Mihail N. Kolountzakis

Analysis of probabilistic roadmaps for path planning

Provides an analysis of a path planning method which uses probabilistic roadmaps. This method has proven very successful in practice, but the theoretical understanding of its performance is still limited. Assuming that a path /spl gamma/ exists between two configurations a and b of the robot, we study the dependence of the failure probability to connect a and b on (i) the length of /spl gamma/, (ii) the distance function of /spl gamma/ from the obstacles, and (iii) the number of nodes N of the probabilistic roadmap constructed. Importantly, our results do not depend strongly on local irregularities of the configuration space, as was the case with previous analysis. These results are illustrated with a simple but illuminating example. In this example, we provide estimates for N, the principal parameter of the method, in order to achieve failure probability within prescribed bounds. We also compare, through this example, the different approaches to the analysis of the planning method.

Tiles with no spectra

Abstract We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their L 2 space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang [Lagarias J. C. and Wang Y.: Spectral sets and factorizations of finite Abelian groups.J. Func. Anal. 145 (1997), 73–98]. Further, we construct a set in some finite Abelian group, which tiles the group but has no spectrum. We extend this last example to the groups ℤ d and ℝ d (for d ≥5 ) thus disproving one direction of the Spectral Set Conjecture of Fuglede [Fuglede B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16 (1974), 101–121]. The other direction was recently disproved by Tao [Tao T.: Fugledes conjecture is false in 5 and higher dimensions. Math. Res. Letters 11 (2004), 251–258].

Fast computation of the Euclidian distance maps for binary images: Fast computations of the Euclidean distance maps for binary images

Abstract A simple algorithm is given for the computation of the Euclidian distance from the set of black points in an N × N black and white image, for all points in the image. The running time is O(N2 log N) and O(N) extra space is required. The algorithm is suitable for implementation on a parallel machine.

Efficient Triangle Counting in Large Graphs via Degree-Based Vertex Partitioning

In this paper we present an efficient triangle counting algorithm which can be adapted to the semistreaming model [12]. The key idea of our algorithm is to combine the sampling algorithm of [31,32] and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in [1], treating each set appropriately. We obtain a running time \(O \left( m + \frac{m^{3/2} \Delta \log{n} }{t \epsilon^2} \right)\) and an e approximation (multiplicative error), where n is the number of vertices, m the number of edges and Δ the maximum number of triangles an edge is contained. Furthermore, we show how this algorithm can be adapted to the semistreaming model with space usage \(O\left(m^{1/2}\log{n} + \frac{m^{3/2} \Delta \log{n}}{t \epsilon^2} \right)\) and a constant number of passes (three) over the graph stream. We apply our methods in various networks with several millions of edges and we obtain excellent results. Finally, we propose a random projection based method for triangle counting and provide a sufficient condition to obtain an estimate with low variance.

The Study of Translational Tiling with Fourier Analysis

In this survey I will try to describe how Fourier analysis is used in the study of translational tiling. Right away I will emphasize two restrictions that separate this area from the general theory of tilings.

Partitioning a planar assembly into two connected parts is NP-complete

Abstract Consider the following decision problem. Given a collection of non-overlapping (but possibly touching) polygons in the plane, is there a proper connected subcollection of it that can be separated from its complement moving as a rigid body, without disturbing the other parts of the collection, and such that the complement is also connected? We show that this decision problem is NP-complete. This had been known to be true without the connectedness requirement, and also with this requirement but in three-dimensional space.

On the Steinhaus tiling problem

Several results are proved related to a question of Steinhaus: is there a set E ⊂ℝ 2 such that the image of E under each rigid motion of IR2 contains exactly one lattice point? Assuming measurability, the analogous question in higher dimensions is answered in the negative, and on the known partial results in the two dimensional case are improved on. Also considered is a related problem involving finite sets of rotations.

Multiple lattice tiles and Riesz bases of exponentials

Suppose

On the Fourier spectrum of symmetric Boolean functions

\Omega\subseteq\RR^d

TURÁN'S EXTREMAL PROBLEM FOR POSITIVE DEFINITE FUNCTIONS ON GROUPS

is a bounded and measurable set and