Publications (61)17.73 Total impact
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ABSTRACT: We study multiple tilings of 3dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $P$ is translated with a discrete multiset $\Lambda$ in such a way that each point of the space gets covered exactly $k$ times, except perhaps the translated copies of the boundary of $P$. It is known that all possible multiple tilers in 3D are zonotopes. In 2D it was known by the work of M. Kolountzakis that, unless $P$ is a parallelogram, the multiset of translation vectors $\Lambda$ must be a finite union of translated lattices (also known as quasi periodic sets). In that work [Kolountzakis, 2002], the author asked whether the same quasiperiodic structure on the translation vectors would be true in 3D. Here we prove that this conclusion is indeed true for 3D. Namely, we show that if $P$ is a convex multiple tiler in 3D, with a discrete multiset $\Lambda$ of translation vectors, then $\Lambda$ has to be a finite union of translated lattices, unless $P$ belongs to a special class of zonotopes. This exceptional class consists of twoflat zonotopes $P$, defined by the Minkowski sum of $n+m$ line segments that lie in the union of two different twodimensional subspaces $H_1$ and $H_2$. Equivalently, a twoflat zonotope $P$ may be thought of as the Minkowski sum of two 2dimensional symmetric polygons one of which may degenerate into a single line segment. It turns out that rational twoflat zonotopes admit a multiple tiling with an aperiodic (nonquasiperiodic) set of translation vectors $\Lambda$. We note that it may be quite difficult to offer a visualization of these 3dimensional nonquasiperiodic tilings, and that we discovered them by using Fourier methods.Discrete and Computational Geometry 08/2012; · 0.65 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Consider the plane as a union of congruent unit squares in a checkerboard pattern, each square colored black or white in an arbitrary manner. The discrepancy of a curve with respect to a given coloring is the difference of its white length minus its black length, in absolute value. We show that for every radius t>1 there exists a full circle of radius either t or 2t with discrepancy greater than ct^(1/2) for some numerical constant c>0. We also show that for every t>1 there exists a circular arc of radius exactly t with discrepancy greater than ct^(1/2). Finally we investigate the corresponding problem for more general curves and their interiors. These results answer questions posed by Kolountzakis and Iosevich.Illinois journal of mathematics 01/2012; · 0.34 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Suppose $\Lambda \subseteq \RR^2$ has the property that any two exponentials with frequency from $\Lambda$ are orthogonal in the space $L^2(D)$, where $D \subseteq \RR^2$ is the unit disk. Such sets $\Lambda$ are known to be finite but it is not known if their size is uniformly bounded. We show that if there are two elements of $\Lambda$ which are distance $t$ apart then the size of $\Lambda$ is $O(t)$. As a consequence we improve a result of Iosevich and Jaming and show that $\Lambda$ has at most $O(R^{2/3})$ elements in any disk of radius $R$.Revista Matematica Iberoamericana 11/2011; · 0.59 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: A bounded measurable set $\Omega$, of Lebesgue measure 1, in the real line is called spectral if there is a set $\Lambda$ of real numbers ("frequencies") such that the exponential functions $e_\lambda(x) = \exp(2\pi i \lambda x)$, $\lambda\in\Lambda$, form a complete orthonormal system of $L^2(\Omega)$. Such a set $\Lambda$ is called a {\em spectrum} of $\Omega$. In this note we prove that any spectrum $\Lambda$ of a bounded measurable set $\Omega\subseteq\RR$ must be periodic.08/2011;  [Show abstract] [Hide abstract]
ABSTRACT: A set $\Omega$, of Lebesgue measure 1, in the real line is called spectral if there is a set $\Lambda$ of real numbers such that the exponential functions $e_\lambda(x) = \exp(2\pi i \lambda x)$ form a complete orthonormal system on $L^2(\Omega)$. Such a set $\Lambda$ is called a spectrum of $\Omega$. In this note we present a simplified proof of the fact that any spectrum $\Lambda$ of a set $\Omega$ which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.Journal of Fourier Analysis and Applications 02/2011; · 1.08 Impact Factor 
Article: Triangle Sparsifiers
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ABSTRACT: In this work, we introduce the notion of triangle sparsifiers, i.e., sparse graphs which are approximately the same to the original graph with respect to the triangle count. This results in a practical triangle counting method with strong theoretical guarantees. For instance, for unweighted graphs we show a randomized algorithm for approximately counting the number of triangles in a graph G, which proceeds as follows: keep each edge independently with probability p, enumerate the triangles in the sparsified graph G and return the number of triangles found in G multiplied by p −3 . We prove that under mild assumptions on G and p our algorithm returns a good approximation for the number of triangles with high probability. Specifically, we show that if p ≥ max (polylog(n)∆ t , polylog(n) t 1/3), where n, t, ∆, and T denote the number of vertices in G, the number of triangles in G, the maximum number of triangles an edge of G is contained and our triangle count estimate respectively, then T is strongly concentrated around t: Pr [T − t ≥ ≤ n −K . We illustrate the efficiency of our algorithm on various large realworld datasets where we obtain significant speedups. Finally, we investigate cut and spectral sparsifiers with respect to triangle counting and show that they are not optimal.Journal of Graph Algorithms and Applications 01/2011; 15(6).  [Show abstract] [Hide abstract]
