A. Naamad

Bio: A. Naamad is an academic researcher from Northwestern University. The author has contributed to research in topics: Nearest neighbor search & Largest empty rectangle. The author has an hindex of 2, co-authored 2 publications receiving 176 citations.

Dynamic Voronoi diagrams

Abstract: A new dynamizing technique is introduced whereby n point Voronoi diagrams (both closest and farthest point) can be updated in O(n) time per insertion or deletion, in the worst case. General properties of these dynamic Voronoi diagrams are explored including a storage/ deletion-time trade-off. In addition, their application to such problems as nearest neighbor search and the 2-minimum spanning circle problem is discussed.

Voronoi diagrams—a survey of a fundamental geometric data structure

Abstract: Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor

Davenport-Schinzel sequences and their geometric applications

Abstract: An $(n,s)$ Davenport--Schinzel sequence, for positive integers $n$ and $s$, is a sequence composed of $n$ symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation $a \cdots b \cdots a \cdots b \cdots$ of length $s+2$ between two distinct symbols $a$ and $b$. The close relationship between Davenport--Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive, because a wide variety of geometric problems can be formulated in terms of lower envelopes. A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems. This paper gives a comprehensive survey on the theory of Davenport--Schinzel sequences and their geometric applications.

Optimal Search in Planar Subdivisions

Abstract: A planar subdivision is any partition of the plane into (possibly unbounded) polygonal regions. The subdivision search problem is the following: given a subdivision $S$ with $n$ line segments and a query point $p$, determine which region of $S$ contains $p$. We present a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage. Our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications.

The farthest point strategy for progressive image sampling

Abstract: A new method of farthest point strategy (FPS) for progressive image acquisition-an acquisition process that enables an approximation of the whole image at each sampling stage-is presented. Its main advantage is in retaining its uniformity with the increased density, providing efficient means for sparse image sampling and display. In contrast to previously presented stochastic approaches, the FPS guarantees the uniformity in a deterministic min-max sense. Within this uniformity criterion, the sampling points are irregularly spaced, exhibiting anti-aliasing properties comparable to those characteristic of the best available method (Poisson disk). A straightforward modification of the FPS yields an image-dependent adaptive sampling scheme. An efficient O(N log N) algorithm for both versions is introduced, and several applications of the FPS are discussed.

Total grain-size distribution and volume of tephra-fall deposits

Abstract: On 17 June 1996, Ruapehu volcano, New Zealand, produced a sustained andesitic sub-Plinian eruption, which generated a narrow tephra-fall deposit extending more than 200 km from the volcano. The extremely detailed data set from this eruption allowed methods for the determination of total grain-size distribution and volume of tephra-fall deposits to be critically investigated. Calculated total grain-size distributions of tephra-fall deposits depend strongly on the method used and on the availability of data across the entire dispersal area. The Voronoi Tessellation method was tested for the Ruapehu deposit and gave the best results when applied to a data set extending out to isomass values of <1 g m−2. The total grain-size distribution of a deposit is also strongly influenced by the very proximal samples, and this can be shown by artificially constructing subsets from the Ruapehu database. Unless the available data set is large, all existing techniques for calculations of total grain-size distribution give only apparent distributions. The tephra-fall deposit from Ruapehu does not show a simple exponential thinning, but can be approximated well by at least three straight-line segments or by a power-law fit on semi-log plots of thickness vs. (area)1/2. Integrations of both fits give similar volumes of about 4×106 m3. Integration of at least three exponential segments and of a power-law fit with at least ten isopach contours available can be considered as a good estimate of the actual volume of tephra fall. Integrations of smaller data sets are more problematic.