743 research outputs found
On lines, joints, and incidences in three dimensions
AbstractWe extend (and somewhat simplify) the algebraic proof technique of Guth and Katz (2010) [9], to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum possible number of incidences between n lines in R3 and m of their joints (points incident to at least three non-coplanar lines) is Θ(m1/3n) for m⩾n, and Θ(m2/3n2/3+m+n) for m⩽n. (ii) In particular, the number of such incidences cannot exceed O(n3/2). (iii) The bound in (i) also holds for incidences between n lines and m arbitrary points (not necessarily joints), provided that no plane contains more than O(n) points and each point is incident to at least three lines. As a preliminary step, we give a simpler proof of (an extension of) the bound O(n3/2), established by Guth and Katz, on the number of joints in a set of n lines in R3. We also present some further extensions of these bounds, and give a trivial proof of Bourgain's conjecture on incidences between points and lines in 3-space, which is an immediate consequence of our incidence bounds, and which constitutes a much simpler alternative to the proof of Guth and Katz (2010) [9]
On the Computational Complexity of Non-dictatorial Aggregation
We investigate when non-dictatorial aggregation is possible from an
algorithmic perspective, where non-dictatorial aggregation means that the votes
cast by the members of a society can be aggregated in such a way that the
collective outcome is not simply the choices made by a single member of the
society. We consider the setting in which the members of a society take a
position on a fixed collection of issues, where for each issue several
different alternatives are possible, but the combination of choices must belong
to a given set of allowable voting patterns. Such a set is called a
possibility domain if there is an aggregator that is non-dictatorial, operates
separately on each issue, and returns values among those cast by the society on
each issue. We design a polynomial-time algorithm that decides, given a set
of voting patterns, whether or not is a possibility domain. Furthermore, if
is a possibility domain, then the algorithm constructs in polynomial time
such a non-dictatorial aggregator for . We then show that the question of
whether a Boolean domain is a possibility domain is in NLOGSPACE. We also
design a polynomial-time algorithm that decides whether is a uniform
possibility domain, that is, whether admits an aggregator that is
non-dictatorial even when restricted to any two positions for each issue. As in
the case of possibility domains, the algorithm also constructs in polynomial
time a uniform non-dictatorial aggregator, if one exists. Then, we turn our
attention to the case where is given implicitly, either as the set of
assignments satisfying a propositional formula, or as a set of consistent
evaluations of an sequence of propositional formulas. In both cases, we provide
bounds to the complexity of deciding if is a (uniform) possibility domain.Comment: 21 page
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
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
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
Specificity prediction of adenylation domains in nonribosomal peptide synthetases (NRPS) using transductive support vector machines (TSVMs)
We present a new support vector machine (SVM)-based approach to predict the substrate specificity of subtypes of a given protein sequence family. We demonstrate the usefulness of this method on the example of aryl acid-activating and amino acid-activating adenylation domains (A domains) of nonribosomal peptide synthetases (NRPS). The residues of gramicidin synthetase A that are 8 Å around the substrate amino acid and corresponding positions of other adenylation domain sequences with 397 known and unknown specificities were extracted and used to encode this physico-chemical fingerprint into normalized real-valued feature vectors based on the physico-chemical properties of the amino acids. The SVM software package SVM(light) was used for training and classification, with transductive SVMs to take advantage of the information inherent in unlabeled data. Specificities for very similar substrates that frequently show cross-specificities were pooled to the so-called composite specificities and predictive models were built for them. The reliability of the models was confirmed in cross-validations and in comparison with a currently used sequence-comparison-based method. When comparing the predictions for 1230 NRPS A domains that are currently detectable in UniProt, the new method was able to give a specificity prediction in an additional 18% of the cases compared with the old method. For 70% of the sequences both methods agreed, for <6% they did not, mainly on low-confidence predictions by the existing method. None of the predictive methods could infer any specificity for 2.4% of the sequences, suggesting completely new types of specificity
Searching edges in the overlap of two plane graphs
Consider a pair of plane straight-line graphs, whose edges are colored red
and blue, respectively, and let n be the total complexity of both graphs. We
present a O(n log n)-time O(n)-space technique to preprocess such pair of
graphs, that enables efficient searches among the red-blue intersections along
edges of one of the graphs. Our technique has a number of applications to
geometric problems. This includes: (1) a solution to the batched red-blue
search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an
algorithm to compute the maximum vertical distance between a pair of 3D
polyhedral terrains one of which is convex in O(n log n) time, where n is the
total complexity of both terrains; (3) an algorithm to construct the Hausdorff
Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n)
time and O(n+m) space, where n is the total number of points in all clusters
and m is the number of crossings between all clusters; (4) an algorithm to
construct the farthest-color Voronoi diagram of the corners of n axis-aligned
rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle
problem for n parallel line segments in the plane in optimal O(n log n) time.
All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure
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
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