Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

Dynamic Convex Hulls under Window-Sliding Updates

We consider the problem of dynamically maintaining the convex hull of a set $S$ of points in the plane under the following special sequence of insertions and deletions (called {\em window-sliding updates}): insert a point to the right of all points of $S$ and delete the leftmost point of $S$. We propose an $O(|S|)$-space data structure that can handle each update in $O(1)$ amortized time, such that standard binary-search-based queries on the convex hull of $S$ can be answered in $O(\log h)$ time, where $h$ is the number of vertices of the convex hull of $S$, and the convex hull itself can be output in $O(h)$ time.Comment: A previous version appeared in WADS 2023, where the query time was O(log |S|). This new version improves the query time to O(log h

Improved time-space trade-offs for computing Voronoi diagrams

Let P be a planar set of n sites in general position. For k∈{1,…,n−1}, the Voronoi diagram of order k for P is obtained by subdividing the plane into cells such that points in the same cell have the same set of nearest k neighbors in P. The (nearest site) Voronoi diagram (NVD) and the farthest site Voronoi diagram (FVD) are the particular cases of k=1 and k=n−1, respectively. For any given K∈{1,…,n−1}, the family of all higher-order Voronoi diagrams of order k=1,…,K for P can be computed in total time O(nK2+nlogn) using O(K2(n−K)) space [Aggarwal et al., DCG'89; Lee, TC'82]. Moreover, NVD and FVD for P can be computed in O(nlogn) time using O(n) space [Preparata, Shamos, Springer'85]. For s∈{1,…,n} , an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words, of Θ(logn) bits each, for reading and writing intermediate data. The output can be written only once and cannot be accessed or modified afterwards. We describe a deterministic s -workspace algorithm for computing NVD and FVD for P that runs in O((n2/s)logs) time. Moreover, we generalize our s-workspace algorithm so that for any given K∈O(s√), we compute the family of all higher-order Voronoi diagrams of order k=1,…,K for P in total expected time O(n2K5s(logs+K2O(log∗K))) or in total deterministic time O(n2K5s(logs+KlogK)). Previously, for Voronoi diagrams, the only known s-workspace algorithm runs in expected time O((n2/s)logs+nlogslog∗s) [Korman et al., WADS'15] and only works for NVD (i.e., k=1). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques

Computational geometry through the information lens

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 111-117).This thesis revisits classic problems in computational geometry from the modern algorithmic perspective of exploiting the bounded precision of the input. In one dimension, this viewpoint has taken over as the standard model of computation, and has led to a powerful suite of techniques that constitute a mature field of research. In two or more dimensions, we have seen great success in understanding orthogonal problems, which decompose naturally into one dimensional problems. However, problems of a nonorthogonal nature, the core of computational geometry, have remained uncracked for many years despite extensive effort. For example, Willard asked in SODA'92 for a o(nlg n) algorithm for Voronoi diagrams. Despite growing interest in the problem, it was not successfully solved until this thesis. Formally, let w be the number of bits in a computer word, and consider n points with O(w)-bit rational coordinates. This thesis describes: * a data structure for 2-d point location with O(n) space, and 0( ... )query time. * randomized algorithms with running time 9 ... ) for 3-d convex hull, 2-d Voronoi diagram, 2-d line segment intersection, and a variety of related problems. * a data structure for 2-d dynamic convex hull, with O ( ... )query time, and O ( ... ) update time. More generally, this thesis develops a suite of techniques for exploiting bounded precision in geometric problems, hopefully laying the foundations for a rejuvenated research direction.by Mihai Pǎtraşcu.S.M