557 research outputs found
The Maximum-Level Vertex in an Arrangement of Lines
Let be a set of lines in the plane, not necessarily in general position. We present an efficient algorithm for finding all the vertices of the arrangement of maximum level, where the level of a vertex is the number of lines of that pass strictly below . The problem, posed in Exercise~8.13 in de Berg etal [BCKO08], appears to be much harder than it seems, as this vertex might not be on the upper envelope of the lines. We first assume that all the lines of are distinct, and distinguish between two cases, depending on whether or not the upper envelope of contains a bounded edge. In the former case, we show that the number of lines of that pass above any maximum level vertex is only . In the latter case, we establish a similar property that holds after we remove some of the lines that are incident to the single vertex of the upper envelope. We present algorithms that run, in both cases, in optimal time. We then consider the case where the lines of are not necessarily distinct. This setup is more challenging, and the best we have is an algorithm that computes all the maximum-level vertices in time . Finally, we consider a related combinatorial question for degenerate arrangements, where many lines may intersect in a single point, but all the lines are distinct: We bound the complexity of the weighted -level in such an arrangement, where the weight of a vertex is the number of lines that pass through the vertex. We show that the bound in this case is , which matches the corresponding bound for non-degenerate arrangements, and we use this bound in the analysis of one of our algorithms
Conflict-Free Coloring Made Stronger
In FOCS 2002, Even et al. showed that any set of discs in the plane can
be Conflict-Free colored with a total of at most colors. That is,
it can be colored with colors such that for any (covered) point
there is some disc whose color is distinct from all other colors of discs
containing . They also showed that this bound is asymptotically tight. In
this paper we prove the following stronger results:
\begin{enumerate} \item [(i)] Any set of discs in the plane can be
colored with a total of at most colors such that (a) for any
point that is covered by at least discs, there are at least
distinct discs each of which is colored by a color distinct from all other
discs containing and (b) for any point covered by at most discs,
all discs covering are colored distinctively. We call such a coloring a
{\em -Strong Conflict-Free} coloring. We extend this result to pseudo-discs
and arbitrary regions with linear union-complexity.
\item [(ii)] More generally, for families of simple closed Jordan regions
with union-complexity bounded by , we prove that there exists
a -Strong Conflict-Free coloring with at most colors.
\item [(iii)] We prove that any set of axis-parallel rectangles can be
-Strong Conflict-Free colored with at most colors.
\item [(iv)] We provide a general framework for -Strong Conflict-Free
coloring arbitrary hypergraphs. This framework relates the notion of -Strong
Conflict-Free coloring and the recently studied notion of -colorful
coloring. \end{enumerate}
All of our proofs are constructive. That is, there exist polynomial time
algorithms for computing such colorings
Finding a needle in an exponential haystack: Discrete RRT for exploration of implicit roadmaps in multi-robot motion planning
We present a sampling-based framework for multi-robot motion planning which
combines an implicit representation of a roadmap with a novel approach for
pathfinding in geometrically embedded graphs tailored for our setting. Our
pathfinding algorithm, discrete-RRT (dRRT), is an adaptation of the celebrated
RRT algorithm for the discrete case of a graph, and it enables a rapid
exploration of the high-dimensional configuration space by carefully walking
through an implicit representation of a tensor product of roadmaps for the
individual robots. We demonstrate our approach experimentally on scenarios of
up to 60 degrees of freedom where our algorithm is faster by a factor of at
least ten when compared to existing algorithms that we are aware of.Comment: Kiril Solovey and Oren Salzman contributed equally to this pape
Efficient Multi-Robot Motion Planning for Unlabeled Discs in Simple Polygons
We consider the following motion-planning problem: we are given unit
discs in a simple polygon with vertices, each at their own start position,
and we want to move the discs to a given set of target positions. Contrary
to the standard (labeled) version of the problem, each disc is allowed to be
moved to any target position, as long as in the end every target position is
occupied. We show that this unlabeled version of the problem can be solved in
time, assuming that the start and target positions are at
least some minimal distance from each other. This is in sharp contrast to the
standard (labeled) and more general multi-robot motion-planning problem for
discs moving in a simple polygon, which is known to be strongly NP-hard
