Open AccessBook
Davenport-Schinzel sequences and their geometric applications
Micha Sharir,Pankaj K. Agarwal +1 more
TLDR
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.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.read more
Citations
More filters
MonographDOI
Planning Algorithms: Introductory Material
TL;DR: This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms, into planning under differential constraints that arise when automating the motions of virtually any mechanical system.
Journal ArticleDOI
Computing the fréchet distance between two polygonal curves
Helmut Alt,Michael Godau +1 more
TL;DR: As a measure for the resemblance of curves in arbitrary dimensions the authors consider the so-called Frechet-distance, which is compatible with parametrizations of the curves, and for polygonal chains P and Q consisting of p and q edges an algorithm of runtime O(pq log( pq))) measuring the Frechet Distance.
Geometric Range Searching and Its Relatives
Pankaj K. Agarwal,Je Erickson +1 more
TL;DR: This volume provides an excellent opportunity to recapitulate the current status of geometric range searching and to summarize the recent progress in this area.
Book
Geometric Approximation Algorithms
TL;DR: This book is the first to cover geometric approximation algorithms in detail, and topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings.
Proceedings ArticleDOI
Data structures for mobile data
TL;DR: This paper develops a concentual framework for kinetic data structures, a number of criteria for the quality of such structures are proposed, and a large number of fundamental techniques for their design are described.
References
More filters
Computational geometry. an introduction
TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.
Journal ArticleDOI
Voronoi diagrams—a survey of a fundamental geometric data structure
TL;DR: The Voronoi diagram as discussed by the authors divides the plane according to the nearest-neighbor points in the plane, and then divides the vertices of the plane into vertices, where vertices correspond to vertices in a plane.
Journal ArticleDOI
Theory of Recursive Functions and Effective Computability.
Solomon Feferman,Hartley Rogers +1 more
TL;DR: Central concerns of the book are related theories of recursively enumerable sets, of degree of un-solvability and turing degrees in particular and generalizations of recursion theory.
Journal ArticleDOI
An algorithm for planning collision-free paths among polyhedral obstacles
TL;DR: A collision avoidance algorithm for planning a safe path for a polyhedral object moving among known polyhedral objects that transforms the obstacles so that they represent the locus of forbidden positions for an arbitrary reference point on the moving object.
Book
Algorithms in Combinatorial Geometry
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.