ABSTRACT: The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering of the hidden thematic structure of the Web and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms which run fast, use small amount of space, provide accurate estimates of the number of triangles and preferably are parallelizable. In this paper we present an efficient triangle counting algorithm which can be adapted to the semistreaming model. The key idea of our algorithm is to combine the sampling algorithm of Tsourakakis et al. and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in the Alon, Yuster and Zwick work 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 $\epsilon$ approximation (multiplicative error), where $n$ is the number of vertices, $m$ the number of edges and $\Delta$ 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. Comment: 1) 12 pages 2) To appear in the 7th Workshop on Algorithms and Models for the Web Graph (WAW 2010)11/2010; 
Article: Tilings by translation
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ABSTRACT: This is a survey about tiling by translation only and related questions and methods, especially those that have to do with Fourier Analysis.09/2010; 
Article: Tesselations by translations
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ABSTRACT: Given Ω,Λ⊆ℝ d we say that Ω tesselates ℝ d with Λ if the copies Ω+λ with λ∈Λ do not overlap and they cover ℝ d . The paper under review discusses the previous problem in different settings using tools such as the Fourier Transform. It also pays attention to tesselations of the integers (which can be defined in the obvious way from the previous definition) and also to tesselations of the cyclic group. This is an interesting survey which can be read without problems by nonespecialists.La Gaceta de la Real Sociedad Matemàtica Española. 01/2010; 13(4).  [Show abstract] [Hide abstract]
ABSTRACT: We study the following question: What is the smallest t such that every symmetric boolean function on k variables (which is not a constant or a parity function), has a nonzero Fourier coecient of order at least 1 and at most t? We exclude the constant functions for which there is no such t and the parity functions for which t has to be k. Let (k) be the smallest such t. Our main result is that (k) 4k/logk. The motivation for our work is to understand the complexity of learning symmetric juntas. A kjunta is a boolean function of n variables that depends only on an unknown subset of k variables. A symmetric kjunta is a junta that is symmetric in the variables it depends on. Our result implies an algorithm to learn the class of symmetric kjuntas, in the uniform PAC learning model, in time no(k). This improves on a result of Mossel, O'Donnell and Servedio in (14), who show that symmetric kjuntas can be learned in time n 2k 3 .Combinatorica 05/2009; 29:363387. · 0.56 Impact Factor 
Article: Approximate Triangle Counting
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ABSTRACT: Triangle counting is an important problem in graph mining. Clustering coefficients of vertices and the transitivity ratio of the graph are two metrics often used in complex network analysis. Furthermore, triangles have been used successfully in several realworld applications. However, exact triangle counting is an expensive computation. In this paper we present the analysis of a practical sampling algorithm for counting triangles in graphs. Our analysis yields optimal values for the sampling rate, thus resulting in tremendous speedups ranging from \emph{2800}x to \emph{70000}x when applied to realworld networks. At the same time the accuracy of the estimation is excellent. Our contributions include experimentation on graphs with several millions of nodes and edges, where we show how practical our proposed method is. Finally, our algorithm's implementation is a part of the \pegasus library (Code and datasets are available at (http://www.cs.cmu.edu/~ctsourak/).) a PetaGraph Mining library implemented in Hadoop, the open source version of Mapreduce. Comment: 1) 16 pages, 2 figures, under submission 2) Removed the erroneous random projection part. Thanks to Ioannis Koutis for pointing out the error. 3) Added experimental session04/2009;  [Show abstract] [Hide abstract]
ABSTRACT: Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. In a previous paper we showed that for any such coloring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their black length, in absolute value, is at least the square root of their length, up to a multiplicative constant. For the corresponding "finite" problem ($N \times N$ checkerboard) we had proved that we can color it in such a way that the above quantity is at most $C \sqrt{N \log N}$, for any placement of the line segment. In this followup we show that it is possible to color the infinite checkerboard with two colors so that for any line segment $I$ the excess of one color over another is bounded above by $C_\epsilon \Abs{I}^{\frac12+\epsilon}$, for any $\epsilon>0$. We also prove lower bounds for the discrepancy of circular arcs. Finally, we make some observations regarding the $L^p$ discrepancies for segments and arcs, $p<2$, for which our $L^2$based methods fail to give any reasonable estimates.12/2008; 
Article: Algorithms for translational tiling
