A lower bound for the Hanf number for joint embedding
- Award ID(s):
- 2137465
- PAR ID:
- 10355935
- Date Published:
- Journal Name:
- Fundamenta Mathematicae
- Volume:
- 258
- Issue:
- 2
- ISSN:
- 0016-2736
- Page Range / eLocation ID:
- 115 to 135
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Beyersdorff, Olaf; Kanté, Mamadou Moustapha; Kupferman, Orna; Lokshtanov, Daniel (Ed.)Given a set P of n points and a set S of n segments in the plane, we consider the problem of computing for each segment of S its closest point in P. The previously best algorithm solves the problem in n^{4/3}2^{O(log^*n)} time [Bespamyatnikh, 2003] and a lower bound (under a somewhat restricted model) Ω(n^{4/3}) has also been proved. In this paper, we present an O(n^{4/3}) time algorithm and thus solve the problem optimally (under the restricted model). In addition, we also present data structures for solving the online version of the problem, i.e., given a query segment (or a line as a special case), find its closest point in P. Our new results improve the previous work.more » « less
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