Vigneron, Antoine E.
KAUST DepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Computer Science Program
Visual Computing Center (VCC)
Geometric Algorithms Group
Permanent link to this recordhttp://hdl.handle.net/10754/575754
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AbstractWe consider the following geometric alignment problem: Given a set of line segments in the plane, find a convex region of smallest area that contains a translate of each input segment. This can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be turned through 360 degrees within this region. Our main result is an optimal Θ(n log n)-time algorithm for our geometric alignment problem, when the input is a set of n line segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then the optimum placement is when the midpoints of the segments coincide. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of any rotated copy of G. © 2012 Springer-Verlag Berlin Heidelberg.
CitationAhn HK., Bae S.W., Cheong O., Gudmundsson J., Tokuyama T., Vigneron A. (2012) A Generalization of the Convex Kakeya Problem. In: Fernández-Baca D. (eds) LATIN 2012: Theoretical Informatics. LATIN 2012. Lecture Notes in Computer Science, vol 7256. Springer, Berlin, Heidelberg
Conference/Event name10th Latin American Symposiumon Theoretical Informatics, LATIN 2012