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We present a generalization of a combinatorial result by Aggarwal, Guibas, Saxe and Shor [Discrete & Computational Geometry, 1989] on a linear-time algorithm that selects a constant fraction of leaves, with pairwise disjoint neighborhoods, from a binary tree embedded in the plane. This result of Aggarwal et al. is essential to the linear-time framework, which they also introduced, that computes certain Voronoi diagrams of points with a tree structure in linear time. An example is the diagram computed while updating the Voronoi diagram of points after deletion of one site. Our generalization allows that only a fraction of the tree leaves is considered, and it is motivated by linear-time Voronoi constructions for non-point sites. We are given a plane tree $T$ of $n$ leaves, $m$ of which have been marked, and each marked leaf is associated with a neighborhood (a subtree of $T$) such that any two topologically consecutive marked leaves have disjoint neighborhoods. We show how to select in linear time a constant fraction of the marked leaves having pairwise disjoint neighborhoods., Comment: 17 pages, 9 figures

We study the Voronoi Diagram of Rotating Rays, a Voronoi structure where the input sites are rays and the distance function between a point and a site/ray, is the counterclockwise angular distance. This novel Voronoi diagram is motivated by illumination or coverage problems, where a domain must be covered by floodlights/wedges of uniform angle, and the goal is to find the minimum angle necessary to cover the domain. We study the diagram in the plane, and we present structural properties, combinatorial complexity bounds, and a construction algorithm. If the rays are induced by a convex polygon, we show how to construct the Voronoi diagram within this polygon in linear time. Using this information, we can find in optimal linear time the Brocard angle, the minimum angle required to illuminate a convex polygon with floodlights of uniform angle.

Any system of bisectors (in the sense of abstract Voronoi diagrams) defines an arrangement of simple curves in the plane. We define Voronoi-like graphs on such an arrangement, which are graphs whose vertices are locally Voronoi. A vertex $v$ is called locally Voronoi, if $v$ and its incident edges appear in the Voronoi diagram of three sites. In a so-called admissible bisector system, where Voronoi regions are connected and cover the plane, we prove that any Voronoi-like graph is indeed an abstract Voronoi diagram. The result can be seen as an abstract dual version of Delaunay's theorem on (locally) empty circles. Further, we define Voronoi-like cycles in an admissible bisector system, and show that the Voronoi-like graph induced by such a cycle $C$ is a unique tree (or a forest, if $C$ is unbounded). In the special case where $C$ is the boundary of an abstract Voronoi region, the induced Voronoi-like graph can be computed in expected linear time following the technique of [Junginger and Papadopoulou SOCG'18]. Otherwise, within the same time, the algorithm constructs the Voronoi-like graph of a cycle $C'$ on the same set (or subset) of sites, which may equal $C$ or be enclosed by $C$. Overall, the technique computes abstract Voronoi (or Voronoi-like) trees and forests in linear expected time, given the order of their leaves along a Voronoi-like cycle. We show a direct application in updating a constraint Delaunay triangulation in linear expected time, after the insertion of a new segment constraint, simplifying upon the result of [Shewchuk and Brown CGTA 2015].

Updating an abstract Voronoi diagram after deletion of one site in linear time has been a well-known open problem; similarly, for concrete Voronoi diagrams of non-point sites. In this paper, we present an expected linear-time algorithm to update an abstract Voronoi diagram after deletion of one site. We introduce the concept of a Voronoi-like diagram, a relaxed version of an abstract Voronoi construct that has a structure similar to an ordinary Voronoi diagram, without, however, being one. We formalize the concept, and prove that it is robust under insertion, therefore, enabling its use in incremental constructions. The time-complexity analysis of the resulting simple randomized incremental construction is non-standard, and interesting in its own right, because the intermediate Voronoi-like structures are order-dependent. We further extend the approach to compute the following structures in expected linear time: the order-(k+1) subdivision within an order-k Voronoi region, and the farthest abstract Voronoi diagram after the order of its regions at infinity is known., Comment: 34 pages, 41 figures

