A duality between small-face problems in arrangements of lines and Heilbronn-type problems

AbstractArrangements of lines in the plane and algorithms for computing extreme features of arrangements are a major topic in computational geometry. Theoretical bounds on the size of these features are also of great interest. Heilbronn's triangle problem is one of the famous problems in discrete geometry. In this paper we show a duality between extreme (small) face problems in line arrangements (bounded in the unit square) and Heilbronn-type problems. We obtain lower and upper combinatorial bounds (some are tight) for some of these problems

Automatic Proofs for Formulae Enumerating Proper Polycubes

This video describes a general framework for computing formulae enumerating polycubes of size n which are proper in n-k dimensions (i.e., spanning all n-k dimensions), for a fixed value of k. (Such formulae are central in the literature of statistical physics in the study of percolation processes and collapse of branched polymers.) The implemented software re-affirmed the already-proven formulae for k <= 3, and proved rigorously, for the first time, the formula enumerating polycubes of size n that are proper in n-4 dimensions

On the complexity of Jensen's algorithm for counting fixed polyominoes

AbstractRecently I. Jensen published a novel transfer-matrix algorithm for computing the number of polyominoes in a rectangular lattice. However, his estimation of the computational complexity of the algorithm (O((2)n), where n is the size of the polyominoes), was based only on empirical evidence. In contrast, our research provides some solid proof. Our result is based primarily on an analysis of the number of some class of strings that plays a significant role in the algorithm. It turns out that this number is closely related to Motzkin numbers. We provide a rigorous computation that roughly confirms Jensen's estimation. We obtain the bound O(n5/2(3)n) on the running time of the algorithm, while the actual number of polyominoes is about C4.06n/n, for some constant C>0

λ > 4

A polyomino (or animal) is an edge-connected set of squares on the regular square lattice. Enumeration of polyominoes is an extremely hard problem in enumerative combinatorics, with important applications in statistical physics. We investigate one of the fundamental problems related to polyominoes, namely, computing their asymptotic growth rate λ=lim A(n+1)/A(n) , where A(n) is the number of polyominoes of size n. λ is also known as Klarner's constant. The best lower and upper bounds on so far were roughly 3.98 and 4.65, respectively, meaning that not even a single decimal digit of was known. Our goal was to settle a long-standing problem: proving that λ>4. To this aim, we developed a computer program which required extremely high computing resources in terms of both running time and main memory. Our program showed rigorously that λ>4.00253

Partial surface matching by using directed footprints

AbstractIn this paper we present a new technique for partial surface and volume matching of images in three dimensions. In this problem, we are given two objects in 3-space, each represented as a set of points, scattered uniformly along its boundary or inside its volume. The goal is to find a rigid motion of one object which makes a sufficiently large portion of its boundary lying sufficiently close to a corresponding portion of the boundary of the second object. This is an important problem in pattern recognition and in computer vision, with many industrial, medical, and chemical applications. Our algorithm is based on assigning a directed footprint to every point of the two sets, and locating all the pairs of points (one of each set) whose undirected components of the footprints are sufficiently similar. The algorithm then computes for each such pair of points all the rigid transformations that map the first point to the second, while making the respective direction components of their footprints coincide. A voting scheme is employed for computing transformations which map significantly large number of points of the first set to points of the second set. Experimental results on various examples are presented and show the accurate and robust performance of our algorithm

Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions

We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions S^(d-1). We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments or lines is O(min{k,n-k} n^(d-1)), which is tight for n-k = O(1). All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d-1)-skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of lines has exactly n^2-n three-dimensional cells, when n >= 2. The Gaussian map of the farthest Voronoi diagram of line segments or lines can be constructed in O(n^(d-1) alpha(n)) time, while if d=3, the time drops to worst-case optimal O(n^2)

Approximating the Maximum Overlap of Polygons under Translation

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

On smallest triangles

Pick n points independently at random in R^2, according to a prescribed probability measure mu, and let D^n_1 <= D^n_2 <= ... be the areas of the binomial n choose 3 triangles thus formed, in non-decreasing order. If mu is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n^3 D^n_i : i >= 1} converges as n --> infinity to a Poisson process with a constant intensity c(mu). This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a triangle. It is proved in addition that, if mu is the uniform probability measure on the region S, then c(mu) <= 2/|S|, where |S| denotes the area of S. Equality holds in that c(mu) = 2/|S| if S is convex, and essentially only then. This work generalizes and extends considerably the conclusions of a recent paper of Jiang, Li, and Vitanyi