This paper provides a full theoretical and experimental analysis of a serial algorithm for the point-in-polygon test, which requires less running time than previous algorithms and can handle all degenerate cases. The serial algorithm can quickly determine whether a point is inside or outside a polygon and accurately determine the contours of input polygon. It describes all degenerate cases and simultaneously provides a corresponding solution to each degenerate case to ensure the stability and reliability. This also creates the prerequisites and basis for our novel boolean operations algorithm that inherits all the benefits of the serial algorithm. Using geometric probability and straight-line equation $F\left(P\right)=(y_i-y_{i+1})(x_p-x_i)-(y_i-y_p)(x_{i+1}-x_i)$ , it optimizes our two algorithms that avoid the division operation and do not need to compute any intersection points. Our algorithms are applicable to any polygon that may be self-intersecting or with holes nested to any level of depth. They do not have to sort the vertices clockwise or counterclockwise beforehand. Consequently, they process all edges one by one in any order for input polygons. This allows a parallel implementation of each algorithm to be made very easily. We also prove several theorems guaranteeing the correctness of algorithms. To speed up the operations, we assign each vector a number code and derive two iterative formulas using differential calculus. However, the experimental results as well as the theoretical proof show that our serial algorithm for the point-in-polygon test is optimal and the time complexities of all algorithms are linear. Our methods can be extended to three-dimensional space, in particular, they can be applied to 3D printing to improve its performance.

Multivariate time series are widely used in industrial equipment monitoring and maintenance, health monitoring, weather forecasting and other fields. Due to abnormal sensors, equipment failures, environmental interference and human errors, the collected multivariate time series usually have certain missing values. Missing values imply the regularity of data, and seriously affect the further analysis and application of multivariate time series. Conventional imputation methods such as statistical imputation and machine learning-based imputation cannot learn the latent relationships of data and are difficult to use for missing values imputation in multivariate time series. This paper proposes a novel Time and Location Gated Recurrent Unit (TLGRU), which takes into account the non-fixed time intervals and location intervals in multivariate time series and effectively deals with missing values. We made necessary modifications to the architecture of the end-to-end imputation model $${E}^{2}$$E 2 GAN and replaced Gated Recurrent Unit for Imputation (GRUI) with TLGRU to make the generated fake sample closer to the original sample. Experiments on a public meteorologic dataset show that our method outperforms the baselines on the imputation accuracy and achieves a new state-of-the-art result.

This paper presents a novel theory and method to calculate the canonical labelings of digraphs whose definition is entirely different from the traditional definition of Nauty. It indicates the mutual relationships that exist between the canonical labeling of a digraph and the canonical labeling of its complement graph. It systematically examines the link between computing the canonical labeling of a digraph and the k-neighborhood and k-mix-neighborhood subdigraphs. To facilitate the presentation, it introduces several concepts including mix diffusion outdegree sequence and entire mix diffusion outdegree sequences. For each node in a digraph G, it assigns an attribute m_NearestNode to enhance the accuracy of calculating canonical labeling. The four theorems proved here demonstrate how to determine the first nodes added into $MaxQ(G)$ . Further, the other two theorems stated below deal with identifying the second nodes added into $MaxQ(G)$ . When computing $C_{max}(G)$ , if $MaxQ(G)$ already contains the first i vertices $u_1,u_2,\cdots ,u_i$ , Diffusion Theorem provides a guideline on how to choose the subsequent node of $MaxQ(G)$ . Besides, the Mix Diffusion Theorem shows that the selection of the $(i+1)$ th vertex of $MaxQ(G)$ for computing $C_{max}(G)$ is from the open mix-neighborhood subdigraph $N^{++}(Q)$ of the nodes set $Q=\{u_1,u_2,\cdots ,u_i\}$ . It also offers two theorems to calculate the $C_{max}(G)$ of the disconnected digraphs. The four algorithms implemented in it illustrate how to calculate $MaxQ(G)$ of a digraph. Through software testing, the correctness of our algorithms is preliminarily verified. Our method can be utilized to mine the frequent subdigraph. We also guess that if there exists a vertex $v\in S^{+}(G)$ satisfying conditions $C_{max}(G-v)⩽C_{max}(G-w)$ for each $w\in S^{+}(G)\wedge w\ne v$ , then $u_1=v$ for $MaxQ(G)$ .

This paper puts forward an innovative theory and method to calculate the canonical labelings of graphs that are distinct to $Nauty$ ’s. It shows the correlation between the canonical labeling of a graph and the canonical labeling of its complement graph. It regularly examines the link between computing the canonical labeling of a graph and the canonical labeling of its $open$ k- $neighborhood$ $subgraph$ . It defines $diffusion$ $degree$ $sequences$ and $entire$ $diffusion$ $degree$ $sequence$ . For each node of a graph G, it designs a characteristic $m\_NearestNode$ to improve the precision for calculating canonical labeling. Two theorems established here display how to compute the first nodes of $MaxQ\left(G\right)$ . Another theorem presents how to determine the second nodes of $MaxQ\left(G\right)$ . When computing $C_{max}\left(G\right)$ , if $MaxQ\left(G\right)$ already holds the first i nodes $u_1,u_2,\cdots ,u_i$ , Diffusion and Nearest Node theorems provide skill on how to pick the succeeding node of $MaxQ\left(G\right)$ . Further, it also establishes two theorems to determine the $C_{max}\left(G\right)$ of disconnected graphs. Four algorithms implemented here demonstrate how to compute $MaxQ\left(G\right)$ of a graph. From the results of the software experiment, the accuracy of our algorithms is preliminarily confirmed. Our method can be employed to mine the frequent subgraph. We also conjecture that if there is a node $v\in S\left(G\right)$ meeting conditions $C_{max}(G-v)$ ⩽ $C_{max}(G-w)$ for each $w\in S\left(G\right)\wedge w\ne v$ , then $u_1=v$ for $MaxQ\left(G\right)$ .