5 research outputs found
Facility location on terrains
Given a terrain defined as a piecewise-linear function with n triangles, and m point sites on it, we
would like to identify the location on the terrain that minimizes the maximum distance to the sites. The
distance is measured as the length of the Euclidean shortest path along the terrain. To simplify the
problem somewhat, we extend the terrain to (the surface of) a polyhedron. To compute the optimum
placement, we compute the furthest-site Voronoi diagram of the sites on the polyhedron. The diagram
has maximum combinatorial complexity Q(mn2), and the algorithm runs in O(mn² log²m log n) time
Geometric algorithms for geographic information systems
A geographic information system (GIS) is a software package for storing geographic data and performing complex operations
on the data. Examples are the reporting of all land parcels that will be flooded when a certain river rises above some level, or
analyzing the costs, benefits, and risks involved with the development of industrial activities at some place. A substantial part
of all activities performed by a GIS involves computing with the geometry of the data, such as location, shape, proximity, and
spatial distribution. The amount of data stored in a GIS is usually very large, and it calls for efficient methods to store,
manipulate, analyze, and display such amounts of data. This makes the field of GIS an interesting source of problems to work on
for computational geometers. In chapters 2-5 of this thesis we give new geometric algorithms to solve four selected GIS
problems.These chapters are preceded by an introduction that provides the necessary background, overview, and definitions
to appreciate the following chapters. The four problems that we study in chapters 2-5 are the following:
Subdivision traversal: we give a new method to traverse planar subdivisions without using mark bits or a stack.
Contour trees and seed sets: we give a new algorithm for generating a contour tree for d-dimensional meshes, and use it
to determine a seed set of minimum size that can be used for isosurface generation. This is the first algorithm that
guarantees a seed set of minimum size. Its running time is quadratic in the input size, which is not fast enough for many
practical situations. Therefore, we also give a faster algorithm that gives small (although not minimal) seed sets.
Settlement selection: we give a number of new models for the settlement selection problem. When settlements, such as
cities, have to be displayed on a map, displaying all of them may clutter the map, depending on the map scale. Choices
have to be made which settlements are selected, and which ones are omitted. Compared to existing selection methods,
our methods have a number of favorable properties.
Facility location: we give the first algorithm for computing the furthest-site Voronoi diagram on a polyhedral terrain, and
show that its running time is near-optimal. We use the furthest-site Voronoi diagram to solve the facility location
problem: the determination of the point on the terrain that minimizes the maximal distance to a given set of sites on the
terrain
Algorithms for Geometric Facility Location: Centers in a Polygon and Dispersion on a Line
We study three geometric facility location problems in this thesis.
First, we consider the dispersion problem in one dimension. We are given an ordered list
of (possibly overlapping) intervals on a line. We wish to choose exactly one point from
each interval such that their left to right ordering on the line matches the input order.
The aim is to choose the points so that the distance between the closest pair of points is
maximized, i.e., they must be socially distanced while respecting the order. We give a new
linear-time algorithm for this problem that produces a lexicographically optimal solution.
We also consider some generalizations of this problem.
For the next two problems, the domain of interest is a simple polygon with n vertices.
The second problem concerns the visibility center. The convention is to think of a polygon
as the top view of a building (or art gallery) where the polygon boundary represents opaque
walls. Two points in the domain are visible to each other if the line segment joining them
does not intersect the polygon exterior. The distance to visibility from a source point to a
target point is the minimum geodesic distance from the source to a point in the polygon
visible to the target. The question is: Where should a single guard be located within the
polygon to minimize the maximum distance to visibility? For m point sites in the polygon,
we give an O((m + n) log (m + n)) time algorithm to determine their visibility center.
Finally, we address the problem of locating the geodesic edge center of a simple polygonâa
point in the polygon that minimizes the maximum geodesic distance to any edge. For a
triangle, this point coincides with its incenter. The geodesic edge center is a generalization
of the well-studied geodesic center (a point that minimizes the maximum distance to any
vertex). Center problems are closely related to farthest Voronoi diagrams, which are well-
studied for point sites in the plane, and less well-studied for line segment sites in the plane.
When the domain is a polygon rather than the whole plane, only the case of point sites has
been addressedâsurprisingly, more general sites (with line segments being the simplest
example) have been largely ignored. En route to our solution, we revisit, correct, and
generalize (sometimes in a non-trivial manner) existing algorithms and structures tailored
to work specifically for point sites. We give an optimal linear-time algorithm for finding
the geodesic edge center of a simple polygon
Facility location on terrains
Given a terrain defined as a piecewise-linear function with n triangles, and m point sites on it, we would like to identify the location on the terrain that minimizes the maximum distance to the sites. The distance is measured as the length of the Euclidean shortest path along the terrain. To simplify the problem somewhat, we extend the terrain to (the surface of) a polyhedron. To compute the optimum placement, we compute the furthest-site Voronoi diagram of the sites on the polyhedron. The diagram has maximum combinatorial complexity Î(mn²), and the algorithm runs in O(mn² log² mlogn) time