Computational Geometry-Theory and Applications最新文献

In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with $\Delta >0$ unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of n axis-parallel affine subspaces of dimension Δ that yields an $(1+ϵ)$-approximate solution in $O\left(nd\right)$ time. The constants hidden behind $O(\cdot )$ depend only on $\Delta ,ϵ$ and k. This improves the $O\left(n^{2}d\right)$-time algorithm by Eiben et al. (2021) [7] by a factor of n.

We study problems related to colouring families of bottomless rectangles in the plane, in an attempt to improve the polychromatic k-colouring number $m_k^{⁎}$. This number is the smallest m such that any collection of bottomless rectangles can be k-coloured so that any m-fold covered point is covered by all k colours. We show that for many families of bottomless rectangles, such as unit-width bottomless rectangles, or bottomless rectangles whose left corners lie on a line, $m_k^{⁎}$ is linear in k. We present the lower bound $m_k^{⁎}\ge 2k-1$ for general families.

We also investigate semi-online colouring algorithms, which need not colour each vertex immediately, but must maintain a proper colouring. We prove that for many sweeping orders, for any positive integers $m,k$, there is no semi-online algorithm that can k-colour bottomless rectangles presented in that order, so that any m-fold covered point is covered by at least two colours. This holds even for translates of quadrants, and is a corollary of a stronger result for arborescence colourings: Any semi-online colouring algorithm that colours an arborescence presented in post-order may produce arbitrarily long monochromatic paths.

We study how to obtain partial matchings using the block function $M_f$, induced by a morphism f between persistence modules. $M_f$ is defined algebraically and is linear with respect to direct sums of morphisms. We study some interesting properties of $M_f$, and provide a way of obtaining $M_f$ using matrix operations.

Let P be a polygon and $C$ a set of shortcuts, where each shortcut is a directed straight-line segment connecting two vertices of P. A shortcut hull of P is another polygon that encloses P and whose oriented boundary is composed of elements from $C$. We require P and the output shortcut hull to be weakly simple polygons, which we define as a generalization of simple polygons. Shortcut hulls find their application in cartography, where a common task is to compute simplified representations of area features. We aim at a shortcut hull that has a small area and a small perimeter. Our optimization objective is to minimize a convex combination of these two criteria. If no holes in the shortcut hull are allowed, the problem admits a straight-forward solution via computation of shortest paths. For the more challenging case in which the shortcut hull may contain holes, we present a polynomial-time algorithm that is based on computing a constrained, weighted triangulation of the input polygon's exterior. We use this problem as a starting point for investigating further variants, e.g., restricting the number of edges or bends. We demonstrate that shortcut hulls can be used for the schematization of polygons.

We study the orthogonal range reporting and rectangle stabbing problems in moderate dimensions, i.e., when the dimension is $c\log \left(n\right)$ for some constant c. In orthogonal range reporting, the input is a set of n points in d dimensions, and the goal is to store these n points in a data structure such that given a query rectangle, we can report all the input points contained in the rectangle. The rectangle stabbing problem is the “dual” problem where the input is a set of rectangles, and the query is a point.

Our main result is the following: assume using $S\left(n\right)$ space, we can solve either problem in $d=c\log n$ dimensions, $c\ge 4$, using $Q\left(n\right)+O\left(t\right)$ time in the pointer machine model of computation where t is the output size. Then, we show that if the query time is small, that is, $Q\left(n\right)=n^{1-\gamma }$, for $\gamma \ge \frac{2}{2+\log c}$, then the space must be $\Omega \left(n^{1-\gamma }{n}^{\sqrt{c\gamma }/e-o\left(\sqrt{c\gamma }\right)}\right)$. Interestingly, we obtain this lower bound using a non-constructive method, and we show the existence of some codes that generalize a specific aspect of error correction codes. Our result overcomes the shortcomings of the previous lower bounds in the pointer machine model for non-constant dimension [3], [4], [5], [13], as the previous results could not be extended for $d=\Omega \left(\sqrt{\log n}\right)$.

The only known lower bounds for rectangle stabbing, when the dimension is non-constant, are based on conditional lower bounds upon the best-known results on CNF-SAT [21]. Therefore, our lower bound is the first non-trivial unconditional lower bound for orthogonal range reporting and rectangle stabbing with non-constant dimension.

Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. We consider this property for their higher-dimensional analogs, the regular polytopes. Three classes of regular polytopes exist for all dimensions (n-simplex, n-cube, n-orthoplex) and three additional regular polytopes appear only in four-dimensions (24-cell, 120-cell, 600-cell). It was recently proven that all unfoldings of the n-cube yield nets. We extend this to the n-simplex and the 4-orthoplex using the geometry of simplicial chains. Finally, we demonstrate failure of this property for any orthoplex of higher dimension, as well as for the 600-cell, providing counterexamples. We conjecture failure for the two remaining open cases, the 24-cell and the 120-cell.

We consider the problem of untangling a given (non-planar) straight-line circular drawing ${\delta }_G$ of an outerplanar graph $G=(V,E)$ into a planar straight-line circular drawing of G by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph G, it is obvious that such a crossing-free circular drawing always exists and we define the circular shifting number ${\mathrm{shift}}^{\circ }\left({\delta }_G\right)$ as the minimum number of vertices that are required to be shifted in order to resolve all crossings of ${\delta }_G$. We show that the problem Circular Untangling, asking whether ${\mathrm{shift}}^{\circ }\left({\delta }_G\right)\le K$ for a given integer K, is NP-complete. For n-vertex outerplanar graphs, we obtain a tight upper bound of ${\mathrm{shift}}^{\circ }\left({\delta }_G\right)\le n-\lfloor \sqrt{n-2}\rfloor -2$. Moreover, we study the Circular Untangling for almost-planar circular drawings, in which a single edge is involved in all of the crossings. For this problem, we provide a tight upper bound ${\mathrm{shift}}^{\circ }\left({\delta }_G\right)\le \lfloor \frac{n}{2}\rfloor -1$ and present an $O\left(n^2\right)$-time algorithm to compute the circular shifting number of almost-planar drawings.

We compute the shortest sequence of local connectivity modifications that transform a genus 0 quad mesh to a polycube. The modification operations are (dual) loop preserving and thus, we are restricted to quad meshes where loops don't self-intersect and two loops intersect at most twice. The intersection patterns of the loops are encoded in a simplicial complex, which we call loop complex. To formulate the modification search over the loop complex, we characterise polycubes combinatorially and determine dependencies between modifications. We show that the full task can be encoded as a mixed-integer problem that is solved by a commodity MIP-solver. We demonstrate the practical feasibility by a number of examples with varying complexity.

Our aim is to develop dynamic data structures that support k-nearest neighbors (k-NN) queries for a set of n point sites in the plane in $O\left(f\right(n)+k)$ time, where $f\left(n\right)$ is some polylogarithmic function of n. The key component is a general query algorithm that allows us to find the k-NN spread over t substructures simultaneously, thus reducing an $O\left(tk\right)$ term in the query time to $O\left(k\right)$. Combining this technique with the logarithmic method allows us to turn any static k-NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, $O({\log }^{2}n/\log \log n+k)$ query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic geodesic k-NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only k-NN data structure. More generally, we obtain a dynamic planar k-NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions.

The paper presents a discrete variation of the Fréchet distance between closed curves, which can be seen as an approximation of the continuous measure. A rather straightforward approach to compute the discrete Fréchet distance between two closed sequences of m and n points using binary search takes $O(mn\log mn)$ time. We present an algorithm that takes $O(mn{\log }^{⁎}mn)$ time, where ${\log }^{⁎}$ is the iterated logarithm.