We consider the problem of automatically cleaning massive sonar data point clouds, that is, the problem of automatically removing noisy points that for example appear as a result of scans of (shoals of) fish, multiple reflections, scanner self-reflections, refraction in gas bubbles, and so on.We describe a new algorithm that avoids the problems of previous local-neighbourhood based algorithms. Our algorithm is theoretically I/O-efficient, that is, it is capable of efficiently processing massive sonar point clouds that do not fit in internal memory but must reside on disk. The algorithm is also relatively simple and thus practically efficient, partly due to the development of a new simple algorithm for computing the connected components of a graph embedded in the plane. A version of our cleaning algorithm has already been incorporated in a commercial product.

Range reporting on categorical (or colored) data is a well-studied generalization of the classical range reporting problem in which each of the N input points has an associated color (category). A query then asks to report the set of colors of the points in a given rectangular query range, which may be far smaller than the set of all points in the query range.We study two-dimensional categorical range reporting in both the word-RAM and I/O-model. For the I/O-model, we present two alternative data structures for three-sided queries. The first answers queries in optimal O(lg B N + K/B) I/Os using O(N lg * N ) space, where K is the number of distinct colors in the output, B is the disk block size, and lg * N is the iterated logarithm of N . Our second data structure uses linear space and answers queries in O(lg B N + lg (h) N + K/B) I/Os for any constant integer h ≥ 1. Here lg (1) N = lg N and lg (h) N = lg(lg (h−1) N ) when h > 1. Both solutions use only comparisons on the coordinates. We also show that the lg B N terms in the query costs can be reduced to optimal lg lg B U when the input points lie on a U × U grid and we allow word-level manipulations of the coordinates. We further reduce the query time to just O(1) if the points are given on an N × N grid. Both solutions also lead to improved data structures for four-sided queries. For the word-RAM, we obtain optimal data structures for three-sided range reporting, as well as improved upper bounds for four-sided range reporting.Finally, we show a tight lower bound on onedimensional categorical range counting using an elegant reduction from (standard) two-dimensional range counting.

Given a set of n points in the plane, range diameter queries ask for the furthest pair of points in a given axis-parallel rectangular range. We provide evidence for the hardness of designing space-efficient data structures that support range diameter queries by giving a reduction from the set intersection problem. The difficulty of the latter problem is widely acknowledged and is conjectured to require nearly quadratic space in order to obtain constant query time, which is matched by known data structures for both problems, up to polylogarithmic factors. We strengthen the evidence by giving a lower bound for an important subproblem arising in solutions to the range diameter problem: computing the diameter of two convex polygons, that are separated by a vertical line and are preprocessed independently, requires almost linear time in the number of vertices of the smaller polygon, no matter how much space is used. We also show that range diameter queries can be answered much more efficiently for the case of points in convex position by describing a data structure of size O(n log n) that supports queries in O(log n) time.

No abstract