng, Shao, Zehui, Rao, Yongsheng, Wu, Pu, and Wang, Shaohui

Subjects
Mathematics - Combinatorics
Abstract

A \emph{perfect Roman dominating function} (PRDF) on a graph $G = (V, E)$ is a function $f : V \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$. The weight of a PRDF is the value $w(f) = \sum_{u \in V}f(u)$. The minimum weight of a PRDF on a graph $G$ is called the \emph{perfect Roman domination number $\gamma_R^p(G)$} of $G$. A graph $G$ is perfect Roman domination domination stable if the perfect Roman domination number of $G$ remains unchanged under the removal of any vertex. In this paper, we characterize all trees that are perfect Roman domination stable.

Published
2018

47. MRI Manifestations of Breast Cancer Stroma and their Role in Predicting Molecular Subtype: A Case-control Study

Author

Wang, Lanyun, primary, Li, Wenjing, additional, Yang, Wenjun, additional, Sun, Xilin, additional, Ding, Yi, additional, Zhao, Qian, additional, Liu, Weiyan, additional, Xie, Xiaoli, additional, Xu, Jingjing, additional, Wei, Ran, additional, Zhu, Shizhen, additional, Ge, Yaqiong, additional, Wu, Pu-Yeh, additional, and Song, Bin, additional

Published
2024
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