This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

Algorithms in Combinatorial Geometry

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.

http://www.math.tau.ac.il/~michas/dssurvey.pdf

Davenport-Schinzel sequences and their geometric applications

There are many problems in computational geometry for which the best know algorithms take time @Q(n^2) (or more) in the worst case while only very low lower bounds are known. In this paper we describe a large class of problems for which we prove that they are all at least as difficult as the following base problem 3sum: Given a set S of n integers, are there three elements of S that sum up to 0. We call such problems 3sum-hard. The best known algorithm for the base problem takes @Q(n^2) time. The class of 3sum-hard problems includes problems like: Given a set of lines in the plane, are there three that meet in a point?; or: Given a set of triangles in the plane, does their union have a hole? Also certain visibility and motion planning problems are shown to be in the class. Although this does not prove a lower bound for these problems, there is no hope of obtaining o(n^2) solutions for them unless we can improve the solution for the base problem.

/pdf/on-a-class-of-o-n2-problems-in-computational-geometry-4e20qjov5v.pdf

On a class of O(n2) problems in computational geometry

Geometric reasoning about mechanical assembly

This paper gives approximation algorithms of solving the following motion planning problem: Given a set of polyhedral obstacles and points s and t, find a shortest path from s to t that avoids the obstacles. The paths found by the algorithms are piecewise linear, and the length of a path is the sum of the lengths of the line segments making up the path. Approximation algorithms will be given for versions of this problem in the plane and in three-dimensional space. The algorithms return an e-short path, that is, a path with length within (1 + e) of shortest. Let n be the total number of faces of the polyhedral obstacles, and e a given value satisfying O e ≤ p. The algorithm for the planar case requires O(n log n)/e time to build a data structure of size O(n/e). Given points s and t, and e-short path from s to t can be found with the use of the data structure in time O(n/e + n log n). The data structure is associated with a new variety of Voronoi diagram. Given obstacles S ⊂ E3 and points s, t e E3, an e-short path between s and t can be found in O(n2l(n) log(n/e)/e4 + n2 lognp log(n logp)) time, where p is the ratio of the length of the longest obstacle edge to the distance between s to t. The function l(n) = a(n)O(a(n)O(1)), where the a(n) is a form of inverse of Ackermann's function. For log(1/e) and log p that are O(log n), this bound is O(log n2(n)l(n)/e4).

Approximation algorithms for shortest path motion planning

The authors show that for every fixed $\delta>0$ the following holds: If $F$ is a union of $n$ triangles, all of whose angles are at least $\delta$, then the complement of $F$ has $O(n)$ connected components and the boundary of $F$ consists of $O(n \log \log n)$ straight segments (where the constants of proportionality depend on $\delta$). This latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges.

Fat Triangles Determine Linearly Many Holes

LetP andQ be two disjoint simple polygons havingm andn sides, respectively. We present an algorithm which determines whetherQ can be moved by a sequence of translations to a position sufficiently far fromP without colliding withP, and which produces such a motion if it exists. Our algorithm runs in timeO(mn?(mn) logm logn) where ?(k) is the extremely slowly growing inverse Ackermann's function. Since in the worst case Ω(mn) translations may be necessary to separateQ fromP, our algorithm is close to optimal.

https://link.springer.com/content/pdf/10.1007%2FBF02187902.pdf

Separating Two Simple Polygons, by a Sequence of Translations

We present efficient algorithms for the following geometric problems: (i) Preprocessing of a 2-D polyhedral terrain so as to support fast ray shooting queries from a fixed point. (ii) Determining whether two disjoint interlocking simple polygons can be separated from one another by a sequence of translations. (iii) Determining whether a given convex polygon can be translated and rotated so as to fit into another given polygonal region. (iv) Motion planning for a convex polygon in the plane amidst polygonal barriers. All our algorithms make use of Davenport Schinzel sequences and on some generalizations of them; these sequences are a powerful combinatorial tool applicable in contexts which involve the calculation of the pointwise maximum or minimum of a collection of functions.

Geometric applications of Davenport-Schinzel sequences

It is shown that for every fixed delta >0 the following holds: if F is a union of n triangles, all of whose angles are at least delta , then the complement of F has O(n) connected components, and the boundary of F consists of O(n log log n) segments. This latter complexity becomes linear if all triangles are of roughly the same size or if they are all infinite wedges. A randomized algorithm that computes F in expected time O(n2/sup alpha (n)/ log n) is given. Several applications of these results are presented. >

Shmuel Sifrony

Papers

Fat Triangles Determine Linearly Many Holes

Separating Two Simple Polygons, by a Sequence of Translations

Geometric applications of Davenport-Schinzel sequences

Fat triangles determine linearly many holes (computational geometry)