Magdalene Grantson

Minimum weight triangulation by cutting out triangles

We describe a fixed parameter algorithm for computing the minimum weight triangulation (MWT) of a simple polygon with (n–k) vertices on the perimeter and k hole vertices in the interior, that is, for a total of n vertices. Our algorithm is based on cutting out empty triangles (that is, triangles not containing any holes) from the polygon and processing the parts or the rest of the polygon recursively. We show that with our algorithm a minimum weight triangulation can be found in time at most O(n3k ! k), and thus in O(n3) if k is constant. We also note that k! can actually be replaced by bk for some constant b. We implemented our algorithm in Java and report experiments backing our analysis.

Covering a set of points with a minimum number of lines

We consider the minimum line covering problem: given a set S of n points in the plane, we want to find the smallest number l of straight lines needed to cover all n points in S. We show that this problem can be solved in O(n log l) time if l ∈ O(log1−en), and that this is optimal in the algebraic computation tree model (we show that the Ω(nlog l) lower bound holds for all values of l up to

A Fixed Parameter Algorithm for Minimum Weight Triangulation: Analysis and Experiments

O(\sqrt n)

Fixed Parameter Algorithms for Minimum Weight Partitions

). Furthermore, a O(log l)-factor approximation can be found within the same O(n log l) time bound if

Bounds on Optimally Triangulating Connected Subsets of the Minimum Weight Convex Partition

l \in O(\sqrt[4]{n})

Tight time bounds for the minimum local convex partition problem

. For the case when l ∈ Ω(log n) we suggest how to improve the time complexity of the exact algorithm by a factor exponential in l.