12,238 research outputs found
The Price of Order
We present tight bounds on the spanning ratio of a large family of ordered
-graphs. A -graph partitions the plane around each vertex into
disjoint cones, each having aperture . An ordered
-graph is constructed by inserting the vertices one by one and
connecting each vertex to the closest previously-inserted vertex in each cone.
We show that for any integer , ordered -graphs with
cones have a tight spanning ratio of . We also show that for any integer , ordered
-graphs with cones have a tight spanning ratio of . We provide lower bounds for ordered -graphs with and cones. For ordered -graphs with and
cones these lower bounds are strictly greater than the worst case spanning
ratios of their unordered counterparts. These are the first results showing
that ordered -graphs have worse spanning ratios than unordered
-graphs. Finally, we show that, unlike their unordered counterparts,
the ordered -graphs with 4, 5, and 6 cones are not spanners
Spanning Properties of Theta-Theta Graphs
We study the spanning properties of Theta-Theta graphs. Similar in spirit
with the Yao-Yao graphs, Theta-Theta graphs partition the space around each
vertex into a set of k cones, for some fixed integer k > 1, and select at most
one edge per cone. The difference is in the way edges are selected. Yao-Yao
graphs select an edge of minimum length, whereas Theta-Theta graphs select an
edge of minimum orthogonal projection onto the cone bisector. It has been
established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio
11.67, for k' >= 6. In this paper we establish a first spanning ratio of
for Theta-Theta graphs, for the same values of . We also extend the class of
Theta-Theta spanners with parameter 6k', and establish a spanning ratio of
for k' >= 5. We surmise that these stronger results are mainly due to a
tighter analysis in this paper, rather than Theta-Theta being superior to
Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta
graphs decreases to 4.64 as k' increases to 8. These are the first results on
the spanning properties of Theta-Theta graphs.Comment: 20 pages, 6 figures, 3 table
Trajectory Planning on Grids: Considering Speed Limit Constraints
Trajectory (path) planning is a well known and thoroughly studied field
of automated planning. It is usually used in computer games, robotics or autonomous
agent simulations. Grids are often used for regular discretization of continuous
space. Many methods exist for trajectory (path) planning on grids, we
address the well known A* algorithm and the state-of-the-art Theta* algorithm.
Theta* algorithm, as opposed to A*, provides ‘any-angle‘ paths that look more realistic.
In this paper, we provide an extension of both these algorithms to enable
support for speed limit constraints.We experimentally evaluate and thoroughly discuss
how the extensions affect the planning process showing reasonability and justification
of our approach
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