This paper applies the randomized incremental construction (RIC) framework to computing the Hausdorff Voronoi diagram of a family of k clusters of points in the plane. The total number of points is n. The diagram is a generalization of Voronoi diagrams based on the Hausdorff distance function. The combinatorial complexity of the Hausdorff Voronoi diagram is O(n+m), where m is the total number of crossings between pairs of clusters. For non-crossing clusters (m=0), our algorithm works in expected O(n log n + k log n log k) time and deterministic O(n) space. For arbitrary clusters (m=O(n^2)), the algorithm runs in expected O((m+n log k) log n) time and O(m +n log k) space. When clusters cross, bisectors are disconnected curves resulting in disconnected Voronoi regions that challenge the incremental construction. This paper applies the RIC paradigm to a Voronoi diagram with disconnected regions and disconnected bisectors, for the first time., Comment: 18 pages, 4 figures

We present an expected linear-time algorithm to construct the farthest-segment Voronoi diagram, given the sequence of its faces at infinity. This sequence forms a Davenport-Schinzel sequence of order 3 and it can be computed in O(n log n) time, where n is the number of input segments. The farthest-segment Voronoi diagram is a tree, with disconnected Voronoi regions, of total complexity Theta(n) in the worst case. Disconnected regions pose a major difficulty in deriving linear-time construction algorithms for such tree-like Voronoi diagrams. In this paper we present a new approach towards this direction., Comment: 20 pages, 13 figures

Surprisingly, the order-$k$ Voronoi diagram of line segments had received no attention in the computational-geometry literature. It illustrates properties surprisingly different from its counterpart for points; for example, a single order-$k$ Voronoi region may consist of $\Omega(n)$ disjoint faces. We analyze the structural properties of this diagram and show that its combinatorial complexity for $n$ non-crossing line segments is $O(k(n-k))$, despite the disconnected regions. The same bound holds for $n$ intersecting line segments, when $k\geq n/2$. We also consider the order-$k$ Voronoi diagram of line segments that form a planar straight-line graph, and augment the definition of an order-$k$ Voronoi diagram to cover non-disjoint sites, addressing the issue of non-uniqueness for $k$-nearest sites. Furthermore, we enhance the iterative approach to construct this diagram. All bounds are valid in the general $L_p$ metric, $1\leq p\leq \infty$. For non-crossing segments in the $L_\infty$ and $L_1$ metrics, we show a tighter $O((n-k)^2)$ bound for $k>n/2$.

In the Hausdorff Voronoi diagram of a family of \emph{clusters of points} in the plane, the distance between a point $t$ and a cluster $P$ is measured as the maximum distance between $t$ and any point in $P$, and the diagram is defined in a nearest-neighbor sense for the input clusters. In this paper we consider %El."non-crossing" \emph{non-crossing} clusters in the plane, for which the combinatorial complexity of the Hausdorff Voronoi diagram is linear in the total number of points, $n$, on the convex hulls of all clusters. We present a randomized incremental construction, based on point location, that computes this diagram in expected $O(n\log^2{n})$ time and expected $O(n)$ space. Our techniques efficiently handle non-standard characteristics of generalized Voronoi diagrams, such as sites of non-constant complexity, sites that are not enclosed in their Voronoi regions, and empty Voronoi regions. The diagram finds direct applications in VLSI computer-aided design., Comment: arXiv admin note: substantial text overlap with arXiv:1306.5838

In the Hausdorff Voronoi diagram of a set of clusters of points in the plane, the distance between a point t and a cluster P is the maximum Euclidean distance between t and a point in P. This diagram has direct applications in VLSI design. We consider so-called "non-crossing" clusters. The complexity of the Hausdorff diagram of m such clusters is linear in the total number n of points in the convex hulls of all clusters. We present randomized incremental constructions for computing efficiently the diagram, improving considerably previous results. Our best complexity algorithm runs in expected time O((n + m(log log(n))^2)log^2(n)) and worst-case space O(n). We also provide a more practical algorithm whose expected running time is O((n + m log(n))log^2(n)) and expected space complexity is O(n). To achieve these bounds, we augment the randomized incremental paradigm for the construction of Voronoi diagrams with the ability to efficiently handle non-standard characteristics of generalized Voronoi diagrams, such as sites of non-constant complexity, sites that are not enclosed in their Voronoi regions, and empty Voronoi regions., Comment: This paper has been withdrawn by the author because the substantially updated version (improved results, major text revision) is now submitted (arXiv:1312.3904)