We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs.

/pdf/spanning-trees-and-spanners-1t65aye0of.pdf

Spanning Trees and Spanners

A set ofn weighted points in general position in źd defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at mostO(nlogn+n[d/2]). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor logn more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coin-flips performed by the algorithm.

Incremental topological flipping works for regular triangulations

1 Partially supported by the Deutsche Forschungsgemeinschaft, grant Kl 655 2-2.

Chapter 5 – Voronoi Diagrams*

We give a linear-time algorithm for computing the medial axis of a simple polygon P . This answers a long-standing open question—previously, the best deterministic algorithm ran in O(n log n) time. We decompose P into pseudonormal histograms, then influence histograms, then xy monotone histograms. We can compute the medial axes for xy monotone histograms and merge to obtain the medial axis for P .

/pdf/finding-the-medial-axis-of-a-simple-polygon-in-linear-time-3j6b6d7c1q.pdf

Finding the Medial Axis of a Simple Polygon in Linear Time

We prove four results on randomized incremental constructions (RICs):


an analysis of the expected behavior under insertion and deletions,


a fully dynamic data structure for convex hull maintenance in arbitrary dimensions,


a tail estimate for the space complexity of RICs,


a lower bound on the complexity of a game related to RICs.

Four results on randomized incremental constructions

We show that the abstract Voronoi diagram ofn sites in the plane can be constructed in timeO(n logn) by a randomized algorithm. This yields an alternative, but simpler,O(n logn) algorithm in many previously considered cases and the firstO(n logn) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of bisecting curves and were recently introduced by Klein [13]. Our algorithm is based on Clarkson and Shor's randomized incremental construction technique [7].

/pdf/on-the-construction-of-abstract-voronoi-diagrams-2nvaicm7id.pdf

St. Meiser

Papers

On the construction of abstract voronoi diagrams