Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number γ with 0 < γ < π, we design an O(n)-time algorithm that constructs a connected spanning subgraph G′ of G whose maximum degree is at most 14 + d2π/γe. If G is the Delaunay triangulation of V , and γ = 2π/3, we show that G′ is a t-spanner of V (for some constant t) with maximum degree at most 17...

Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number γ with 0 < γ < π, we design an O(n)-time algorithm that constructs a connected subgraph G′ of G with vertex set V whose maximum degree is at most 14 + d2π/γe. If G is the Delaunay triangulation of V , and γ = 2π/3, we show that G′ is a t-spanner of V (for some constant t) with maximum degree a...

In this paper we describe the architecture of a simple evacuation model which is based on a graph representation of the scene. Such graphs are typically constructed using Medial Axis Transform (MAT) or Straight Medial Axis Transform (S-MAT) transformations, the former being a part of the Voronoi diagram (Dirichlet tessellation) of the floor plan. In our work we construct such graphs for floor p...

A localized Delaunay triangulation owns the following interesting properties for sensor and wireless ad hoc networks: it can be built with localized information, the communication cost imposed by control information is limited, and it supports geographical routing algorithms that offer guaranteed convergence. This paper presents two localized algorithms, FLDT1 and FLDT2, that build a graph call...

We show how to delete a vertex q from a three-dimensional Delaunay triangulation DT(S) in expected O(C⊗(P )) time, where P is the set of vertices neighboring q in DT(S) and C⊗(P ) is an upper bound on the expected number of tetrahedra whose circumspheres enclose q that are created during the randomized incremental construction of DT(P ). Experiments show that our approach is significantly faste...

Given a point set P and a class C of geometric objects, GC(P ) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C ∈ C containing both p and q but no other points from P . We study G5(P ) graphs where 5 is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the...

Given a class C of geometric objects and a point set P , a C-matching of P is a set M = {C1, . . . , Ck} ⊆ C of elements of C such that each Ci contains exactly two elements of P and each element of P lies in at most one Ci. If all of the elements of P belong to some Ci, M is called a perfect matching. If, in addition, all of the elements of M are pairwise disjoint, we say that this matching M ...

We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set P of points in general position in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle 5, and there is an edge between two points in P if and only if there is an empty homothet of 5 having the two points on its boundary. We consider higher-order triangular-di...

We present a new and simple randomized algorithm for constructing the Delaunay triangulation using nearest neighbor graphs for point location. It runs in linear expected time for points in the plane with polynomially bounded spread, i.e., if the ratio between the largest and smallest pointwise distance is polynomially bounded. This also holds for point sets with bounded spread in higher dimensi...

Some of the first routing algorithms for geographically aware wireless networks used the Delaunay triangulation among the network’s nodes as the underlying connectivity graph [4]. These solutions were considered impractical, however, because in general the Delaunay triangulation may contain arbitrarily long edges, and because calculating the Delaunay triangulation generally requires a global vi...