Fast Frechet Distance Between Curves With Long Edges

Computing the Fr\'echet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fr\'echet distance computations become easier. Let $P$ and $Q$ be two polygonal curves in $\mathbb{R}^d$ with $n$ and $m$ vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fr\'echet distance between them: (1) a linear-time algorithm for deciding the Fr\'echet distance between two curves, (2) an algorithm that computes the Fr\'echet distance in $O((n+m)\log (n+m))$ time, (3) a linear-time $\sqrt{d}$-approximation algorithm, and (4) a data structure that supports $O(m\log^2 n)$-time decision queries, where $m$ is the number of vertices of the query curve and $n$ the number of vertices of the preprocessed curve

Hierarchical Decompositions and Circular Ray Shooting in Simple Polygons

A hierarchical decomposition of a simple polygon is introduced. The hierarchy has depth O(logn), linear size, and its regions have at most three neighbors. Using this hierarchy, circular ray shooting queries in a simple polygon can be answered in O(log 2 n) query time and O(n log n) space. If the radius of the circle is fLxed, the query time can be improved to O(log n) and the space to O(n)

Hierarchical Decompositions and Circular Ray Shooting in Simple Polygons

A hierarchical decomposition of a simple polygon is introduced. The hierarchy has logarithmic depth, linear size, and its regions have at most three neighbors. Using this hierarchy, circular ray shooting queries in a simple polygon on n vertices can be answered in O(log(2) n) query time and O(n log n) space. If the radius of the circle is fixed, the query time can be improved to O(log n) and the space to O(n)