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ABSTRACT: In this paper we study algorithms for tiling problems. We show that the conditions $(T1)$ and $(T2)$ of Coven and Meyerowitz, conjectured to be necessary and sufficient for a finite set $A$ to tile the integers, can be checked in time polynomial in ${diam}(A)$. We also give heuristic algorithms to find all nonperiodic tilings of a cyclic group $Z_N$. In particular we carry out a full classification of all nonperiodic tilings of $Z_{144}$.Journal of Mathematics and Music 11/2008; · 0.52 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We show that there are polynomials $p_N$ of arbitrarily large degree $N$, with coefficients equal to 0 or 1 (Newman polynomials), such that $$ \liminf_{N \to \infty} N \Linf{p_N^2} \bigl / p_N^2(1) < 1, $$ where $\Linf{q}$ denotes the maximum coefficient of the polynomial $q$ and which, at the same time, are sparse: $p_N(1)/N \to 0$. This disproves a conjecture of Yu \cite{yu}. We build on some previous results of Berenhaut and Saidak \cite{berenhautsaidak} and Dubickas \cite{dubickas} whose examples lacked the sparsity. This sparsity we create from these examples by randomization.07/2008;  [Show abstract] [Hide abstract]
ABSTRACT: Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. We show that for any such coloring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their black length, in absolute value, is at least the square root of their length, up to a multiplicative constant. For the corresponding ``finite'' problem ($N \times N$ checkerboard) we also prove that we can color it in such a way that the above quantity is at most $C \sqrt{N \log N}$, for any placement of the line segment.12/2007;  [Show abstract] [Hide abstract]
ABSTRACT: Suppose we put an $\epsilon$disk around each lattice point in the plane, and then we rotate this object around the origin for a set $\Theta$ of angles. When do we cover the whole plane, except for a neighborhood of the origin? This is the problem we study in this paper. It is very easy to see that if $\Theta = [0,2\pi]$ then we do indeed cover. The problem becomes more interesting if we try to achieve covering with a small closed set $\Theta$.12/2006;  [Show abstract] [Hide abstract]
ABSTRACT: Let $G$ be a finite abelian group and $E$ a subset of it. Suppose that we know for all subsets $T$ of $G$ of size up to $k$ for how many $x \in G$ the translate $x+T$ is contained in $E$. This information is collectively called the $k$deck of $E$. One can naturally extend the domain of definition of the $k$deck to include functions on $G$. Given the group $G$ when is the $k$deck of a set in $G$ sufficient to determine the set up to translation? The 2deck is not sufficient (even when we allow for reflection of the set, which does not change the 2deck) and the first interesting case is $k=3$. We further restrict $G$ to be cyclic and determine the values of $n$ for which the 3deck of a subset of $\ZZ_n$ is sufficient to determine the set up to translation. This completes the work begun by Gr\"unbaum and Moore as far as the 3deck is concerned. We additionally estimate from above the probability that for a random subset of $\ZZ_n$ there exists another subset, not a translate of the first, with the same 3deck. We give an exponentially small upper bound when the previously known one was $O(1\bigl / \sqrt{n})$.04/2006;  [Show abstract] [Hide abstract]
ABSTRACT: We give an algorithm for learning symmetric kjuntas (boolean functions of $n$ boolean variables which depend only on an unknown set of $k$ of these variables) in the PAC model under the uniform distribution, which runs in time n^{O(k/\log k)}. Our bound is obtained by proving the following result: Every symmetric boolean function on k variables, except for the parity and the constant functions, has a nonzero Fourier coefficient of order at least 1 and at most O(k/\log k). This improves the previously best known bound of (3/31)k, and provides the first n^{o(k)} time algorithm for learning symmetric juntas.05/2005;  [Show abstract] [Hide abstract]
ABSTRACT: By analyzing the connection between complex Hadamard matrices and spectral sets we prove the direction ``spectral > tile'' of the Sectral Set Conjecture for all sets A of size at most 5 in any finite Abelian group. This result is then extended to the infinite grid $\Z^d$ for any dimension d, and finally to Euclidean space. It was pointed out recently by Tao that the corresponding statement fails for A=6 in the group $\Z_3^5$, and this observation quickly led to the failure of the Spectral Set Conjecture in $\R^5$ (Tao), and subsequently in $\R^4$ (Matolcsi). In the second part of this note we reduce this dimension further, showing that the direction ``spectral > tile'' of the Spectral Set Conjecture is false already in dimension 3. In a computational search for counterexamples in lower dimension (one and two) one needs, at the very least, to be able to decide efficiently if a set is a tile (in, say, a cyclic group) and if it is spectral. Such efficient procedures are lacking however and we make a few comments for the computational complexity of some related problems.Collectanea Mathematica 12/2004; · 0.79 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We give a new proof of the following interesting fact recently proved by Bower and Michael: if a ddimensional rectangular box can be tiled using translates of two types of rectangular bricks, then it can also be tiled in the following way. We can cut the box across one of its sides into two boxes, one of which can be tiled with the first brick only and the other one with the second brick. Our proof relies on the Fourier Transform. We also show that no such result is true for three, or more, types of bricks.The electronic journal of combinatorics 10/2004; · 0.53 Impact Factor
Publication Stats
497  Citations  
17.73  Total Impact Points  
Top Journals
Institutions

1998–2011

University of Crete
 Department of Mathematics
Retimo, Crete, Greece


1995–1996

Stanford University
 • Department of Mathematics
 • Department of Computer Science
Palo Alto, California, United States
