We revisit the problem of computing Fréchet distance between polygonal curves under L 1 , L 2 , and L ∞ norms, focusing on discrete Fréchet distance, where only distance between vertices is considered. We develop efficient algorithms for two natural classes of curves. In particular, given two polygonal curves of n vertices each, a ε-approximation of their discrete Fréchet distance can be computed in roughly O(nκ 3 log n/ε 3 ) time in three dimensions, if one of the curves is κ-bounded. Previously, only a κ-approximation algorithm was known. If both curves are the socalled backbone curves, which are widely used to model protein backbones in molecular biology, we can ε-approximate their Fréchet distance in near linear time in two dimensions, and in roughly O(n 4/3 log nm) time in three dimensions. In the second part, we propose a pseudo-output-sensitive algorithm for computing Fréchet distance exactly. The complexity of the algorithm is a function of a quantity we call the number of switching cells, which is quadratic in the worst case, but tends to be much smaller in practice.
Let F be a collection of n d-variate, possibly part ially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2-faces of the lower envelope (i.e., pointwise minimum) of .F in expected time O(n~+c), for any c > 0. For d = 3, by combining this algorithm with the point location technique of Preparataand Tamassia, we can compute, in randomized expected time 0(n3+E ), for any E > 0, a data structure of size O(n3+') that, given any query point q, can determine in 0(log2 n) time whether q lies above, below or on the envelope.As a consequence, we obtain improved algorithmic solutions to many problems in computational geometry, including (a) computing the width of a point set in 3-space, (b) computing the biggest stick in *
Nearest-neighbor queries, which ask for returning the nearest neighbor of a query point in a set of points, are important and widely studied in many fields because of a wide range of applications. In many of these applications, such as sensor databases, location based services, face recognition, and mobile data, the location of data is imprecise. We therefore study nearest neighbor queries in a probabilistic framework in which the location of each input point and/or query point is specified as a probability density function and the goal is to return the point that minimizes the expected distance, which we refer to as the expected nearest neighbor (ENN). We present methods for computing an exact ENN or an ε-approximate ENN, for a given error parameter 0 < ε < 1, under different distance functions. These methods build an index of near-linear size and answer ENN queries in polylogarithmic or sublinear time, depending on the underlying function. As far as we know, these are the first nontrivial methods for answering exact or ε-approximate ENN queries with provable performance guarantees.
Guth [13] showed that given a family S of n g-dimensional semi-algebraic sets in R d and an integer parameter D ≥ 1, there is a d-variate partitioning polynomial P of degree at most D, so that each connected component of R d \ Z(P ) intersects O(n/D d−g ) sets from S. Such a polynomial is called a generalized partitioning polynomial. We present a randomized algorithm that efficiently computes such a polynomial P . Specifically, the expected running time of our algorithm is only linear in |S|, where the constant of proportionality depends on d, D, and the complexity of the description of S. Our approach exploits the technique of quantifier elimination combined with that of ε-samples.We present four applications of our result. The first is a data structure for answering pointlocation queries among a family of semi-algebraic sets in R d in O(log n) time; the second is data structure for answering range search queries with semi-algebraic ranges in O(log n) time; the third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in R d in O(log 2 n) time; and the fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments, i.e., into Jordan arcs, each pair of which intersect at most once.
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