48,306 research outputs found
Incremental and Decremental Maintenance of Planar Width
We present an algorithm for maintaining the width of a planar point set
dynamically, as points are inserted or deleted. Our algorithm takes time
O(kn^epsilon) per update, where k is the amount of change the update causes in
the convex hull, n is the number of points in the set, and epsilon is any
arbitrarily small constant. For incremental or decremental update sequences,
the amortized time per update is O(n^epsilon).Comment: 7 pages; 2 figures. A preliminary version of this paper was presented
at the 10th ACM/SIAM Symp. Discrete Algorithms (SODA '99); this is the
journal version, and will appear in J. Algorithm
Approximation Algorithm for Line Segment Coverage for Wireless Sensor Network
The coverage problem in wireless sensor networks deals with the problem of
covering a region or parts of it with sensors. In this paper, we address the
problem of covering a set of line segments in sensor networks. A line segment `
is said to be covered if it intersects the sensing regions of at least one
sensor distributed in that region. We show that the problem of finding the
minimum number of sensors needed to cover each member in a given set of line
segments in a rectangular area is NP-hard. Next, we propose a constant factor
approximation algorithm for the problem of covering a set of axis-parallel line
segments. We also show that a PTAS exists for this problem.Comment: 16 pages, 5 figures
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Fully dynamic maintenance of euclidean minimum spanning trees
We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in time O(n^5/6 log1^2/2 n) per update operation. No nontrivial dynamic geometric minimum spanning tree algorithm was previously known. We reduce the problem to maintaining bichromatic closest pairs, which we also solve in the same time bounds. Our algorithm uses a novel construction, the ordered nearest neighbors of a sequence of points. Any point set or bichromatic point set can be ordered so that this graph is a simple path
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page
Automatic Reconstruction of Fault Networks from Seismicity Catalogs: 3D Optimal Anisotropic Dynamic Clustering
We propose a new pattern recognition method that is able to reconstruct the
3D structure of the active part of a fault network using the spatial location
of earthquakes. The method is a generalization of the so-called dynamic
clustering method, that originally partitions a set of datapoints into
clusters, using a global minimization criterion over the spatial inertia of
those clusters. The new method improves on it by taking into account the full
spatial inertia tensor of each cluster, in order to partition the dataset into
fault-like, anisotropic clusters. Given a catalog of seismic events, the output
is the optimal set of plane segments that fits the spatial structure of the
data. Each plane segment is fully characterized by its location, size and
orientation. The main tunable parameter is the accuracy of the earthquake
localizations, which fixes the resolution, i.e. the residual variance of the
fit. The resolution determines the number of fault segments needed to describe
the earthquake catalog, the better the resolution, the finer the structure of
the reconstructed fault segments. The algorithm reconstructs successfully the
fault segments of synthetic earthquake catalogs. Applied to the real catalog
constituted of a subset of the aftershocks sequence of the 28th June 1992
Landers earthquake in Southern California, the reconstructed plane segments
fully agree with faults already known on geological maps, or with blind faults
that appear quite obvious on longer-term catalogs. Future improvements of the
method are discussed, as well as its potential use in the multi-scale study of
the inner structure of fault zones
Explicit linear kernels via dynamic programming
Several algorithmic meta-theorems on kernelization have appeared in the last
years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of
bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding
a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed
topological minor. Typically, these results guarantee the existence of linear
or polynomial kernels on sparse graph classes for problems satisfying some
generic conditions but, mainly due to their generality, it is not clear how to
derive from them constructive kernels with explicit constants. In this paper we
make a step toward a fully constructive meta-kernelization theory on sparse
graphs. Our approach is based on a more explicit protrusion replacement
machinery that, instead of expressibility in CMSO logic, uses dynamic
programming, which allows us to find an explicit upper bound on the size of the
derived kernels. We demonstrate the usefulness of our techniques by providing
the first explicit linear kernels for -Dominating Set and -Scattered Set
on apex-minor-free graphs, and for Planar-\mathcal{F}-Deletion on graphs
excluding a fixed (topological) minor in the case where all the graphs in
\mathcal{F} are connected.Comment: 32 page
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