Deletion in Abstract Voronoi Diagrams in Expected Linear Time

Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open problem for a long time. Similarly for various concrete Voronoi diagrams of generalized sites, other than points. In this paper we present a simple, expected linear-time algorithm to update an abstract Voronoi diagram after deletion. We introduce the concept of a Voronoi-like diagram, a relaxed version of a Voronoi construct that has a structure similar to an abstract Voronoi diagram, without however being one. Voronoi-like diagrams serve as intermediate structures, which are considerably simpler to compute, thus, making an expected linear-time construction possible. We formalize the concept and prove that it is robust under an insertion operation, thus, enabling its use in incremental constructions

New Results on Abstract Voronoi Diagrams

Voronoi diagrams are a fundamental structure used in many areas of science. For a given set of objects, called sites, the Voronoi diagram separates the plane into regions, such that points belonging to the same region have got the same nearest site. This definition clearly depends on the type of given objects, they may be points, line segments, polygons, etc. and the distance measure used. To free oneself from these geometric notions, Klein introduced abstract Voronoi diagrams as a general construct covering many concrete Voronoi diagrams. Abstract Voronoi diagrams are based on a system of bisecting curves, one for each pair of abstract sites, separating the plane into two dominance regions, belonging to one site each. The intersection of all dominance regions belonging to one site p defines its Voronoi region. The system of bisecting curves is required to fulfill only some simple combinatorial properties, like Voronoi regions to be connected, the union of their closures cover the whole plane, and the bisecting curves are unbounded. These assumptions are enough to show that an abstract Voronoi diagram of n sites is a planar graph of complexity O(n) and can be computed in expected time O(n log n) by a randomized incremental construction. In this thesis we widen the notion of abstract Voronoi diagrams in several senses. One step is to allow disconnected Voronoi regions. We assume that in a diagram of a subset of three sites each Voronoi region may consist of at most s connected components, for a constant s, and show that the diagram can be constructed in expected time O(s2 n ∑3 ≤ j ≤ n mj / j), where mj is the expected number of connected components of a Voronoi region over all diagrams of a subset of j sites. The case that all Voronoi regions are connected is a subcase, where this algorithm performs in optimal O(n log n) time, because here s = mj =1. The next step is to additionally allow bisecting curves to be closed. We present an algorithm constructing such diagrams which runs in expected time O(s2 n log(max{s,n}) ∑2 ≤ j≤ n mj / j). This algorithm is slower by a log n-factor compared to the one for disconnected regions and unbounded bisectors. The extra time is necessary to be able to handle special phenomenons like islands, where a Voronoi region is completely surrounded by another region, something that can occur only when bisectors are closed. However, this algorithm solves many open problems and improves the running time of some existing algorithms, for example for the farthest Voronoi diagram of n simple polygons of constant complexity. Another challenge was to study higher order abstract Voronoi diagrams. In the concrete sense of an order-k Voronoi diagram points are collected in the same Voronoi region, if they have the same k nearest sites. By suitably intersecting the dominance regions this can be defined also for abstract Voronoi diagrams. The question arising is about the complexity of an order-k Voronoi diagram. There are many subsets of size k but fortunately many of them have an empty order-k region. For point sites it has already been shown that there can be at most O(k (n-k)) many regions and even though order-k regions may be disconnected when considering line segments, still the complexity of the order-k diagram remains O(k(n-k)). The proofs used to show this strongly depended on the geometry of the sites and the distance measure, and were thus not applicable for our abstract higher order Voronoi diagrams. The proofs used to show this strongly depended on the geometry of the sites and the distance measure, and were thus not applicable for our abstract higher order Voronoi diagrams. Nevertheless, we were able to come up with proofs of purely topological and combinatorial nature of Jordan curves and certain permutation sequences, and hence we could show that also the order-k abstract Voronoi diagram has complexity O(k (n-k)), assuming that bisectors are unbounded, and the order-1 regions are connected. Finally, we discuss Voronoi diagrams having the shape of a tree or forest. Aggarwal et. al. showed that if points are in convex position, then given their ordering along the convex hull, their Voronoi diagram, which is a tree, can be computed in linear time. Klein and Lingas have generalized this idea to Hamiltonian abstract Voronoi diagrams, where a curve is given, intersecting each Voronoi region with respect to any subset of sites exactly once. If the ordering of the regions along the curve is known in advance, all Voronoi regions are connected, and all bisectors are unbounded, then the abstract Voronoi diagram can be computed in linear time. This algorithm also applies to diagrams which are trees for all subsets of sites and the ordering of the unbounded regions around the diagram is known. In this thesis we go one step further and allow the diagram to be a forest for subsets of sites as long as the complete diagram is a tree. We show that also these diagrams can be computed in linear time

An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams

Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O(k(n-k) log^2 n +n log^3 n) steps, where O(k(n-k)) is the number of faces in the worst case. Due to those axioms, this result applies to disjoint line segments in the L_p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, this kind of run time with a polylog factor to the number of faces was only achieved for point sites in the L_1 or Euclidean metric before

Vesicle computers: Approximating Voronoi diagram on Voronoi automata

Irregular arrangements of vesicles filled with excitable and precipitating chemical systems are imitated by Voronoi automata --- finite-state machines defined on a planar Voronoi diagram. Every Voronoi cell takes four states: resting, excited, refractory and precipitate. A resting cell excites if it has at least one excited neighbour; the cell precipitates if a ratio of excited cells in its neighbourhood to its number of neighbours exceed certain threshold. To approximate a Voronoi diagram on Voronoi automata we project a planar set onto automaton lattice, thus cells corresponding to data-points are excited. Excitation waves propagate across the Voronoi automaton, interact with each other and form precipitate in result of the interaction. Configuration of precipitate represents edges of approximated Voronoi diagram. We discover relation between quality of Voronoi diagram approximation and precipitation threshold, and demonstrate feasibility of our model in approximation Voronoi diagram of arbitrary-shaped objects and a skeleton of a planar shape.Comment: Chaos, Solitons & Fractals (2011), in pres

On the Complexity of Randomly Weighted Voronoi Diagrams

In this paper, we provide an $O(n \mathrm{polylog} n)$ bound on the expected complexity of the randomly weighted Voronoi diagram of a set of $n$ sites in the plane, where the sites can be either points, interior-disjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not their location. This compares favorably with the worst case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal etal [AHKS13] of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems