Diamonds are not a minimum weight triangulation's best friend

Two recent methods have increased hopes of finding a polynomial time solution to the problem of computing the minimum weight triangulation of a set S of n points in the plane. Both involve computing what was believed to be a connected or nearly connected subgraph of the minimum weight triangulation, and then completing the triangulation optimally. The first method uses the light graph of S as its initial subgraph. The second method uses the LMT-skeleton of S. Both methods rely, for their polynomial time bound, on the initial subgraphs having only a constant number of components. Experiments performed by the authors of these methods seemed to confirm that randomly chosen point sets displayed this desired property. We show that there exist point sets where the number of components is linear in n. In fact, the expected number of components in either graph on a randomly chosen point set is linear in n, and the probability of the number of components exceeding some constant times n tends to one

Studies of several tetrahedralization problems

The main purpose of decomposing an object into simpler components is to simplify a problem involving the complex object into a number of subproblems having simpler components. In particular, a tetrahedralization is a partition of the input domain in R3 into a number of tetrahedra that meet only at shared faces. Tetrahedralizations have applications in the finite element method, mesh generation, computer graphics, and robotics. This thesis investigates four problems in tetrahedralizations and triangulations. The first problem is on the computational complexity of tetrahedralization detections. We present an O(nm log n) algorithm to determine whether a set of line segments .C is the edge set of a tetrahedralization, where m is the number of segments and n is the number of endpoints in .C. We show that it is NP-complete to decide whether .C contains the edge set of a tetrahedralization. We also show that it is NP-complete to decide whether .C is tetrahedralizable. The second problem is on minimal tetrahedralizations. After deriving some properties of the graph of polyhedra, we identify a class of polyhedra and show that this class of polyhedra can be minimally tetrahedralized in O(n²) time. The third problem is on the tetrahedralization of two nested convex polyhedra. We give a method to tetrahedralize the region between two nested convex polyhedra into a linear number of tetrahedra without introducing Steiner points. This result answers an open problem raised by Bern [16]. The fourth problem is on the lower bound for β-skeletons belonging to minimum weight triangulations. We prove a lower bound on β (β = [one sixth times the square root of two times the square root of 3] + 45 such that if β is less than this value, the β-skeleton of a point set may not always be a subgraph of the minimum weight triangulation of this point set. This result settles Keil's conjecture [62]