In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing. 1 Introduction The matching and analysis of geometric patterns and shapes is of importance in various application areas, in particular in computer vision and pattern recognition, but also in other disciplines concerned with the form of objects such as cartography, molecular biology, and computer animation. The general situation is that we are given two objects A, B and want to know how much they resemble each other. Usually one of the objects may undergo certain transformations like translations, rotations or scalings in order to be matched with th...

The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

We consider rule sets for internet packet routing and filtering, where each rule consists of a range of source addresses, a range of destination addresses, a priority, and an action. A given packet should be handled by the action from the maximum priority rule that matches its source and destination. We describe new data structures for quickly finding the rule matching an incoming packet, in near-linear space, and a new algorithm for determining whether a rule set contains any conflicts, in time O(n 3/2 ). 1 Introduction The working of the current Internet and its posited evolution depend on efficient packet filtering mechanisms: databases of rules, maintained at various parts of the network, which use patterns to filter out sets of IP packets and specify actions to be performed on those sets. Typical filter patterns are based on packet header information such as the source or destination IP addresses. The actions to be performed depend on where the packet filtering is performed i...

Weintroduce a new method for finding several types of optimal k-point sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex k-vertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area k-point sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.

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Patterns used for supersampling in graphics have been analyzed from statistical and signal-processing viewpoints. We present an analysis based on a type of isotropic discrepancy---how good patterns are at estimating the area in a region of defined type. We present algorithms for computing discrepancy relative to regions that are defined by rectangles, halfplanes, and higher-dimensional figures. Experimental evidence shows that popular supersampling patterns have discrepancies with better asymptotic behavior than random sampling, which is not inconsistent with theoretical bounds on discrepancy.

Analytic placement methods that simultaneously minimize wire length and spread cells are receiving renewed attention from both academia and industry. In this paper, we describe the implementation details of a force-directed placer, FDP. Specifically, we provide (1) a description of efficient force computation for spreading cells, (2) an illustration of numerical instability in these methods and a means by which these instabilities are avoided, (3) spread metrics for measuring cell distribution throughout the placement region and (4) a complementary technique which aids in directly minimizing HPWL. We present results comparing our analytic placer to other academic tools for both standard cell and mixed-size designs. Compared to Kraftwerk and Capo 8.7, our tool produces results with an average improvement of 9 % and 3%, respectively.

We present in this paper a simple yet efficient algorithm for stabbing a set S of n axis-parallel boxes in d-dimensional space with c(S) points in output-sensitive time O(dn log c(S)) and linear space. Let c ∗ (S) and b ∗ (S) be, respectively, the minimum number of points required to stab S and the maximum number of pairwise disjoint boxes of S. We prove that b ∗ (S)6c ∗ (S)6c(S)6b ∗ (S)(1+log2 b ∗ (S)) d−1. Since nding a minimal set of c ∗ (S) points is NP-complete as soon as d¿1, we obtain a fast precision-sensitive heuristic for stabbing S whose quality does not depend on the input size. In the case of congruent or constrained isothetic boxes, our algorithm reports, respectively, c(S)62 d−1 b ∗ (S) and c(S)=Od(b ∗ (S)) stabbing points. Moreover, we show that the bounds we get on c(S) are asymptotically tight and corroborate our results with some experiments. We also describe an optimal output-sensitive algorithm for nding a minimal-size optimal stabbing point-set of intervals. Finally, we conclude with

We consider a system that allows n networked users to share control over a robotic webcamera.

We give the first nontrivial worst-case results for dynamic versions of various basic geometric optimization and measure problems under the semi-online model, where during the insertion of an object we are told when the object is to be deleted. Problems that we can solve with sublinear update time include the Hausdorff distance of two point sets, discrete 1-center, largest empty circle, convex hull volume in three dimensions, volume of the union of axis-parallel cubes, and minimum enclosing rectangle. The decision versions of the Hausdorff distance and discrete 1-center problems can be solved fully dynamically. Some applications are mentioned.