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We revisit a new type of a Voronoi diagram, in which distance is measured from a point to a pair of points. We consider a few more such distance functions, based on geometric primitives, and analyze the structure and complexity of the…
We study Voronoi diagrams for distance functions that add together two convex functions, each taking as its argument the difference between Cartesian coordinates of two planar points. When the functions do not grow too quickly, then the…
In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi…
Given a set of sites in a simple polygon, a geodesic Voronoi diagram of the sites partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point,…
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…
Voronoi diagrams appear in many areas in science and technology and have numerous applications. They have been the subject of extensive investigation during the last decades. Roughly speaking, they are a certain decomposition of a given…
Searching for optimal ways in a network is an important task in multiple application areas such as social networks, co-citation graphs or road networks. In the majority of applications, each edge in a network is associated with a certain…
We present a general framework for computing two-dimensional Voronoi diagrams of different classes of sites under various distance functions. The framework is sufficiently general to support diagrams embedded on a family of two-dimensional…
We consider the Voronoi diagram of lines in $\mathbb{R}^3$ under the Euclidean metric, and give a full classification of its structure in the base case of four lines in general position. We first show that the number of vertices in the…
We study the geodesic Voronoi diagram of a set $S$ of $n$ linearly moving sites inside a static simple polygon $P$ with $m$ vertices. We identify all events where the structure of the Voronoi diagram changes, bound the number of such…
Given a set $S$ of $n$ colored sites, each $s\in S$ associated with a distance-to-site function $\delta_s \colon \mathbb{R}^2 \to \mathbb{R}$, we consider two distance-to-color functions for each color: one takes the minimum of $\delta_s$…
We consider the following variant of the Monge-Kantorovich transportation problem. Let S be a finite set of point sites in d dimensions. A bounded set C in d-dimensional space is to be distributed among the sites p in S such that (i) each p…
In a crowd, individuals make different motion choices such as "moving to destination", "following another pedestrian", and "making a detour". For the sake of convenience, the three direction choices are respectively called destination…
The structure of road networks impacts various urban dynamics, from traffic congestion to environmental sustainability and access to essential services. Recent studies reveal that most roads are underutilized, faster alternative routes are…
In this paper, we propose to compute Voronoi diagrams over mesh surfaces driven by an arbitrary geodesic distance solver, assuming that the input is a triangle mesh as well as a collection of sites $P=\{p_i\}_{i=1}^m$ on the surface. We…
We present an extension of Voronoi diagrams where when considering which site a client is going to use, in addition to the site distances, other site attributes are also considered (for example, prices or weights). A cell in this diagram is…
We present an edge labeling of order-$k$ Voronoi diagrams, $V_k(S)$, of point sets $S$ in the plane, and study properties of the regions defined by them. Among them, we show that $V_k(S)$ has a small orientable cycle and path double cover,…
The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define…
The Voronoi diagram-based dual-front active contour models are known as a powerful and efficient way for addressing the image segmentation and domain partitioning problems. In the basic formulation of the dual-front models, the evolving…
This paper studies real-world road networks from an algorithmic perspective, focusing on empirical studies that yield useful properties of road networks that can be exploited in the design of fast algorithms that deal with geographic data.…