Let $\mathcal{P}$ be a simple polygon with $m$ vertices and let $P$ be a set of $n$ points inside~$\mathcal{P}$. We show that for any $\varepsilon>0$ there exists a subset $C \subseteq P$ of size $O(1/\varepsilon^2)$ such that, for every query point $q \in \mathcal{P}$, the geodesic distance from $q$ to its furthest neighbor in $C$ is at least $(1-\varepsilon)$ times the distance to its furthest neighbor in $P$. We call such a set $C$ an $\varepsilon$-coreset for furthest-neighbor queries. Using $C$, we can answer $\varepsilon$-approximate furthest-neighbor queries with a data structure whose storage requirement is independent of $n$. The coreset can be constructed in $O\left(\frac{1}{\varepsilon} \left(n\log(1/\varepsilon) + (n+m)\log(n+m)\right) \right)$ time.
A set is star-shaped if there is a point in the set that can see every other point in the set in the sense that the line-segment connecting the points lies within the set. We show that testing whether a non-empty compact smooth region is star-shaped is $\forall\mathbb{R}$-complete. Since the obvious definition of star-shapedness has logical form $\exists\forall$, this is a somewhat surprising result, based on Krasnosel'skiĭ's theorem from convex geometry; we study several related complexity classifications in the real hierarchy based on other results from convex geometry.
Given a set P of n weighted points and a set H of n half-planes in the plane, the hitting set problem is to compute a subset P ' of points from P such that each half-plane contains at least one point from P ' and the total weight of the points in P ' is minimized. The previous best algorithm solves the problem in O(n7/2 log2 n) time. In this paper, we present a new algorithm with runtime O(n5/2 log2 n). For the unweighted case where all points have the same weight, our algorithm runs in O(n log n) time. This improves the previous best algorithm of O(n3 log n) time by a quadratic factor and matches an ohm(n log n) time lower bound under the algebraic decision tree model.
The Planar Separator Theorem, which states that any planar graph G has a separator consisting of O(root n) nodes whose removal partitions G into components of size at most 2n/3, is a widely used tool to obtain fast algorithms on planar graphs. Intersection graphs of disks, which generalize planar graphs, do not admit such separators. It has recently been shown that disk graphs do admit so-called clique-based separators that consist of O(root n) cliques. This result has been generalized to intersection graphs of various other types of disk-like objects. Unfortunately, segment intersection graphs do not admit small clique-based separators, because they can contain arbitrarily large bicliques. This is true even in the simple case of axis-aligned segments. In this paper we therefore introduce biclique-based separators (and, in particular, star-based separators), which are separators consisting of a small number of bicliques (or stars). We prove that any c-oriented set of n segments in the plane, where c is a constant, admits a star-based separator consisting of O(root n) stars. In fact, our result is more general, as it applies to any set of n pseudo-segments that is partitioned into c subsets such that the pseudo-segments in the same subset are pairwise disjoint. We extend our result to intersection graphs of c-oriented polygons. These results immediately lead to an almost-exact distance oracle for such intersection graphs, which has O(n root n) storage and O(root n) query time, and that can report the hop-distance between any two query nodes in the intersection graph with an additive error of at most 2. This is the first distance oracle for such types of intersection graphs that has subquadratic storage and sublinear query time and that only has an additive error. 2012 ACM Subject Classification Theory of computation -> Design and analysis of algorithms
A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if we know, for each vertex, the number of neighbors in each of the four cardinal directions, the triangulation is not completely determined. In fact, we show that counting such triangulations is #P-hard via a reduction from #3-regular bipartite planar vertex cover.
