809 research outputs found
Fast Frechet Distance Between Curves With Long Edges
Computing the Fr\'echet distance between two polygonal curves takes roughly
quadratic time. In this paper, we show that for a special class of curves the
Fr\'echet distance computations become easier. Let and be two polygonal
curves in with and vertices, respectively. We prove four
results for the case when all edges of both curves are long compared to the
Fr\'echet distance between them: (1) a linear-time algorithm for deciding the
Fr\'echet distance between two curves, (2) an algorithm that computes the
Fr\'echet distance in time, (3) a linear-time
-approximation algorithm, and (4) a data structure that supports
-time decision queries, where is the number of vertices of
the query curve and the number of vertices of the preprocessed curve
Can Computer Algebra be Liberated from its Algebraic Yoke ?
So far, the scope of computer algebra has been needlessly restricted to exact
algebraic methods. Its possible extension to approximate analytical methods is
discussed. The entangled roles of functional analysis and symbolic programming,
especially the functional and transformational paradigms, are put forward. In
the future, algebraic algorithms could constitute the core of extended symbolic
manipulation systems including primitives for symbolic approximations.Comment: 8 pages, 2-column presentation, 2 figure
Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables
We prove weak laws of large numbers and central limit theorems
of Lindeberg type for empirical centres of mass (empirical Fréchet means)
of independent nonidentically distributed random variables taking values in
Riemannian manifolds. In order to prove these theorems we describe and
prove a simple kind of Lindeberg–Feller central approximation theorem for
vector-valued random variables, which may be of independent interest and
is therefore the subject of a self-contained section. This vector-valued result
allows us to clarify the number of conditions required for the central limit
theorem for empirical Fréchet means, while extending its scope
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