As a measure for the resemblance of curves in arbitrary dimensions we consider the so-called Frechet-distance, which is compatible with parametrizations of the curves. For polygonal chains P and Q consisting of p and q edges an algorithm of runtime O(pq log(pq)) measuring the Frechet-distance between P and Q is developed. Then some important variants are considered, namely the Frechet-distance for closed curves, the nonmonotone Frechet-distance and a distance function derived from the Frechet-distance measuring whether P resembles some part of the curve Q.

Computing the fréchet distance between two polygonal curves

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

A brief historical perspective relating the discovery of dendrimers and other dendritic polymers is presented. Dendritic polymers are recognized as the fourth major class of macromolecular architecture consisting of four sub- classes, namely, (1) random hyperbranched, (2) dendrigrafts, (3) dendrons, and (4) dendrimers. The previous literature is reviewed with anecdotal events leading to implications for dendrimers in the emerging science of nanotechnology. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 2719–2728, 2002

Discovery of dendrimers and dendritic polymers: A brief historical perspective*

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

Unmanned aerial vehicles (UAVs), or aerial drones, are an emerging technology with significant market potential. UAVs may lead to substantial cost savings in, for instance, monitoring of difficult-to-access infrastructure, spraying fields and performing surveillance in precision agriculture, as well as in deliveries of packages. In some applications, like disaster management, transport of medical supplies, or environmental monitoring, aerial drones may even help save lives. In this article, we provide a literature survey on optimization approaches to civil applications of UAVs. Our goal is to provide a fast point of entry into the topic for interested researchers and operations planning specialists. We describe the most promising aerial drone applications and outline characteristics of aerial drones relevant to operations planning. In this review of more than 200 articles, we provide insights into widespread and emerging modeling approaches. We conclude by suggesting promising directions for future research.

Optimization approaches for civil applications of unmanned aerial vehicles (UAVs) or aerial drones: A survey

Let A and B be two sets of n objects in \reals

 d
 , and let Match be a (one-to-one) matching between A and B . Let min(Match ), max(Match ), and Σ(Match) denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of Match , respectively. Bottleneck matching— a matching that minimizes max(Match )— is suggested as a convenient way for measuring the resemblance between A and B . Several algorithms for computing, as well as approximating, this resemblance are proposed. The running time of all the algorithms involving planar objects is roughly O(n

 1.5
 
) . For instance, if the objects are points in the plane, the running time of the exact algorithm is O(n

 1.5
 log  n ) . A semidynamic data structure for answering containment problems for a set of congruent disks in the plane is developed. This data structure may be of independent interest. Next, the problem of finding a translation of B that maximizes the resemblance to A under the bottleneck matching criterion is considered. When A and B are point-sets in the plane, an O(n

 5
 log  n) -time algorithm for determining whether for some translated copy the resemblance gets below a given ρ is presented, thus improving the previous result of Alt, Mehlhorn, Wagener, and Welzl by a factor of almost n . This result is used to compute the smallest such ρ in time O(n

 5
 log 
 2
 n ) , and an efficient approximation scheme for this problem is also given. The uniform matching problem (also called the balanced assignment problem, or the fair matching problem) is to find Match
*
U , a matching that minimizes max (Match)-min(Match) . A minimum deviation matching
Match
*
D is a matching that minimizes (1/n)Σ(Match) - min(Match) . Algorithms for computing Match
*
U and Match
*
D in roughly O(n

 10/3
 
) time are presented. These algorithms are more efficient than the previous O(n

 4
 
) -time algorithms of Martello, Pulleyblank, Toth, and de Werra, and of Gupta and Punnen, who studied these problems for general bipartite graphs.

Geometry helps in bottleneck matching and related problems

The traditional worst-case analysis often fails to predict the actual behavior of the running time of geometric algorithms in practical situations. One reason is that worst-case scenarios are often very contrived and do not occur in practice. To avoid this, models are needed that describe the properties that realistic inputs have, so that the analysis can take these properties into account.

/pdf/realistic-input-models-for-geometric-algorithms-13v9e8pog3.pdf

Realistic input models for geometric algorithms

The traditional worst-case analysis often fails to predict the actual behavior of the running time of geometric algorithms in practical situations. One reason is that worst-case scenarios are often very contrived and do not occur in practice. To avoid this, models are needed that describe the properties that realistic inputs have, so that the analysis can take these properties into account. We try to bring some structure to this emerging research direction. In particular, we present the following results: • We show the relations between various models that have been proposed in the literature. • For several of these models, we give algorithms to compute the model parameter(s) for a given (planar) scene; these algorithms can be used to verify whether a model is appropriate for typical scenes in some application area. • As a case study, we give some experimental results on the appropriateness of some of the models for one particular type of scene often encountered in geographic information systems, namely certain triangulated irregular networks.

/pdf/realistic-input-models-for-geometric-algorithms-25j1hhy9sv.pdf

https://www.cs.bgu.ac.il/~matya/Papers/tspn.pdf

TSP with neighborhoods of varying size

Given a set P of points in the plane, and a set D of unit disks of fixed location, the discrete unit disk cover problem is to find a minimum-cardinality subset D′ ⊆ D that covers all points of P. This problem is a geometric version of the general set cover problem, where the sets are defined by a collection of unit disks. It is still NP-hard, but while the general set cover problem is not approximable within c log |P|, for some constant c, the discrete unit disk cover problem was shown to admit a constant-factor approximation. Due to its many important applications, e.g., in wireless network design, much effort has been invested in trying to reduce the constant of approximation of the discrete unit disk cover problem. In this paper we significantly improve the best known constant from 72 to 38, using a novel approach. Our solution is based on a 4-approximation that we devise for the subproblem where the points of P are located below a line l and contained in the subset of disks of D centered above l. This problem is of independent interest.

/pdf/covering-points-by-unit-disks-of-fixed-location-v01cjpdyr6.pdf

Matthew J. Katz

Papers

Geometry helps in bottleneck matching and related problems

Realistic input models for geometric algorithms

Realistic input models for geometric algorithms

TSP with neighborhoods of varying size

Covering points by unit disks of fixed location