The Voronoi diagram is a fundamental geometric structure that encodes proximity information. Given a set of geometric objects, called sites, their Voronoi diagram is a subdivision of the underlying space into maximal regions, such that all points within one region have the same nearest site. Problems in diverse application domains (such as VLSI CAD, robotics, facility location, etc.) demand var...

Given n point sites in the plane each painted in one of k colors, the region of a c-colored site p in the Farthest Color Voronoi Diagram (FCVD) contains all points of the plane for which c is the farthest color and p the nearest c-colored site. This novel structure generalizes both the standard Voronoi diagram (k = 1) and the farthest site Voronoi diagram (k = n). We show that the FCVD has stru...

We present linear-time algorithms to construct tree-like Voronoi diagrams with disconnected regions after the sequence of their faces along an enclosing boundary (or at infinity) is known. We focus on the farthest-segment Voronoi diagram, however, our techniques are also applicable to constructing the order-(k+1) subdivision within an order-k Voronoi region of segments and updating a nearest-ne...

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log n) time algorithm to compute it. We also prove a number of structural properties of this ...

Consider the continuum of points along the edges of a network, an embedded undirected graph with positive edge weights. Distance between these points can be measured as shortest path distance along the edges of the network. We introduce two new concepts to capture farthest-point information in this metric space. The first, eccentricity diagrams, are used to encode the distance towards farthest ...

We present an algorithm to compute the geodesic $$L_1$$ farthest-point Voronoi diagram of m point sites in presence n rectangular obstacles plane. It takes $$O(nm+n \log + m\log m)$$ construction time using O(nm) space. This is first optimal for constructing obstacles. can construct a data structure same and space that answers farthest-neighbor query $$O(\log (n+m))$$ time.

We show that the planar dual to the Euclidean farthest point Voronoi diagram for the set of vertices of a convex polygon has the lexicographic minimum possible sequence of triangle angles, sorted from sharpest to least sharp. As a consequence, the sharpest angle determined by three vertices of a convex polygon can be found in linear time.

We investigate higher-order Voronoi diagrams in the city metric. This metric is induced by quickest paths in the L1 metric in the presence of an accelerating transportation network of axis-parallel line segments. For the structural complexity of k-order city Voronoi diagrams of n point sites, we show an upper bound of O(k(n − k) + kc) and a lower bound of Ω(n+ kc), where c is the complexity of ...

In this paper, we propose an algorithm for computing the farthest-segment Voronoi diagram for the edges of a convex polygon and apply this to obtain an O(n logn) algorithm for the following proximity problem: Given a set P of n (>2) points in the plane, we have O(n2) implicitly defined segments on pairs of points. For each point p ∈ P , find a segment from this set of implicitly defined segment...

We present linear-time algorithms to construct tree-structured Voronoi diagrams, after the sequence of their regions at infinity or along a given boundary is known. We focus on Voronoi diagrams of line segments, including the farthest-segment Voronoi diagram, the order-(k+1) subdivision within a given order-k Voronoi region, and deleting a segment from a nearest-neighbor diagram. Although tree-...