Mikko Koivisto, Valentin Polishchuk
We show that the geodesic diameter of a polygonal domain with n vertices can be computed in O(n^4 log n) time by considering O(n^3) candidate diameter endpoints; the endpoints are a subset of vertices of the overlay of shortest path maps from vertices of the domain.
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For a polygonal domain with $h$ holes and a total of $n$ vertices, we present algorithms that compute the $L_1$ geodesic diameter in $O(n^2+h^4)$ time and the $L_1$ geodesic center in $O((n^4+n^2 h^4)α(n))$ time, respectively, where $α(\cdot)$ denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in $O(n^{7.73})$ or $O(n^7(h+\log n))$ time, and compute the geodesic center in $O(n^{11}\log n)$ time. Therefore, our algorithms are significantly faster than the algorithms for the …
In this paper, we study the problem of computing Euclidean geodesic centers of a polygonal domain $\mathcal{P}$ with a total of $n$ vertices. We discover many interesting observations. We give a necessary condition for a point being a geodesic center. We show that there is at most one geodesic center among all points of $\mathcal{P}$ that have topologically-equivalent shortest path maps. This implies that the total number of geodesic centers is bounded by the combinatorial size of the shortest path map equivalence decomposition of $\mathcal{P}$, which is known to be $O(n^{10})$. One key observ…
We present an algorithm that computes the geodesic center of a given polygonal domain. The running time of our algorithm is $O(n^{12+ε})$ for any $ε>0$, where $n$ is the number of corners of the input polygonal domain. Prior to our work, only the very special case where a simple polygon is given as input has been intensively studied in the 1980s, and an $O(n \log n)$-time algorithm is known by Pollack et al. Our algorithm is the first one that can handle general polygonal domains having one or more polygonal holes.
This paper studies the geodesic diameter of polygonal domains having h holes and n corners. For simple polygons (i.e., h = 0), the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time, as known by Hershberger and Suri. For general polygonal domains with h >= 1, however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time O(n^7.73) or O(n^7 (log n + h)). The main difficulty unlike the si…
We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of $n$ vertices and $h$ holes. We introduce a \emph{graph of oriented distances} to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in $\min \{\,O(n^ω), O(n^2 + nh \log h + χ^2)\,\}$ time, where $ω<2.373$ denotes the matrix multiplication exponent and $χ\in Ω(n)\cap O(n^2)$ is the number of edges of the gra…
We unveil an alluring alternative to parametric search that applies to both the non-geodesic and geodesic Fréchet optimization problems. This randomized approach is based on a variant of red-blue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search. We present the first algorithm for the geodesic Fréchet distance between two polygonal curves $A$ and $B$ inside a simple bounding polygon $P$. The geodesic Fréchet decision problem is solved almost as fast as its non-geodesic sibling and requires $O(N^{2\log k)$ time and $O(k+N)$ space afte…
Maximal Distortion of Geodesic Diameters in Polygonal Domains
研究了多边形域中测地直径与欧几里得直径的比值，证明了在具有凸孔的凸多边形中，该比值的上界随孔数变化而变化，并探讨了不同条件下该比值的性质。
The geodesic between two points $a$ and $b$ in the interior of a simple polygon~$P$ is the shortest polygonal path inside $P$ that connects $a$ to $b$. It is thus the natural generalization of straight line segments on unconstrained point sets to polygonal environments. In this paper we use this extension to generalize the concept of the order type of a set of points in the Euclidean plane to geodesic order types. In particular, we show that, for any set $S$ of points and an ordered subset $\mathcal{B} \subseteq S$ of at least four points, one can always construct a polygon $P$ such that the p…
In this paper, we show that the $L_1$ geodesic diameter and center of a simple polygon can be computed in linear time. For the purpose, we focus on revealing basic geometric properties of the $L_1$ geodesic balls, that is, the metric balls with respect to the $L_1$ geodesic distance. More specifically, in this paper we show that any family of $L_1$ geodesic balls in any simple polygon has Helly number two, and the $L_1$ geodesic center consists of midpoints of shortest paths between diametral pairs. These properties are crucial for our linear-time algorithms, and do not hold for the Euclidean …
Given a point $s$ and a set of $h$ pairwise disjoint polygonal obstacles of totally $n$ vertices in the plane, we present a new algorithm for building an $L_1$ shortest path map of size O(n) in $O(T)$ time and O(n) space such that for any query point $t$, the length of the $L_1$ shortest obstacle-avoiding path from $s$ to $t$ can be reported in $O(\log n)$ time and the actual shortest path can be found in additional time proportional to the number of edges of the path, where $T$ is the time for triangulating the free space. It is currently known that $T=O(n+h\log^{1+ε}h)$ for an arbitrarily sm…
Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ defined with $n$ vertices, we compute a sketch $Ω$ of $\mathcal{P}$ whose size is independent of $n$, depending only on $h$ and the input parameter $ε$. We utilize $Ω$ to compute a $(1+ε)$-approximate geodesic shortest path between the two given points in $O(n + h((\lg{n}) + (\lg{h})^{1+δ} + (\frac{1}ε\lg{\frac{h}ε})))$ time. Here, $ε$ is a user parameter, and $δ$ is a small positive constant (resulting from the time for triangulating the free space of $\cal P$ using the algorithm in \cite{journals/…
We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other grap…
The Complexity of Geodesic Spanners
该研究针对受限二维环境下的几何t-生成树问题，提出生成树复杂度的新概念，给出简单多边形和带洞多边形域下不同松弛程度测地线生成树的构造算法与复杂度上界，并证明了生成树复杂度的下界。
The Geodesic Fréchet Distance Between Two Curves Bounding a Simple Polygon
本文研究界定简单多边形的两条曲线之间的测地弗雷歇距离，提出了任意精度下的近线性时间近似算法，并在凸多边形情形下给出线性时间精确算法，显著改进了此前的2-近似结果。
Motivated by the $k$-center problem in location analysis, we consider the \emph{polygon burning} (PB) problem: Given a polygonal domain $P$ with $h$ holes and $n$ vertices, find a set $S$ of $k$ vertices of $P$ that minimizes the maximum geodesic distance from any point in $P$ to its nearest vertex in $S$. Alternatively, viewing each vertex in $S$ as a site to start a fire, the goal is to select $S$ such that fires burning simultaneously and uniformly from $S$, restricted to $P$, consume $P$ entirely as quickly as possible. We prove that PB is NP-hard when $k$ is arbitrary. We show that the di…
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