Reverse nearest neighbor queries in fixed dimension

Reverse nearest neighbor queries are defined as follows: Given an input point-set P, and a query point q, find all the points p in P whose nearest point in P U {q} \ {p} is q. We give a data structure to answer reverse nearest neighbor queries in fixed-dimensional Euclidean space. Our data structure uses O(n) space, its preprocessing time is O(n log n), and its query time is O(log n).Comment: 7 pages, 3 figures; typos corrected; more background material on compressed quadtree

A deterministic algorithm for fitting a step function to a weighted point-set

Given a set of n points in the plane, each point having a positive weight, and an integer k>0, we present an optimal O(n \log n)-time deterministic algorithm to compute a step function with k steps that minimizes the maximum weighted vertical distance to the input points. It matches the expected time bound of the best known randomized algorithm for this problem. Our approach relies on Cole's improved parametric searching technique.Comment: 5 pages, 2 figure

An Elementary Algorithm for Reporting Intersections of Red/Blue Curve Segments

Let E_r and E_b be two sets of x-monotone and non-intersecting curve segments, E=E_r \cup E_b and |E|=n. We give a new sweep-line algorithm that reports the $k$ intersecting pairs of segments of E. Our algorithm uses only three simple predicates that allow to decide if two segments intersect, if a point is left or right to another point, and if a point is above, below or on a segment. These three predicates seem to be the simplest predicates that lead to subquadratic algorithms. Our algorithm is almost optimal in this restricted model of computation. Its time complexity is $O((n+k)\logn) and it requires O(n) space. The same algoritm has been described in our previous report [5]. That report presented also an algoritm for the general case but its analysis was not not correct

Elementary Algorithms for Reporting Intersections of Curve Segments

We propose several algorithms to report the k intersecting pairs among a set of n curve segments. Apart from the intersection predicate, our algorithms only use two simple predicates : the predicate that compares the coordinates of two points and the predicate that says if a point is below, on, or above a segment. In particular, the predicates we use do not allow to count the number of intersection points nor to sort them, and the time complexity of our algorithms depends on the number of intersectin- g pairs, not on the number of intersection points (differently from the other non trivial algorithms). We present an algorithm for the red-blue variant of the problem where we have a set of blue segments and a set of red segments so that no two segments of the same set intersect. The time complexity is O((n+k)\log n). This algorithm is then used to solve the general case in O(n\sqrtk\log n) time. In the case of pseudo-segments (i.e. segments that intersect in at most one point) we propose a better algorithm whose time complexity is O((k+n)\log n+ n\sqrt k). All our time complexity results are a log factor from optimal

Fluid force and symmetry breaking modes of a 3D bluff body with a base cavity

International audienceA cavity at the base of the squareback Ahmed model at Re 4 × 10 5 is able to reduce the base suction by 18% and the drag coefficient by 9%, while the geometry at the separation remains unaffected. Instantaneous pressure measurements at the body base, fluid force measurements and wake velocity measurements are investigated varying the cavity depth from 0 to 35% of the base height. Due to the reflectional symmetry of the rectangular base, there are two Reflectional Symmetry Breaking (RSB) mirror modes present in the natural wake that switch from one to the other randomly in accordance with the recent findings of Grandemange et al. (2013b). It is shown that these modes exhibit an energetic 3D static vortex system close to the base of the body. A sufficiently deep cavity is able to stabilize the wake toward a symmetry preserved wake, thus suppressing the RSB modes and leading to a weaker elliptical toric recirculation. The stabilization can be modelled with a Langevin equation. The plausible mechanism for drag reduction with the base cavity is based on the interaction of the static 3D vortex system of the RSB modes with the base and their suppression by stabilization. There are some strong evidences that this mechanism may be generalized to axisym-metric bodies with base cavity

Sparse geometric graphs with small dilation

Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position