Discrete Geometric Path Analysis in Computational Geometry

In this thesis, first, we study the weighted region problem (WRP), which is to compute a geometric shortest path on a weighted partitioning of a plane. Recent results show that WRP is not solvable in any algebraic computation model over rational numbers. Thus, scientists have focused on approximate solutions. We first study the WRP when the input partitioning of space is an arrangement of lines. We provide a technique that makes it possible to apply the existing approximation algorithms for triangulations to arrangements of lines. Then, we formulate two qualitative criteria for weighted short paths. We show how to produce a path that is quantitatively close-to-optimal and qualitatively satisfactory. The results of our experiments carried out on triangular irregular networks show that the proposed algorithm could save, on average, 51% in query time and 69% in memory usage, in comparison with the existing method. We also study some variants of the Frechet distance. The Frechet distance is a commonly used measure to capture the similarity of polygonal curves. Firstly, we study a robust variant of the Frechet distance since the standard Frechet distance exhibits a high sensitivity to the presence of outliers. Secondly, we propose a new measure to capture similarity between polygonal curves, called the Minimum Backward Frechet Distance (MBFD). More specifically, for a given threshold epsilon, we are searching for a pair of walks for two entities on the two input polygonal curves such that the union of the portions of required backward movements is minimized and the distance between the two entities, at any time during the walk, is less than or equal to epsilon. Thirdly, we generalize MBFD to capture scenarios when the cost of backtracking on the input polygonal curves is not homogeneous. Lastly, for a given graph H, a polygonal curve T, and a threshold epsilon, we propose a geometric algorithm that computes a path, P, in H, and a parameterization of T, that minimize the sum of the length of walks on T and P whereby the distance between the entities moving along P and T is at most epsilon.