An efficient method to improve the quality of tetrahedron mesh with MFRC

In this paper, we proposed a novel operation to reconstruction tetrahedrons within a certain region, which we call MFRC (Multi-face reconstruction). During the existing tetrahedral mesh improvement methods, the flip operation is one of the very important components. However, due to the limited area affected by the flip, the improvement of the mesh quality by the flip operation is also very limited. The proposed MFRC algorithm solves this problem. MFRC can reconstruct the local mesh in a larger range and can find the optimal tetrahedron division in the target area within acceptable time complexity. Therefore, based on the MFRC algorithm, we combined other operations including smoothing, edge removal, face removal, and vertex insertion/deletion to develop an effective mesh quality improvement method. Numerical experiments of dozens of meshes show that the algorithm can effectively improve the low-quality elements in the tetrahedral mesh, and can effectively reduce the running time, which has important significance for the quality improvement of large-scale mesh.

REOPTIMIZATION UNDER VERTEX INSERTION: MAX Pk-FREE SUBGRAPH AND MAX PLANAR SUBGRAPH

The reoptimization issue studied in this paper can be described as follows: given an instance I of some problem Π, an optimal solution OPT in I and an instance I' resulting from a local perturbation of I that consists of insertions or removals of a small number of data, we wish to use OPT to solve Π in I', either optimally or by guaranteeing an approximation ratio better than that guaranteed by an ex nihilo computation and with running time better than that needed for such a computation. We use this setting to study weighted versions of MAX Pk-FREE SUBGRAPH and MAX PLANAR SUBGRAPH, which are representatives of a broad class of problems known in the literature as maximum induced hereditary subgraph problems. We also show that the techniques presented allow us to handle BIN PACKING.

DYNAMIZATION OF THE TRAPEZOID METHOD FOR PLANAR POINT LOCATION IN MONOTONE SUBDIVISIONS

We present a fully dynamic data structure for point location in a monotone subdivision, based on the trapezoid method. The operations supported are insertion and deletion of vertices and edges, and horizontal translation of vertices. Let n be the current number of vertices of the subdivision. Point location queries take O( log n) time, while updates take O ( log 2 n) time (amortized for vertex insertion/deletion and worst-case for the other updates). The space requirement is O(n log n). This is the first fully dynamic point location data structure for monotone subdivisions that achieves optimal query time.