Browse Theses

We present solutions to three types of geometric problems in the thesis. The first two essentially deal with the visibility inside simple polygons. Two points in a simple polygon $P$ are said to be visible from each other if the line segment that connects them nowhere intersects the exterior of $P$. A line segment can see another point if at least one point of the segment can see, or is visible from, that point. We give efficient algorithms for the following three problems: (1) Find a line segment in $P$, if there is, that can see each point of $P$. Such a line segment can be intuitively called a viewer; (2) Preprocess $P$ such that given a query line segment, find out very quickly if it can see each point of $P$; (3) Compute the shortest line segment in $P$, if there is, that can see each point of $P$.

Furthermore, we place a set of geometric objects (e.g., points or line segments) in $P$ and require our viewer, which can be either a point or a line segment, to see these objects instead of all points of $P$. Under the assumption that the placed objects do not block the visibility of the viewer, we have obtained fast algorithms for several related problems.

The third type of problems is concerned with multi-link paths in simple polygons, a concept that generalizes the notion of visibility into the motion planning area. The link distance between two points inside a simple polygon $P$ is defined to be the minimum number of edges required to form a polygonal path in $P$ that connects the points. We thoroughly explore this notion and investigate several problems of great interest. In particular, we present efficient algorithms for the following problems: (1) Compute the collection of points in $P$ whose maximum link distance to all points of $P$ is minimized over all other points. This collection of points is called the link center of $P$; (2) Preprocess $P$ such that given a point $p$ in $P$, find very quickly a point $q$ of $P$ that is furthest away from $p$ in terms of the link distance. This point $q$ is called a link furthest neighbor of p; (3) Compute a link furthest neighbor for each vertex of $P$;

We conclude with a discussion and some open problems.