Drago Krznaric

Quasi-greedy triangulations approximating the minimum weight triangulation

This article settles the following two longstanding open problems:?What is the worst case approximation ratio between the greedy triangulation and the minimum weight triangulation??Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from the minimum?The answer to the first question is that the known lower bound is tight. The second question is answered in the affirmative by using a slight modification of anO(nlogn) algorithm for the greedy triangulation. We also derive some other interesting results. For example, we show that a constant-factor approximation of the minimum weight convex partition can be obtained within the same time bounds.

Minimum spanning trees in d dimensions

It is shown that a minimum spanning tree of n points in ℝd can be computed in optimal O(Td(n,n)) time under any fixed Lt−metric, where Td (n, m) denotes the time to find a bichromatic closest pair between n red points and m blue points. The previous bound was O(Td (n, n) log n) and it was proved only for the L2 (Euclidean) metric. Furthermore, for d = 3 it is shown that a minimum spanning tree can be found in optimal O(n log n) time under the L1 and L∞-metric. The previous bound was O(n log n log log n).

Optimal algorithms for complete linkage clustering in d dimensions

It is shown that the complete linkage clustering of n points in Rd, where d ≥ 1 is a constant, can be computed in optimal O(nlogn) time and linear space, under the L1 and L∞-metrics. Furthermore, for every other fixed Lt-metric, it is shown that it can be approximated within an arbitrarily small constant factor in O(nlogn) time and linear space.

Fast Algorithms for Complete Linkage Clustering

Abstract. It is shown that the complete linkage clustering of n points can be computed in O(n log2 n) time. Furthermore, it is shown that the complete linkage clustering can be approximated within an arbitrarily small constant factor in O(n log n) time.

Tight lower bounds for minimum weight triangulation heuristics

Abstract The minimum spanning tree heuristic is obtained by optimally triangulating a subgraph of the Delaunay triangulation, whereas the greedy spanning tree heuristic is obtained by optimally triangulating a subgraph of the greedy triangulation. In this paper it is shown that these two known heuristics can produce triangulations that are Ω(n) , respectively Ω(√n) , times longer than the optimum, which are tight bounds.

The greedy triangulation can be computed from the Delaunay triangulation in linear time

Abstract The greedy triangulation of a finite planar point set is obtained by repeatedly inserting a shortest diagonal that does not cross those already in the plane. The Delaunay triangulation, which is the straight-line dual of the Voronoi diagram, can be produced in O( n log n ) worst-case time, and often even faster, by several practical algorithms. In this paper we show that for any planar point set S, if the Delaunay triangulation of S is given, then the greedy triangulation of S can be computed in linear worst-case time (and linear space).

A Linear-Time Approximation Scheme for Minimum Weight Triangulation of Convex Polygons

Abstract. A linear-time heuristic for minimum weight triangulation of convex polygons is presented. This heuristic produces a triangulation of length within a factor 1 + ε from the optimum, where ε is an arbitrarily small positive constant. This is the first subcubic algorithm that guarantees such an approximation factor, and it has interesting applications.