956 research outputs found
A Fire Fighter's Problem
Suppose that a circular fire spreads in the plane at unit speed. A single
fire fighter can build a barrier at speed . How large must be to
ensure that the fire can be contained, and how should the fire fighter proceed?
We contribute two results.
First, we analyze the natural curve \mbox{FF}_v that develops when the
fighter keeps building, at speed , a barrier along the boundary of the
expanding fire. We prove that the behavior of this spiralling curve is governed
by a complex function , where and are real
functions of . For all zeroes are complex conjugate
pairs. If denotes the complex argument of the conjugate pair nearest to
the origin then, by residue calculus, the fire fighter needs
rounds before the fire is contained. As decreases towards these two
zeroes merge into a real one, so that argument goes to~0. Thus, curve
\mbox{FF}_v does not contain the fire if the fighter moves at speed .
(That speed is sufficient for containing the fire has been proposed
before by Bressan et al. [7], who constructed a sequence of logarithmic spiral
segments that stay strictly away from the fire.)
Second, we show that any curve that visits the four coordinate half-axes in
cyclic order, and in inreasing distances from the origin, needs speed
, the golden ratio, in order to contain the fire.
Keywords: Motion Planning, Dynamic Environments, Spiralling strategies, Lower
and upper boundsComment: A preliminary version of the paper was presented at SoCG 201
09171 Abstracts Collection -- Adaptive, Output Sensitive, Online and Parameterized Algorithms
From 19.01. to 24.04.2009, the Dagstuhl Seminar
09171 ``Adaptive, Output Sensitive, Online and Parameterized Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Searching with an Autonomous Robot
We discuss online strategies for visibility-based
searching for an object hidden behind a corner,
using Kurt3D, a real autonomous mobile robot. This task
is closely related to a number of well-studied problems.
Our robot uses a three-dimensional laser scanner in a
stop, scan, plan, go fashion for building a virtual
three-dimensional environment.
Besides planning trajectories and avoiding obstacles, Kurt3D is capable of identifying objects like a chair.
We derive a practically useful and asymptotically
optimal strategy that guarantees a competitive ratio of 2,
which differs remarkably from the well-studied scenario
without the need of stopping for surveying the environment. Our strategy is used by Kurt3D, documented in a separate video
Online Searching with an Autonomous Robot
We discuss online strategies for visibility-based searching for an object
hidden behind a corner, using Kurt3D, a real autonomous mobile robot. This task
is closely related to a number of well-studied problems. Our robot uses a
three-dimensional laser scanner in a stop, scan, plan, go fashion for building
a virtual three-dimensional environment. Besides planning trajectories and
avoiding obstacles, Kurt3D is capable of identifying objects like a chair. We
derive a practically useful and asymptotically optimal strategy that guarantees
a competitive ratio of 2, which differs remarkably from the well-studied
scenario without the need of stopping for surveying the environment. Our
strategy is used by Kurt3D, documented in a separate video.Comment: 16 pages, 8 figures, 12 photographs, 1 table, Latex, submitted for
publicatio
Growth through the Bottleneck of Limited Budgets
The interplay between supply and demand in the context of growth processes is examined. It is demonstrated that economic development is also significantly influenced by demand under classic, exchange-based hypotheses. Growth assumes that economic operators are willing to change the composition of their budgets. Any additional product must pass through the confines of an existing budget. Growth is not possible without a change in the preferences of solvent buyers. Factors like inventions, changes in society, new fashions or amended regulations can facilitate the integration of additional products. The willingness or resistance to change of potential buyers is thus a possible element in explaining the development of economies that do not fully utilize their production potential. Changes in the structure of demand can cause an economy to grow or shrink. Increases in demand funded by broadly spread savings that are ultimately recompensed – i.e. in form of a new matching of exchange relationships – create growth. Conversely, savings concentrated on certain products that lead to money saved being spent diffusely result in negative growth. In order to assess whether changes in state or state-influenced demand, technical innovations, shifts in society or wage adjustments are conducive to growth, the question has to be asked whether the integration of additional products into the budget is ultimately being encouraged. Government decisions, whether as a fiscal body or setter of standards, can influence the structure of demand and hence economic developments, strategically or unintentionally.Wachstum; Nachfrage; Nachfragestruktur; Fiskalpolitik; neoklassisch
A tight upper bound for the path length of AVL trees
AbstractWe prove that the internal path length of an AVL tree of size N is bounded from above by 1.4404N(log2 N-log2log2N)+O(N) and show that this bound is achieved by an infinite family of AVL trees, each tree of which is not of maximal height. These results carry over to the comparison cost of brother trees
The limit of Voronoi diagrams as is the bounding-box-area Voronoi diagram
We consider the Voronoi diagram of points in the real plane when the distance
between two points and is given by where We prove that the Voronoi diagram has a limit as
converges to zero from above or from below: it is the diagram that corresponds
to the distance function . In this diagram, the bisector of
two points in general position consists of a line and two branches of a
hyperbola that split the plane into three faces per point. We propose to name
as defined above the "geometric distance".Comment: 15 pages, 13 figure
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