We study the question whether a crossing-free 3D morph between two straight-line drawings of an n-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with O(log n) steps, while for the latter Θ(n) steps are always sufficient and sometimes necessary. * We here refer to pole dancing as a fitness and competitive sport. The authors hope that many of our readers try this activity themselves, and will in return introduce many pole dancers to Graph Drawing, thereby alleviating the gender imbalance in both communities. The authors do not condone any pole activity used for sexual exploitation or abuse of women or men. between any two topologically-equivalent † planar straight-line ‡ drawings of the same planar graph always exists; this was proved for maximal planar graphs by Cairns [8] back in 1944, and then for all planar graphs by Thomassen [16] almost forty years later. Note that a planar morph between two planar graph drawings that are not topologically equivalent does not exist.It has lately been well investigated whether a planar morph between any two topologically-equivalent planar straight-line drawings of the same planar graph always exists such that the vertex trajectories have low complexity. This is usually formalized as follows. Let Γ and Γ be two topologically-equivalent planar straight-line drawings of the same planar graph G. Then a morph M is a sequence Γ 1 , Γ 2 , . . . , Γ k of planar straight-line drawings of G such that Γ 1 = Γ , Γ k = Γ , and Γ i , Γ i+1 is a planar linear morph, for each i = 1, . . . , k − 1. A linear morph Γ i , Γ i+1 is such that each vertex moves along a straight-line segment at uniform speed; that is, assuming that the morph happens between time t = 0 and time t = 1, the position of a vertex v at any time t ∈The complexity of a morph M is then measured by the number of its steps, i.e., by the number of linear morphs it consists of.A recent sequence of papers [3,4,5,6] culminated in a proof [2] that a planar morph between any two topologically-equivalent planar straight-line drawings of the same n-vertex planar graph can always be constructed consisting of Θ(n) steps. This bound is asymptotically optimal in the worst case, even for paths.The question we study in this paper is whether morphs with sub-linear complexity can be constructed if a third dimension is allowed to be used. That is: Given two topologically-equivalent planar straight-line drawings Γ and Γ of the same n-vertex planar graph G does a morph M = Γ = Γ 1 , Γ 2 , . . . , Γ k = Γ exist such that: (i) for i = 1, . . . , k, the drawing Γ i is a crossing-free straight-line 3D drawing of G, i.e., a straight-line drawing of G in R 3 such that no two edges cross; (ii) for i = 1, . . . , k − 1, the step Γ i , Γ i+1 is a crossing-free linear morph, i.e., no two edges cross throughout the tr...
We study the question whether a crossing-free 3D morph between two straight-line drawings of an $n$-vertex tree $T$ can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with $O({rpw}(T))\subseteq O(\log n)$ steps, where ${rpw}(T)$ is the rooted pathwidth or Strahler number of $T$, while for the latter setting $\Theta(n)$ steps are always sufficient and sometimes necessary.
Which convex 3D polyhedra can be obtained by gluing several regular hexagons edgeto-edge? It turns out that there are only 15 possible types of shapes, 5 of which are doubly-covered 2D polygons. We give examples for most of them, including all simplicial and all flat shapes, and give a characterization for the latter ones. It is open whether the remaining can be realized.
A data structure is presented that explicitly maintains the graph of a Voronoi diagram of N point sites in the plane or the dual graph of a convex hull of points in three dimensions while allowing insertions of new sites/points. Our structure supports insertions in Õ(N 3/4 ) expected amortized time, where Õ suppresses polylogarithmic terms. This is the first result to achieve sublinear time insertions; previously it was shown by Allen et al. that Θ( √ N) amortized combinatorial changes per insertion could occur in the Voronoi diagram but a sublinear-time algorithm was only presented for the special case of points in convex position.
The Hausdorff Voronoi diagram of clusters of points in the plane is a generalization of Voronoi diagrams based on the Hausdorff distance function. Its combinatorial complexity is O(n + m), where n is the total number of points and m is the number of crossings between the input clusters (m = O(n 2 )); the number of clusters is k. We present efficient algorithms to construct this diagram following the randomized incremental construction (RIC) framework [Clarkson et al. 89, 93]. Our algorithm for non-crossing clusters (m = 0) runs in expected O(n log n + k log n log k) time and deterministic O(n) space. The algorithm for arbitrary clusters runs in expected O((m + n log k) log n) time and O(m + n log k) space. The two algorithms can be combined in a crossing-oblivious scheme within the same bounds. We show how to apply the RIC framework to handle non-standard characteristics of generalized Voronoi diagrams, including sites (and bisectors) of non-constant complexity, sites that are not enclosed in their Voronoi regions, empty Voronoi regions, and finally, disconnected bisectors and disconnected Voronoi regions. The Hausdorff Voronoi diagram finds direct applications in VLSI CAD.
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