An \~{O}(n2)(n^2) Time Matrix Multiplication Algorithm

We show, for the input vectors $(a_0, a_1, ..., a_{n-1})$ and $(b_0, b_1, ..., b_{n-1})$, where $a_i$'s and $b_j$'s are real numbers, after \~{O}$(n)$ time preprocessing for each of them, the vector multiplication $(a_0, a_1, ..., a_{n-1})(b_0, b_1, ..., b_{n-1})^T$ can be computed in \~{O}$(1)$ time. This enables the matrix multiplication of two $n\times n$ matrices to be computed in \~{O}$(n^2)$ time.Comment: Version 11 and Version 12 section 2 laid the foundation of this algorithm but has a problem unresolved. This version corrects the problem in Version 11 and Section 2 of Version 1

Stochastic Block Coordinate Frank-Wolfe Algorithm for Large-Scale Biological Network Alignment

With increasingly "big" data available in biomedical research, deriving accurate and reproducible biology knowledge from such big data imposes enormous computational challenges. In this paper, motivated by recently developed stochastic block coordinate algorithms, we propose a highly scalable randomized block coordinate Frank-Wolfe algorithm for convex optimization with general compact convex constraints, which has diverse applications in analyzing biomedical data for better understanding cellular and disease mechanisms. We focus on implementing the derived stochastic block coordinate algorithm to align protein-protein interaction networks for identifying conserved functional pathways based on the IsoRank framework. Our derived stochastic block coordinate Frank-Wolfe (SBCFW) algorithm has the convergence guarantee and naturally leads to the decreased computational cost (time and space) for each iteration. Our experiments for querying conserved functional protein complexes in yeast networks confirm the effectiveness of this technique for analyzing large-scale biological networks

Quantum Image Matching

Quantum image processing (QIP) means the quantum based methods to speed up image processing algorithms. Many quantum image processing schemes claim that their efficiency are theoretically higher than their corresponding classical schemes. However, most of them do not consider the problem of measurement. As we all know, measurement will lead to collapse. That is to say, executing the algorithm once, users can only measure the final state one time. Therefore, if users want to regain the results (the processed images), they must execute the algorithms many times and then measure the final state many times to get all the pixels' values. If the measurement process is taken into account, whether or not the algorithms are really efficient needs to be reconsidered. In this paper, we try to solve the problem of measurement and give a quantum image matching algorithm. Unlike most of the QIP algorithms, our scheme interests only one pixel (the target pixel) instead of the whole image. It modifies the probability of pixels based on Grover's algorithm to make the target pixel to be measured with higher probability, and the measurement step is executed only once. An example is given to explain the algorithm more vividly. Complexity analysis indicates that the quantum scheme's complexity is $O(2^{n})$ in contradistinction to the classical scheme's complexity $O(2^{2n+2m})$, where $m$ and $n$ are integers related to the size of images.Comment: 29 page