计算单位圆弧交点数
Counting Unit Circular Arc Intersections
AI总结：
该研究提出了一种更高效的算法，用于计算平面中单位圆弧的交点数，针对小K情况进一步优化了时间复杂度。
摘要：
给定平面中n个半径相同的圆弧，我们考虑计算这些圆弧之间交点数的问题。该问题此前已被研究， previously best algorithm 解决该问题的时间复杂度为O(n^{4/3+ε}) [Agarwal, Pellegrini, and Sharir, SIAM J. Comput., 1993]，对于任何常数ε>0。在超过30年后，该问题未取得进展。我们提出一个新的算法，时间为O(n^{4/3}log^{16/3}n)，并改进为O(n^{1+ε}+K^{1/3}n^{2/3}(n²/(n+K))^εlog^{16/3}n)时间，其中K是所有圆弧的交点数。
英文摘要：
Given a set of $n$ circular arcs of the same radius in the plane, we consider the problem of computing the number of intersections among the arcs. The problem was studied before and the previously best algorithm solves the problem in $O(n^{4/3+ε})$ time [Agarwal, Pellegrini, and Sharir, SIAM J. Comput., 1993], for any constant $ε>0$. No progress has been made on the problem for more than 30 years. We present a new algorithm of $O(n^{4/3}\log^{16/3}n)$ time and improve it to $O(n^{1+ε}+K^{1/3}n^{2/3}(\frac{n^2}{n+K})^ε\log^{16/3}n)$ time for small $K$, where $K$ is the number of intersections of all arcs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
