Decomposition of multiple packings with subquadratic union complexity

Suppose $k$ is a positive integer and $\mathcal{X}$ is a $k$-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most $k$ sets. Suppose there is a function $f(n)=o(n^2)$ with the property that any $n$ members of $\mathcal{X}$ determine at most $f(n)$ holes, which means that the complement of their union has at most $f(n)$ bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that $\mathcal{X}$ can be decomposed into at most $p$ ($1$-fold) packings, where $p$ is a constant depending only on $k$ and $f$.Comment: Small generalization of the main result, improvements in the proofs, minor correction

Reporting flock patterns

Data representing moving objects is rapidly getting more available, especially in the area of wildlife GPS tracking. It is a central belief that information is hidden in large data sets in the form of interesting patterns. One of the most common spatio-temporal patterns sought after is flocks. A flock is a large enough subset of objects moving along paths close to each other for a certain pre-defined time. We give a new definition that we argue is more realistic than the previous ones, and by the use of techniques from computational geometry we present fast algorithms to detect and report flocks. The algorithms are analysed both theoretically and experimentally

Vertical ray shooting and computing depth orders of fat objects

We present new results for three problems dealing with a set $\mathcal{P}$ of $n$ convex constant-complexity fat polyhedra in 3-space. (i) We describe a data structure for vertical ray shooting in $\mathcal{P}$ that has $O(\log^2 n)$ query time and uses $O(n\log^2 n)$ storage. (ii) We give an algorithm to compute in $O(n\log^3 n)$ time a depth order on $\mathcal{P}$ if it exists. (iii) We give an algorithm to verify in $O(n\log^3 n)$ time whether a given order on $\mathcal{P}$ is a valid depth order. All three results improve on previous results

On fat partitioning, fat covering and the union size of polygons

AbstractThe complexity of the contour of the union of simple polygons with n vertices in total can be O(n2) in general. A notion of fatness for simple polygons is introduced that extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity O(n log log n), which is a generalization of a similar result for fat triangles (Matoušek et al., 1994). Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, and more. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(l) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any exact cover of a fat simple polygon by a linear number of fat triangles

Decomposing and packing polygons / Dania el-Khechen.

In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop