Showing papers by "Jeff Erickson published in 2021"
Proceedings Article•
10 Jan 2021
TL;DR: In this article, the first algorithm to morph graphs on the torus was presented, which computes a continuous deformation from one drawing to the other, such that all edges are geodesics at all times.
Abstract: We present the first algorithm to morph graphs on the torus. Given two isotopic essentially 3-connected embeddings of the same graph on the Euclidean flat torus, where the edges in both drawings are geodesics, our algorithm computes a continuous deformation from one drawing to the other, such that all edges are geodesics at all times. Previously even the existence of such a morph was not known. Our algorithm runs in $O(n^{1+\omega/2})$ time, where $\omega$ is the matrix multiplication exponent, and the computed morph consists of $O(n)$ parallel linear morphing steps. Existing techniques for morphing planar straight-line graphs do not immediately generalize to graphs on the torus; in particular, Cairns' original 1944 proof and its more recent improvements rely on the fact that every planar graph contains a vertex of degree at most 5. Our proof relies on a subtle geometric analysis of 6-regular triangulations of the torus. We also make heavy use of a natural extension of Tutte's spring embedding theorem to torus graphs.
4 citations
Posted Content•
TL;DR: In this paper, the authors presented simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater's asymmetric extension of Tutte's classical spring embedding theorem.
Abstract: We present simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater's asymmetric extension of Tutte's classical spring-embedding theorem. First, we give a much simpler algorithm to construct piecewise-linear morphs between planar straight-line graphs. Specifically, given isomorphic straight-line drawings $\Gamma_0$ and $\Gamma_1$ of the same 3-connected planar graph $G$, with the same convex outer face, we construct a morph from $\Gamma_0$ to $\Gamma_1$ that consists of $O(n)$ unidirectional morphing steps, in $O(n^{1+\omega/2})$ time. Our algorithm entirely avoids the classical edge-collapsing strategy dating back to Cairns; instead, in each morphing step, we interpolate the pair of weights associated with a single edge. Second, we describe a natural extension of barycentric interpolation to geodesic graphs on the flat torus. Barycentric interpolation cannot be applied directly in this setting, because the linear systems defining intermediate vertex positions are not necessarily solvable. We describe a simple scaling strategy that circumvents this issue. Computing the appropriate scaling requires $O(n^{\omega/2})$ time, after which we can can compute the drawing at any point in the morph in $O(n^{\omega/2})$ time. Our algorithm is considerably simpler than the recent algorithm of Chambers et al. (arXiv:2007.07927) and produces more natural morphs. Our techniques also yield a simple proof of a conjecture of Connelly et al. for geodesic torus triangulations.
1 citations
Posted Content•
TL;DR: In this article, a human and a dog can be tracked on a simple closed curve in the plane, and the human can walk along the curve at bounded speed and change direction as desired.
Abstract: We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others' work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time.
29 Jun 2021
TL;DR: In this article, it was shown that the set of fusible numbers, ordered by the usual order on R, is well-ordered, with order type e 0, and that the density of the Fusible Numbers along the real line grows at an incredibly fast rate.
Abstract: Inspired by a mathematical riddle involving fuses, we define the fusible numbers as follows: 0 is fusible, and whenever x, y are fusible with |y - x| x + y + 1)/2 is also fusible. We prove that the set of fusible numbers, ordered by the usual order on R, is well-ordered, with order type e0. Furthermore, we prove that the density of the fusible numbers along the real line grows at an incredibly fast rate: Letting g(n) be the largest gap between consecutive fusible numbers in the interval [n, ∞), we have g(n)-1 ≥ Fe0(n - c) for some constant c, where Fα denotes the fast-growing hierarchy. Finally, we derive some true statements that can be formulated but not proven in Peano Arithmetic, of a different flavor than previously known such statements: PA cannot prove the true statement "For every natural number n there exists a smallest fusible number larger than n." Also, consider the algorithm "M(x): if x x, else return M(x - M(x - 1))/2." Then M terminates on real inputs, although PA cannot prove the statement "M terminates on all natural inputs."