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No abstract uploaded!

A circle family is a group of separate or tangent circles in the plane. In this paper, we study how many parts at most a plane can be divided by several circle families if the circles in a same family must be separate (resp. if the circles can be tangent). We also study the necessary conditions for the intersection of two circle families. Then we primarily discuss the similar problems in higher dimensional space and in the end,raise some conjectures.

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Dijkstra and Bellman-ford are the two basic methods to solve Single-source shortest paths problem，and SPFA is the optimized version of Bellman-ford. However, since the algorithm time complexity is 0 (kn), SPFA’s time efficiency has some instabilities. This paper will compare the time efficiency between Dijkstra optimized with heap and three kinds of SPFA in different density graphs. Meanwhile, random data is created to test optimized Dijkstra and 3 types of SPFA. Finally, according to the calculation result of two methods’ time efficiency, some suggestion is given for their application under different situation.