We revisit several maximization problems for geometric networks design under the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM Symposium on Computational Geometry, 1993). Given a set of $n$ points in the plane in general position (no three points collinear), compute a longest non-crossing configuration composed of straight line segments that is: (a) a matching (b) a Hamiltonian path (c) a spanning tree. Here we obtain new results for (b) and (c), as well as for the Hamiltonian cycle problem:
(i) For the longest non-crossing Hamiltonian path problem, we give an approximation algorithm with ratio $\frac{2}{π+1} \approx 0.4829$. The previous best ratio, due to Alon et al., was $1/π\approx 0.3183$. Moreover, the ratio of our algorithm is close to $2/π$ on a relatively broad class of instances: for point sets whose perimeter (or diameter) is much shorter than the maximum length matching. The algorithm runs in $O(n^{7/3}\log{n})$ time.
(ii) For the longest non-crossing spanning tree problem, we give an approximation algorithm with ratio 0.502 which runs in $O(n \log{n})$ time. The previous ratio, 1/2, due to Alon et al., was achieved by a quadratic time algorithm. Along the way, we first re-derive the result of Alon et al. with a faster $O(n \log{n})$-time algorithm and a very simple analysis.
(iii) For the longest non-crossing Hamiltonian cycle problem, we give an approximation algorithm whose ratio is close to $2/π$ on a relatively broad class of instances: for point sets with the product $\bf{<}$diameter$\times$ convex hull size $\bf{>}$ much smaller than the maximum length matching. The algorithm runs in $O(n^{7/3}\log{n})$ time. No previous approximation results were known for this problem.
A Randomized Incremental Algorithm for the Hausdorff Voronoi Diagram of Non-crossing Clusters
在平面中的一族\emph{clusters of points}的豪斯多夫沃罗诺伊图中，点$t$与簇$P$之间的距离被测量为$t$与$P$中的任何点之间的最大距离，并且该图是在输入簇的最近邻意义上定义的。在本文中，我们考虑%El。平面中的“非交叉”\emph{non-crossing}簇，其中Hausdorff沃罗诺伊图的组合复杂度在所有簇的凸包上的点的总数$n$中是线性的。我们提出了一个随机增量建设，基于点的位置，计算这个图在预期的$O(n\log^2{n})$时间和预期的$O(n)$空间。我们的技术有效地处理非标准特性的广义沃罗诺伊图，如网站的非常数复杂性，网站不封闭在其沃罗诺伊地区，空沃罗诺伊地区。该图在VLSI计算机辅助设计中有直接应用。
Structure and generation of crossing-critical graphs
Charles University, Prague（布拉格查理大学）；Faculty of Informatics, Masaryk University, Brno（布尔诺马萨里克大学信息学院）；Department of Mathematics, Simon Fraser University, Burnaby（本拿比西蒙菲莎大学数学系）
