On the Planar Two-Center Problem and Circular Hulls
On the Planar Two-Center Problem and Circular Hulls
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关于平面二中心问题和圆壳
DOI：
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发表时间：
2020
影响因子：
0.8
通讯作者：
Haitao Wang
中科院分区：
文献类型：
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作者：
Haitao Wang
鉴于欧几里得平面中的一组n个点，两个中心的问题是找到两个最小半径的一致磁盘，它们的结合覆盖了S的所有S. Eppstein，Eppstein（Soda'97）给出了O的随机算法O（nlog2n ）文档class [12pt] {minimal} usepackage {amsmath} usepackage {wasysym} usepackage {amsfonts} usepackage {amssymb} {文档} $ $ o（nlog ^2！n）$$ end {document}预期时间和chan（cgta'99）提出了O（nlog2nlog2logn）documentClass [12pt] {12pt] {minimal} usepackage {amsmath} usepackage {amsmath} usepackage {wasysym} } usepackage {amssymb} usepackage {amsbsy} usepackage {mathrsfs} usepackage {upgreek} setLength {oddSidemargin} { - 69pt} { - 69pt}在本文中，我们提出了一个o（nlog2n）文档classclass [12pt] {minimal} usepackage {amsmath} usepackage {wasySym} usepackage {amsfonts} usepackage} usepackage {amssymb emargin} {-69pt} egin {document} $$ o（nlog ^2！n）$$ end {document}时间确定性算法，它可以改善Chan的确定性算法并匹配Eppstein的随机界限。解决O（nlognloglogn）中的问题[12pt] {minimal} usepackage {amsmath} usepackage {wasysym} useymym} usepackage {amsfonts} usepackage {amssymb} argin} {-69pt } egin {document} $$ o（nlog nlog log n）$$ end {document}确定时间。
Given a set S of n points in the Euclidean plane, the two-center problem is to find two congruent disks of smallest radius whose union covers all points of S. Previously, Eppstein (SODA’97) gave a randomized algorithm of O(nlog2n)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$O(nlog ^2!n)$$end{document} expected time and Chan (CGTA’99) presented a deterministic algorithm of O(nlog2nlog2logn)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$O(nlog ^2!nlog ^2log n)$$end{document} time. In this paper, we propose an O(nlog2n)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$O(nlog ^2!n)$$end{document} time deterministic algorithm, which improves Chan’s deterministic algorithm and matches the randomized bound of Eppstein. If S is in convex position, then we solve the problem in O(nlognloglogn)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$O(nlog nlog log n)$$end{document} deterministic time. Our results rely on new techniques for dynamically maintaining circular hulls under point insertions and deletions, which are of independent interest.