Abstract
Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points. If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n log n) time, which is optimal, even in the presence of collinear points. If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. Finally, our results imply that a k-angulation of a set of points can be achieved with the addition of at most k-3 extra points within the same time bound.
research supported by Killam and NSERC postdoctorate fellowships and grants NSERC-OGP0009293 and FCAR-93ER0291.
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© 1995 Springer-Verlag Berlin Heidelberg
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Bose, P., Toussaint, G. (1995). No quadrangulation is extremely odd. In: Staples, J., Eades, P., Katoh, N., Moffat, A. (eds) Algorithms and Computations. ISAAC 1995. Lecture Notes in Computer Science, vol 1004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015443
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DOI: https://doi.org/10.1007/BFb0015443
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