Duru Türkoğlu | Searcharxiv

arXiv2020

The Stretch Factor of Hexagon-Delaunay Triangulations

The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation is one of the longstanding open problems in computational geometry. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for computing the exact stretch factor of ``easier'' types of Delaunay triangulations, in particular those defined using regular-polygons instead of a circle. Tight bounds exist for Delaunay triangulations defined using an equilateral triangle and a square. In this paper, we determine the exact stretch factor of Delaunay triangulations defined using a regular hexagon: It is 2. We think that the main contribution of this paper are the two techniques we have developed to compute tight upper bounds for the stretch factor of Hexagon-Delaunay triangulations.

cs.CG↗

arXiv2017

Edge-Based Wedge Sampling to Estimate Triangle Counts in Very Large Graphs

The number of triangles in a graph is useful to deduce a plethora of important features of the network that the graph is modeling. However, finding the exact value of this number is computationally expensive. Hence, a number of approximation algorithms based on random sampling of edges, or wedges (adjacent edge pairs) have been proposed for estimating this value. We argue that for large sparse graphs with power-law degree distribution, random edge sampling requires sampling large number of edges before providing enough information for accurate estimation, and existing wedge sampling methods lead to biased samplings, which in turn lead to less accurate estimations. In this paper, we propose a hybrid algorithm between edge and wedge sampling that addresses the deficiencies of both approaches. We start with uniform edge sampling and then extend each selected edge to form a wedge that is more informative for estimating the overall triangle count. The core estimate we make is the number of triangles each sampled edge in the first phase participates in. This approach provides accurate approximations with very small sampling ratios, outperforming the state-of-the-art up to 8 times in sample size while providing estimations with 95% confidence.

cs.DS↗