Abstract
Let S be a set of n points in d-space. A convex Steiner partition is a tiling of conv(S) with empty convex bodies. For every integer d, we show that S admits a convex Steiner partition with at most ⌈(n − 1)/d⌉ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d ≥ 3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position.
Establishing a tight lower bound for the maximum volume of a tile in a Steiner partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n) in any dimension d ≥ 2. Here we give a (1 − ε)-approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d-dimensional unit box [0,1]d.
Dumitrescu is supported in part by the NSF grant DMS-1001667; Har-Peled is supported in part by the NSF grant CCF-0915984; Tóth acknowledges support from the NSERC grant RGPIN 35586.
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Dumitrescu, A., Har-Peled, S., Tóth, C.D. (2012). Minimum Convex Partitions and Maximum Empty Polytopes. In: Fomin, F.V., Kaski, P. (eds) Algorithm Theory – SWAT 2012. SWAT 2012. Lecture Notes in Computer Science, vol 7357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31155-0_19
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