828 research outputs found
Relative Convex Hull Determination from Convex Hulls in the Plane
A new algorithm for the determination of the relative convex hull in the
plane of a simple polygon A with respect to another simple polygon B which
contains A, is proposed. The relative convex hull is also known as geodesic
convex hull, and the problem of its determination in the plane is equivalent to
find the shortest curve among all Jordan curves lying in the difference set of
B and A and encircling A. Algorithms solving this problem known from
Computational Geometry are based on the triangulation or similar decomposition
of that difference set. The algorithm presented here does not use such
decomposition, but it supposes that A and B are given as ordered sequences of
vertices. The algorithm is based on convex hull calculations of A and B and of
smaller polygons and polylines, it produces the output list of vertices of the
relative convex hull from the sequence of vertices of the convex hull of A.Comment: 15 pages, 4 figures, Conference paper published. We corrected two
typing errors in Definition 2: has to be defined based on , and
has to be defined based on (not just using ). These errors
appeared in the text of the original conference paper, which also contained
the pseudocode of an algorithm where and appeared as correctly
define
Pruning Algorithms for Pretropisms of Newton Polytopes
Pretropisms are candidates for the leading exponents of Puiseux series that
represent solutions of polynomial systems. To find pretropisms, we propose an
exact gift wrapping algorithm to prune the tree of edges of a tuple of Newton
polytopes. We prefer exact arithmetic not only because of the exact input and
the degrees of the output, but because of the often unpredictable growth of the
coordinates in the face normals, even for polytopes in generic position. We
provide experimental results with our preliminary implementation in Sage that
compare favorably with the pruning method that relies only on cone
intersections.Comment: exact, gift wrapping, Newton polytope, pretropism, tree pruning,
accepted for presentation at Computer Algebra in Scientific Computing, CASC
201
Approximate Data Structures with Applications
In this paper we introduce the notion of approximate
data structures, in which a small amount of error is
tolerated in the output. Approximate data structures
trade error of approximation for faster operation, leading to theoretical and practical speedups for a wide variety of algorithms. We give approximate variants of the van Emde Boas data structure, which support the same dynamic operations as the standard van Emde Boas data structure [28, 201, except that answers to queries are approximate. The variants support all operations in constant time provided the error of approximation is l/polylog(n), and in O(loglog n) time provided the error
is l/polynomial(n), for n elements in the data structure.
We consider the tolerance of prototypical algorithms to approximate data structures. We study in particular Prim’s minimumspanning tree algorithm, Dijkstra’s single-source shortest paths algorithm, and an on-line variant of Graham’s convex hull algorithm. To obtain output which approximates the desired output
with the error of approximation tending to zero, Prim’s algorithm requires only linear time, Dijkstra’s algorithm requires O(mloglogn) time, and the on-line variant of Graham’s algorithm requires constant amortized time per operation
Output-Sensitive Tools for Range Searching in Higher Dimensions
Let be a set of points in . A point is
\emph{-shallow} if it lies in a halfspace which contains at most points
of (including ). We show that if all points of are -shallow, then
can be partitioned into subsets, so that any hyperplane
crosses at most subsets. Given such
a partition, we can apply the standard construction of a spanning tree with
small crossing number within each subset, to obtain a spanning tree for the
point set , with crossing number . This allows us to extend the construction of Har-Peled
and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set
of points in (without the shallowness assumption), a
spanning tree with {\em small relative crossing number}. That is, any
hyperplane which contains points of on one side, crosses
edges of . Using a
similar mechanism, we also obtain a data structure for halfspace range
counting, which uses space (and somewhat higher
preprocessing cost), and answers a query in time , where is the output size
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