International Journal of Computational Geometry & Applications

Given a set of n points (known as terminals) and a set of λ ≥ 2 uniformly distributed (legal) orientations in the plane, the uniform orientation Steiner tree problem asks for a minimum-length network that interconnects the terminals with the restriction that the network is composed of line segments using legal orientations only. This problem is also known as the λ-geometry Steiner tree problem. We show that for any fixed λ > 2 the uniform orientation Steiner tree problem is NP-hard. In fact we prove a strictly stronger result, namely that the problem is NP-hard even when the terminals are restricted to lying on two parallel lines.

In this paper, we consider the k-center problem for streaming points in ℝ^d. More precisely, we consider the single-pass streaming model, where each point in the stream is allowed to be examined only once and a small amount of information can be stored in a device. Since the size of memory is much smaller than the size of the data in the streaming model, it is important to develop an algorithm whose space complexity does not depend on the number of input data. We present an approximation algorithm for k = 2 that guarantees a (2 + ε)-factor using O(d/ε) space and update time in arbitrary dimensions for any metric. We show that our algorithm can be extended to approximate an optimal k-center within factor (2 + ε) for k > 2.

We present an algorithm that takes as input a finite point set in ℝ^m, and performs a perturbation that guarantees that the Delaunay triangulation of the resulting perturbed point set has quantifiable stability with respect to the metric and the point positions. There is also a guarantee on the quality of the simplices: they cannot be too at. The algorithm provides an alternative tool to the weighting or refinement methods to re-move poorly shaped simplices in Delaunay triangulations of arbitrary dimension, but in addition it provides a guarantee of stability for the resulting triangulation.

Delaunay and Gabriel graphs are widely studied geometric proximity structures. Motivated by applications in wireless routing, relaxed versions of these graphs known as Locally Delaunay Graphs (LDGs) and Locally Gabriel Graphs (LGGs) were proposed. We propose another generalization of LGGs called Generalized Locally Gabriel Graphs (GLGGs) in the context when certain edges are forbidden in the graph. Unlike a Gabriel Graph, there is no unique LGG or GLGG for a given point set because no edge is necessarily included or excluded. This property allows us to choose an LGG/GLGG that optimizes a parameter of interest in the graph. We show that computing an edge maximum GLGG for a given problem instance is NP-hard and also APX-hard. We also show that computing an LGG on a given point set with dilation ≤ k is NP-hard. Finally, we give an algorithm to verify whether a given geometric graph G = (V,E) is a valid LGG.