Flood susceptibility analysis (FSAn) using multi-criteria evaluation (MCE) technique for landuse planning: A case from Penampang, Sabah, Malaysia
Flooding is one of the main natural disasters in Sabah, Malaysia. Several current cases of disastrous flooding were recorded particularly in Penampang area, Sabah (e.g. July 1999; October 2010; April 2013; October & December 2014). Substantial downpour has triggered floods and caused extreme loss in Penampang area. The 2014 floods have affected 40,000 people from 70 villages. The objectives of this research are (i) To determine the factors contributing to the flood occurrences; (ii) To analyst the Flood Susceptibility Level (FSL); and (iii) and to produce the flood hazard map for the study area. In this study, eight (8) parameters were considered in relation to the causative factors to flooding, which are: rainfall, slope gradient, elevation, drainage density, land use, soil textures, slope curvatures and flow accumulation. Flood Susceptibility Analysis (FSAn) map was produced based on the data collected from the field survey, laboratory analysis, high resolution digital radar images (IFSAR) acquisition, and secondary data in year 2014. FSL was defined using Multi Criteria Evaluation (MCE) technique integrated with GIS software. Based on the FSAn, approximately 3.17% of total study area classified as Very High Hazard (VHH), 4.55% as High Hazard (HH), 15.52% as Moderate Hazard (MH), 15.72% as Low Hazard (LH) dan 61.04% as Very Low Hazard (VLH) respectively. Based on the risk rate, requirements for the development procedure has been recommended in this paper. The map produced will be a very useful source for consulting, planning agencies and local governments in managing risk, land-use zoning and redressal efforts to mitigate risks. Besides, the method used in this study can easily be applied to other areas, where other factors may be considered, depending on the convenience of data
Star Unfolding Convex Polyhedra via Quasigeodesic Loops
We extend the notion of star unfolding to be based on a quasigeodesic loop Q
rather than on a point. This gives a new general method to unfold the surface
of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut
along one shortest path from each vertex of P to Q, and cut all but one segment
of Q.Comment: 10 pages, 7 figures. v2 improves the description of cut locus, and
adds references. v3 improves two figures and their captions. New version v4
offers a completely different proof of non-overlap in the quasigeodesic loop
case, and contains several other substantive improvements. This version is 23
pages long, with 15 figure
Simulation-Based Design of Bicuspidization of the Aortic Valve
Objective: Severe congenital aortic valve pathology in the growing patient
remains a challenging clinical scenario. Bicuspidization of the diseased aortic
valve has proven to be a promising repair technique with acceptable durability.
However, most understanding of the procedure is empirical and retrospective.
This work seeks to design the optimal gross morphology associated with surgical
bicuspidization with simulations, based on the hypothesis that modifications to
the free edge length cause or relieve stenosis.
Methods: Model bicuspid valves were constructed with varying free edge
lengths and gross morphology. Fluid-structure interaction simulations were
conducted in a single patient-specific model geometry. The models were
evaluated for primary targets of stenosis and regurgitation. Secondary targets
were assessed and included qualitative hemodynamics, geometric height,
effective height, orifice area and prolapse.
Results: Stenosis decreased with increasing free edge length and was
pronounced with free edge length less than or equal to 1.3 times the annular
diameter d. With free edge length 1.5d or greater, no stenosis occurred. All
models were free of regurgitation. Substantial prolapse occurred with free edge
length greater than or equal to 1.7d.
Conclusions: Free edge length greater than or equal to 1.5d was required to
avoid aortic stenosis in simulations. Cases with free edge length greater than
or equal to 1.7d showed excessive prolapse and other changes in gross
morphology. Cases with free edge length 1.5-1.6d have a total free edge length
approximately equal to the annular circumference and appeared optimal. These
effects should be studied in vitro and in animal studies
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
Bounded Model Checking for Probabilistic Programs
In this paper we investigate the applicability of standard model checking
approaches to verifying properties in probabilistic programming. As the
operational model for a standard probabilistic program is a potentially
infinite parametric Markov decision process, no direct adaption of existing
techniques is possible. Therefore, we propose an on-the-fly approach where the
operational model is successively created and verified via a step-wise
execution of the program. This approach enables to take key features of many
probabilistic programs into account: nondeterminism and conditioning. We
discuss the restrictions and demonstrate the scalability on several benchmarks
- …