A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon
A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon
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简单多边形点测地线Voronoi图的近乎最优算法
DOI：
10.1007/s00453-019-00624-2
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发表时间：
2018
期刊：
影响因子：
1.1
通讯作者：
Chih
中科院分区：
文献类型：
--
作者：
Chih
N顶点的简单多边形内的M点位点的测量伏罗尼亚图是多边形的一个细分，一个对M细胞，以使每个点的所有点在大地距离下共享相同的位点施工时间的下限为ω（n+mlogm）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasySym} \ useym} \ usepackage {amsfonts} \ usepackage {mathrsfs} \ usepackage {upgreek} \ setLength {\ oddSidemargin} { - 69pt} \ begin是一个长期的开放式问题。 } \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ use-package {Mathrsfs} \ usepackage {upgreek} \ setLength {\ oddSideMargin} { - 69pt} \ begin o（n+mlogmlog2n）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasySym} \ useym} \ usepackage {amsfonts} \ usepackage {amssymb}希腊语} \ setLength {\ oddSidemargin} { - 69pt} \ begin {document} $$ o（n+m \ log m \ log ^2n）$ \ end {document}时间，对于m =ω（n）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasySym} {amssymb} \ usepackage {amsbsy} \ usepackage {mathrsfs} \ usepackage {upgreek} \ setLength {\ oddSidemargin} { - 69pt} { - 69pt} m = o（nlog3n）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=O （\ frac {n} {\ log ^3n}）$$ \ end {document {document}。最小值} \ usepackage {amsmath} \ usepackage {asysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {Mathrsfs} \ usepackage { n））$$ \ end {document}时间，并且从某种意义上说，如果可以在o（logn）\ documentclass [12pt] {minimal} \ usepackage {amsmath} {amsmath} \ usepackage {\ usepackage { asysym} \ usepackage {amsfonts} \ usepackage {amssymb} \ usepackage {amsbsy} \ usepackage {Mathrsfs} \ usepackage {upgreek} \ setLength {\ oddSideMargin} { - 69pt} \ 69pt} \ begin时间将成为最佳的O（n+mlogm）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasySym} \ useymym} \ usepackage {amsfonts} \ usepackage {amssymb} usepackage {upgreek} \ setLength {\ oddSidemargin} { - 69pt} \ begin {document} $$ o（n+m \ log m）$ teen $ end \ end {document {document}，我们减少了在最佳时间构建图表的问题。要在O（logn）\ documentClass [12pt] {minimal} \ usepackage {amsmath} \ usepackage {wasySym} \ usepackage {amsmssfonts} {amsfonts} \ usepackage {amssmssmbage {amssmbbage {amssmssmbaikepapk {amsypack {amsypack} mathrsfs} \ usepackage {upgreek} \ setLength {\ oddSideMargin} { - 69pt} \ begin {document} $$ o（\ log n）$ n）$$ \ end \ end {document} time。
The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is Ω(n+mlogm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (n+m\log m)$$\end{document}, and a matching upper bound is a long-standing open question. The state-of-the-art construction algorithms achieve O((n+m)log(n+m))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O( (n+m) \log (n+m) )$$\end{document} and O(n+mlogmlog2n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n+m\log m\log ^2n)$$\end{document} time, which are optimal for m=Ω(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=\varOmega (n)$$\end{document} and m=O(nlog3n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=O(\frac{n}{\log ^3n})$$\end{document}, respectively. In this paper, we give a construction algorithm with O(n+m(logm+log2n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O( n + m ( \log m+ \log ^2 n ) )$$\end{document} time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\log n)$$\end{document} time, then the construction time will become the optimal O(n+mlogm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n+m\log m)$$\end{document}. In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in O(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\log n)$$\end{document} time.