Computational geometry

Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

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

In 1973, Victor Klee posed the following question: How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naive question of combinatorial geometry has, since its formulation, stimulated a plethora of papers, surveys and a book, most of them written in the last fifteen years. The first result in this area, due to V. Chvatal, asserts that n 3 guards are occasionally necessary and always sufficient to guard an art gallery represented by a simple polygon with n vertices. Since ChvataFs result, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this paper, we survey most of these results.

Art Gallery and Illumination Problems

: Significant progress has been made with solution of location problems and in preprocessing and decomposition for discrete optimization. There has also been research on the application of techniques from combinational optimization to nonlinear problems.

http://www.dtic.mil/dtic/tr/fulltext/u2/a273552.pdf

Discrete Applied Mathematics

In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least $\lceil (n-4)/2 \rceil$ edges that can be flipped. We also prove that O(n + k

 2
 
) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T' such that to transform T into T' requires Ω(n

 2
 
) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips.

/pdf/flipping-edges-in-triangulations-1uukalckf5.pdf

Flipping edges in triangulations

We review results concerning edge flips in planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another? We overview both the combinatorial perspective (where only a combinatorial embedding of the graph is specified) and the geometric perspective (where the graph is embedded in the plane, vertices are points and edges are straight-line segments). We highlight the similarities and differences of the two settings, describe many extensions and generalizations, highlight algorithmic issues, outline several applications and mention open problems.

Flips in planar graphs

Define a graph G T ( n ) with one node for each triangulation of a convex n-gon. Place an edge between each pair of nodes that differ by a single flip: two triangles forming a quadrilateral are exchanged for the other pair of triangles forming the same quadrilateral. In this paper we introduce a tree of all triangulations of polygons with any number of vertices which gives a unified framework in which several results on G T ( n ) admit new and simple proofs.

Graph of triangulations of a convex polygon and tree of triangulations

We investigate the number of plane geometric, i.e., straight-line, graphs, a set S of n points in the plane admits. We show that the number of plane geometric graphs and connected plane geometric graphs as well as the number of cycle-free plane geometric graphs is minimized when S is in convex position. Moreover, these results hold for all these graphs with an arbitrary but fixed number of edges. Consequently, we provide a unified proof that the cardinality of any family of acyclic graphs (for example spanning trees, forests, perfect matchings, spanning paths, and more) is minimized for point sets in convex position.

In addition we construct a new maximizing configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ* $${{(\sqrt{72}\,}^n)}$$ = Θ*(8.4853n) triangulations (omitting polynomial factors), improving the previously known best maximizing examples.

On the Number of Plane Geometric Graphs

In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least $\lceil (n-4)/2 \rceil$ edges that can be flipped. We also prove that O(n + k2) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T' such that to transform T into T' requires Ω(n2) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips.

/pdf/flipping-edges-in-triangulations-58r53b3u2z.pdf

Ferran Hurtado

Papers

Flipping edges in triangulations

Flips in planar graphs

Graph of triangulations of a convex polygon and tree of triangulations

On the Number of Plane Geometric Graphs

Flipping edges in triangulations