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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 $O((n^4+n^2 h^4)alpha(n))$ time, respectively, where $alpha(cdot)$ 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 Euclidean problems. Our algorithms are based on several interesting observations on $L_1$ shortest paths in polygonal domains.
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.
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 poin
Let $mathcal{P}$ be a polygonal domain of $h$ holes and $n$ vertices. We study the problem of constructing a data structure that can compute a shortest path between $s$ and $t$ in $mathcal{P}$ under the $L_1$ metric for any two query points $s$ and $
Given a simple polygon $P$ and a set $Q$ of points contained in $P$, we consider the geodesic $k$-center problem where we want to find $k$ points, called emph{centers}, in $P$ to minimize the maximum geodesic distance of any point of $Q$ to its close
In 2015, Driemel, Krivov{s}ija and Sohler introduced the $(k,ell)$-median problem for clustering polygonal curves under the Frechet distance. Given a set of input curves, the problem asks to find $k$ median curves of at most $ell$ vertices each that