Maintaining the Union of Unit Discs under Insertions with Near-Optimal Overhead
Maintaining the Union of Unit Discs under Insertions with Near-Optimal Overhead
复制标题
DOI：
10.1145/3527614
复制
发表时间：
2019-03
期刊：
影响因子：
--
通讯作者：
P. Agarwal;Ravid Cohen;D. Halperin;Wolfgang Mulzer
中科院分区：
文献类型：
--
作者：
P. Agarwal;Ravid Cohen;D. Halperin;Wolfgang Mulzer
我们提出了有效的动态数据结构，用于维持单位盘的结合和平面中的伪线的较低信封。插入下的一组单元光盘的结合。联盟的结构变化插入新的光盘也可以在插入每个盘后的联合区域内计算。一组X-Monotone伪线可以处理o（log2n）时间的插入/删除。在带有X坐标X0的下部包膜上；对于查询点q∈ℝ2伪线位于Q下方O（log n+klog2 n）（iii）。圆盘），因此，对于查询单元圆盘D，所有输入弧都可以在O（N1/2 + + K）时间中报告，其中K是输出尺寸，并且ɛ> 0是任意的较小常数可以将单位圆形弧插入或删除O（log2 n）时间。
We present efficient dynamic data structures for maintaining the union of unit discs and the lower envelope of pseudo-lines in the plane. More precisely, we present three main results in this paper: (i) We present a linear-size data structure to maintain the union of a set of unit discs under insertions. It can insert a disc and update the union in O((k+1)log2 n) time, where n is the current number of unit discs and k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. It can also compute, within the same time bound, the area of the union after the insertion of each disc. (ii) We propose a linear-size data structure for maintaining the lower envelope of a set of x-monotone pseudo-lines. It can handle insertion/deletion of a pseudo-line in O(log2n) time; for a query point x0∈ ℝ, it can report, in O(log n) time, the point on the lower envelope with x-coordinate x0; and for a query point q∈ ℝ2, it can return all k pseudo-lines lying below q in time O(log n+klog2 n). (iii) We present a linear-size data structure for storing a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), so that for a query unit disc D, all input arcs intersecting D can be reported in O(n1/2+ɛ + k) time, where k is the output size and ɛ > 0 is an arbitrarily small constant. A unit-circle arc can be inserted or deleted in O(log2 n) time.