Decomposing Polygons into Fat Components

关于脂肪多边形中测地凸集的倍数维度和周长

On the Doubling Dimension and the Perimeter of Geodesically Convex Sets in Fat Polygons

研究了脂肪多边形的倍数维度特性及测地凸集的周长，证明了局部脂肪多边形不一定有有限倍数维度，而(α,β)覆盖多边形则有，且周长与欧几里得直径成正比，进而提出新的算法。

用测地单位球分解简单多边形

Decomposing a Simple Polygon with Geodesic Unit-Balls

研究如何用\(\ell_2\)度量将简单多边形覆盖和分割为具有单位测地半径或直径的部分，针对半径问题给出覆盖和分割的9近似算法，针对直径问题的分割版本给出15近似算法，改进了前人复杂算法。

Decomposing Coverings and the Planar Sensor Cover Problem

We show that a $k$-fold covering using translates of an arbitrary convex polygon can be decomposed into $Ω(k)$ covers (using an efficient algorithm). We generalize this result to obtain a constant factor approximation to the sensor cover problem where the ranges of the sensors are translates of a given convex polygon. The crucial ingredient in this generalization is a constant factor approximation algorithm for a one-dimensional version of the sensor cover problem, called the Restricted Strip Cover (RSC) problem, where sensors are intervals of possibly different lengths. Our algorithm for RSC …

Fat Polygonal Partitions with Applications to Visualization and Embeddings

Let $\mathcal{T}$ be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in $\mathcal{T}$ is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather…

On conflict-free chromatic guarding of simple polygons

We study the problem of colouring the vertices of a polygon, such that every viewer in it can see a unique colour. The goal is to minimise the number of colours used. This is also known as the conflict-free chromatic guarding problem with vertex guards, and is motivated, e.g., by the problem of radio frequency assignment to sensors placed at the polygon vertices. We first study the scenario in which viewers can be all points of the polygon (such as a mobile robot which moves in the interior of the polygon). We efficiently solve the related problem of minimising the number of guards and approxi…

How Many Potatoes are in a Mesh?

We consider the combinatorial question of how many convex polygons can be made by using the edges taken from a fixed triangulation of n vertices. For general triangulations, there can be exponentially many: we show a construction that has Omega(1.5028^n) convex polygons, and prove an O(1.62^n) upper bound in the worst case. If the triangulation is fat (every triangle has its angles lower-bounded by a constant delta>0), then there can be only polynomially many. We also consider the problem of counting convex outerplanar polygons (i.e., they contain no vertices of the triangulation in their inte…