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This paper presents a heuristic algorithm for solving a specific NP-hard 2D rectangular packing problem in which a rectangle called central rectangle is required to be placed in the center of the final layout, and the aspect ratio of the container is also required to be in a given range. The key component of the proposed algorithm is a greedy constructive procedure, according to which, the rectangles are packed into the container one by one and each rectangle is packed into the container by an angle-occupying placement with maximum fit degree. The proposed algorithm is evaluated on two groups of 35 well-known benchmark instances. Computational results disclose that the proposed algorithm outperforms the previous algorithm for the packing problem. For the first group of test instances, solutions with average filling rate 99.31% can be obtained; for the real-world layout problem in the second group, the filling rate of the solution is 94.75%.

Evaluating influential spreaders in networks is of great significance for promoting the dissemination of beneficial information or inhibiting the spreading of harmful information. Currently, there are some central indices that can be used to evaluate spreading influence of {nodes}. However, most of them ignore the spreading probability and take into consideration only the network topology or the location of source node, so the excellent results can be achieved only when the spreading probability is in a specified range. For example, the degree centrality is appropriate for a minor spreading probability, but to ensure the accuracy, semi-local and closeness centralities are more suitable for a slightly larger one. To solve the sensitivity problem of spreading probability, a novel algorithm is proposed based on the extension of degree. In this algorithm, the coverage area of degree is recursively extended by the overlapping of degree of neighbors, which makes different extension levels correspond to different spreading probabilities. For a certain spreading probability, the proper level index is calculated by finding the most correlate ranking sequences of sampling {nodes}, which is obtained by matching the results of different spreading levels and SIR simulation. In this paper, the relationship between extension level and spreading probability is explained by the theory of fitting the weight and infected possibility of {nodes}, and the feasibility of the sampling method is verified by the computational experiments. The experimental results on both real and computer-generated datasets show that the proposed algorithm can effectively evaluate the spreading influences of {nodes} under different spreading probabilities, and the performance is close or even superior to that evaluated by using other central indices.

Dynamic geometry technology has been extensively used in geometry education. To aid the teaching and learning of geometry reasoning, many efficient automated geometry reasoning algorithms have been integrated into the dynamic geometry system. However, the reasoning functionalities in most current dynamic geometry system cannot fully meet the requirement of the geometry reasoning education. To tap the educational potential of automated geometry reasoning to the maximum, we proposed the concept and implementation mechanism of constrained automated geometry reasoning in this paper, and developed a constrained automated geometry reasoning system with a convenient dynamic geometry interface. Experimental results show that the proposed system could successfully solve 107 out of 132 geometry proof problems, which means it can meet the basic requirement of the geometry education.