51 research outputs found
Computational Geometry Column 35
The subquadratic algorithm of Kapoor for finding shortest paths on a
polyhedron is described
Geodesic-Preserving Polygon Simplification
Polygons are a paramount data structure in computational geometry. While the
complexity of many algorithms on simple polygons or polygons with holes depends
on the size of the input polygon, the intrinsic complexity of the problems
these algorithms solve is often related to the reflex vertices of the polygon.
In this paper, we give an easy-to-describe linear-time method to replace an
input polygon by a polygon such that (1)
contains , (2) has its reflex
vertices at the same positions as , and (3) the number of vertices
of is linear in the number of reflex vertices. Since the
solutions of numerous problems on polygons (including shortest paths, geodesic
hulls, separating point sets, and Voronoi diagrams) are equivalent for both
and , our algorithm can be used as a preprocessing
step for several algorithms and makes their running time dependent on the
number of reflex vertices rather than on the size of
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
Efficient motion planning for problems lacking optimal substructure
We consider the motion-planning problem of planning a collision-free path of
a robot in the presence of risk zones. The robot is allowed to travel in these
zones but is penalized in a super-linear fashion for consecutive accumulative
time spent there. We suggest a natural cost function that balances path length
and risk-exposure time. Specifically, we consider the discrete setting where we
are given a graph, or a roadmap, and we wish to compute the minimal-cost path
under this cost function. Interestingly, paths defined using our cost function
do not have an optimal substructure. Namely, subpaths of an optimal path are
not necessarily optimal. Thus, the Bellman condition is not satisfied and
standard graph-search algorithms such as Dijkstra cannot be used. We present a
path-finding algorithm, which can be seen as a natural generalization of
Dijkstra's algorithm. Our algorithm runs in time, where~ and are the number of vertices and
edges of the graph, respectively, and is the number of intersections
between edges and the boundary of the risk zone. We present simulations on
robotic platforms demonstrating both the natural paths produced by our cost
function and the computational efficiency of our algorithm
An Efficient Algorithm for Computing High-Quality Paths amid Polygonal Obstacles
We study a path-planning problem amid a set of obstacles in
, in which we wish to compute a short path between two points
while also maintaining a high clearance from ; the clearance of a
point is its distance from a nearest obstacle in . Specifically,
the problem asks for a path minimizing the reciprocal of the clearance
integrated over the length of the path. We present the first polynomial-time
approximation scheme for this problem. Let be the total number of obstacle
vertices and let . Our algorithm computes in time
a path of total cost
at most times the cost of the optimal path.Comment: A preliminary version of this work appear in the Proceedings of the
27th Annual ACM-SIAM Symposium on Discrete Algorithm
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