Abstract
Let \(T=\{\triangle _1,\ldots ,\triangle _n\}\) be a set of n triangles in \(\mathbb {R}^3\) with pairwise-disjoint interiors, and let B be a convex polytope in \(\mathbb {R}^3\) with a constant number of faces. For each i, let \(C_i = \triangle _i \oplus r_i B\) denote the Minkowski sum of \(\triangle _i\) with a copy of B scaled by \(r_i>0\). We show that if the scaling factors \(r_1, \ldots , r_n\) are chosen randomly then the expected complexity of the union of \(C_1, \ldots , C_n\) is \(O(n^{2+{\varepsilon }})\), for any \({\varepsilon }> 0\); the constant of proportionality depends on \({\varepsilon }\) and on the complexity of B. The worst-case bound can be \(\Theta (n^3)\). We also consider a special case of this problem in which T is a set of points in \(\mathbb {R}^3\) and B is a unit cube in \(\mathbb {R}^3\), i.e., each \(C_i\) is a cube of side-length \(2r_i\). We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is \(O(n\log ^2n)\), and it improves to \(O(n\log n)\) if the scaling factors are chosen randomly from a “well-behaved” probability density function (pdf). We also extend the latter results to higher dimensions. For any fixed odd value of \(d>3\), we show that the expected complexity of the union of the hypercubes is \(O(n^{{\lfloor d/2\rfloor }}\log n)\) and the bound improves to \(O(n^{{\lfloor d/2\rfloor }})\) if the scaling factors are chosen from a “well-behaved” pdf. The worst-case bounds are \(\Theta (n^2)\) in \(\mathbb {R}^3\), and \(\Theta (n^{\lceil d/2\rceil })\) in higher odd dimensions.
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Notes
Here the bound is cubic in the number of polytopes but is only near linear in the number of facets.
In contrast, the complexity of the union of congruent balls in \(\mathbb {R}^3\) is quadratic in the worst case; see, e.g., [3].
Given a point set \(P=\{p_1,\ldots ,p_n\}\) and a weight \(w_i > 0\) for each point \(p_i\), the Voronoi cell of \(p_i\) in the multiplicatively weighted Voronoi diagram is \(\{x\in \mathbb {R}^d \mid w_i\Vert x-p_i\Vert \le w_j \Vert x-p_j\Vert \; \forall j=1,\ldots ,n\}\).
This monotonicity is easy to prove: For example, by looking only at sequences of n radii of which at least n/2 are 0, we essentially eliminate half of the hypercubes. So \(\mathcal {O}_0(n)\), even when restricted to these sequences, is at least as large as \(\mathcal {O}_0(n/2)\).
If w lies on more than one such \((d-2)\)-face, it cannot be charged at all.
Clearly, \(\bar{\mathcal {U}}\) and \(\bar{\mathcal {F}}\), as well as the sets \(T_\tau ^<\), are uniquely determined. For \(\bar{\mathcal {F}}^\nabla \), the statement means that if we follow an agreed-upon (and deterministic) implementation of the construction in the proof of Lemma 3.1, \(\bar{\mathcal {F}}^\nabla \) is also uniquely determined.
Throughout this section we assume \(\log x\) to be \(\log _2x\).
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The authors thank Sariel Har-Peled for helpful discussions and an anonymous reviewer for many useful comments on the paper.
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Work by P.A. and M.S. was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, and work by H.K. and M.S. was supported by Grant 1367/2017 from the German-Israeli Foundation for Scientific Research and Development, and by the Blavatnik Research Fund in Computer Science at Tel Aviv University. Work by P.A. was also supported by NSF under grants IIS-14-08846, CCF-13-31133, and CCF-15-13816, and by an ARO grant W911NF-15-1-0408. Work by H.K. was also supported by Grants 1841/14 and 1595/19 from the Israel Science Foundation, work by M.S. was also supported by Grants 892/13 and 260/18 from the Israel Science Foundation and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11).
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Agarwal, P.K., Kaplan, H. & Sharir, M. Union of Hypercubes and 3D Minkowski Sums with Random Sizes. Discrete Comput Geom 65, 1136–1165 (2021). https://doi.org/10.1007/s00454-020-00274-0
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DOI: https://doi.org/10.1007/s00454-020-00274-0