934 research outputs found
Query-points visibility constraint minimum link paths in simple polygons
We study the query version of constrained minimum link paths between two
points inside a simple polygon with vertices such that there is at
least one point on the path, visible from a query point. The method is based on
partitioning into a number of faces of equal link distance from a point,
called a link-based shortest path map (SPM). Initially, we solve this problem
for two given points , and a query point . Then, the proposed
solution is extended to a general case for three arbitrary query points ,
and . In the former, we propose an algorithm with preprocessing
time. Extending this approach for the latter case, we develop an algorithm with
preprocessing time. The link distance of a - path between
, as well as the path are provided in time and , respectively, for the above two cases, where is the number of links
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Diffuse Reflection Diameter in Simple Polygons
We prove a conjecture of Aanjaneya, Bishnu, and Pal that the minimum number
of diffuse reflections sufficient to illuminate the interior of any simple
polygon with walls from any interior point light source is . Light reflecting diffusely leaves a surface in all directions,
rather than at an identical angle as with specular reflections.Comment: To appear in Discrete Applied Mathematic
Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams
We consider preprocessing a set of points in convex position in the
plane into a data structure supporting queries of the following form: given a
point and a directed line in the plane, report the point of that
is farthest from (or, alternatively, nearest to) the point among all points
to the left of line . We present two data structures for this problem.
The first data structure uses space and preprocessing
time, and answers queries in time, for any . The second data structure uses space and
polynomial preprocessing time, and answers queries in time. These
are the first solutions to the problem with query time and
space.
The second data structure uses a new representation of nearest- and
farthest-point Voronoi diagrams of points in convex position. This
representation supports the insertion of new points in clockwise order using
only amortized pointer changes, in addition to -time
point-location queries, even though every such update may make
combinatorial changes to the Voronoi diagram. This data structure is the first
demonstration that deterministically and incrementally constructed Voronoi
diagrams can be maintained in amortized pointer changes per operation
while keeping -time point-location queries.Comment: 17 pages, 6 figures. Various small improvements. To appear in
Algorithmic
Shortest Path Problems on a Polyhedral Surface
We develop algorithms to compute shortest path edge sequences, Voronoi diagrams, the FrƩchet distance, and the diameter for a polyhedral surface
The Visibility Freeze-Tag Problem
In the Freeze-Tag Problem, we are given a set of robots at points inside some metric space. Initially, all the robots are frozen except one. That robot can awaken (or āunfreezeā) another robot by moving to its position, and once a robot is awakened, it can move and help to awaken other robots. The goal is to awaken all the robots in the shortest time. The Freeze-Tag Problem has been studied in different metric spaces: graphs and Euclidean spaces.
In this thesis, we look at the Freeze-Tag Problem in polygons, and we introduce the Visibility Freeze-Tag Problem, where one robot can awaken another robot by āseeingā it. Furthermore, we introduce a variant of the Visibility Freeze-Tag Problem, called the Line/Point Freeze Tag Problem, where each robot lies on an awakening line, and one robot can awaken another robot by touching its awakening line.
We survey the current results for the Freeze-Tag Problem in graphs, Euclidean spaces and polygons. Since the Visibility Freeze-Tag Problem bears some resemblance to the Watchman Route Problem, we also survey the background literature on the Watchman Route Problem. We show that the Freeze-Tag Problem in polygons and the Visibility Freeze-Tag Problem are NP-hard, and we present an O(n)-approximation algorithm for the Visibility Freeze-Tag Problem. For the Line/Point Freeze-Tag Problem, we give a polynomial time algorithm for the special case where all the awakening lines are parallel to each other. We prove that the general case is NP-hard, and we present an O(1)- approximation algorithm
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