Renyi Institute of Mathematics

Budapest, Hungary

Renyi Institute of Mathematics

Budapest, Hungary
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Kerner D.,Ben - Gurion University of the Negev | Nemethi A.,Renyi Institute of Mathematics
Journal of Combinatorial Theory. Series A | Year: 2017

We prove a Fortuin–Kasteleyn–Ginibre-type inequality for the lattice of compositions of the integer n with at most r parts. As an immediate application we get a wide generalization of the classical Alexandrov–Fenchel inequality for mixed volumes and of Teissier's inequality for mixed covolumes. © 2016 Elsevier Inc.

Harremoes P.,Copenhagen Business School | Tusnady G.,Renyi Institute of Mathematics
IEEE International Symposium on Information Theory - Proceedings | Year: 2012

For testing goodness of fit it is very popular to use either the χ 2-statistic or G 2-statistics (information divergence). Asymptotically both are χ 2-distributed so an obvious question is which of the two statistics that has a distribution that is closest to the χ 2-distribution. Surprisingly, when there is only one degree of freedom it seems like the distribution of information divergence is much better approximated by a χ 2-distribution than the χ 2-statistic. For random variables we introduce a new transformation that transform several important distributions into new random variables that are almost Gaussian. For the binomial distributions and the Poisson distributions we formulate a general conjecture about how close their transform are to the Gaussian. The conjecture is proved for Poisson distributions. © 2012 IEEE.

Ahlswede R.,Renyi Institute of Mathematics | Csiszar I.,Renyi Institute of Mathematics
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2013

Upper and lower bounds to the oblivious transfer (OT) capacity of discrete memoryless channels and multiple sources are obtained, for 1 of 2 strings OT with honest but curious participants. The upper bounds hold also for one-string OT. The results provide the exact value of OT capacity for a specified class of models, and the necessary and sufficient condition of its positivity, in general. © Springer-Verlag Berlin Heidelberg 2013.

Matus F.,Czech Institute of Information Theory And Automation | Csirmaz L.,Central European University | Csirmaz L.,Debrecen University | Csirmaz L.,Renyi Institute of Mathematics
IEEE Transactions on Information Theory | Year: 2016

The entropy region is constructed from vectors of random variables by collecting Shannon entropies of all subvectors. Its shape is studied here by means of polymatroidal constructions, notably by convolution. The closure of the region is decomposed into the direct sum of tight and modular parts, reducing the study to the tight part. The relative interior of the reduction belongs to the entropy region. Behavior of the decomposition under self-adhesivity is clarified. Results are specialized and extended to the region constructed from four tuples of random variables. This and computer experiments help to visualize approximations of a symmetrized part of the entropy region. The four-atom conjecture on the minimal Ingleton score is refuted. © 2016 IEEE.

Jowhari H.,Simon Fraser University | Saglam M.,Simon Fraser University | Tardos G.,Simon Fraser University | Tardos G.,Renyi Institute of Mathematics
Proceedings of the ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems | Year: 2011

In this paper, we present near-optimal space bounds for Lp- samplers. Given a stream of updates (additions and subtraction) to the coordinates of an underlying vector x ε Rn, a perfect L p sampler outputs the i-th coordinate with probability |x i|p/||x||pp. In SODA 2010, Monemizadeh and Woodru showed polylog space upper bounds for approximate Lp-samplers and demonstrated various applications of them. Very recently, Andoni, Krauthgamer and Onak improved the upper bounds and gave a O(ε-p log3 n) space ε relative error and constant failure rate Lp-sampler for p ε [1; 2]. In this work, we give another such algorithm requiring only O(ε-p log 3 n) space for p ε (1; 2). For p ε (0; 1), our space bound is O(ε-p log3 n), while for the p = 1 case we have an O(log(1=ε) ε-1 log2 n) space algorithm. We also give a O(log2 n) bits zero relative error L0-sampler, improving the O(log3 n) bits algorithm due to Frahling, Indyk and Sohler. As an application of our samplers, we give better upper bounds for the problem of finding duplicates in data streams. In case the length of the stream is longer than the alphabet size, L1 sampling gives us an O(log2 n) space algorithm, thus improving the previous O(log3 n) bound due to Gopalan and Radhakrishnan. In the second part of our work, we prove an (log 2 n) lower bound for sampling from 0, ±1 vectors (in this special case, the parameter p is not relevant for Lp sampling). This matches the space of our sampling algorithms for constant ε > 0. We also prove tight space lower bounds for the finding duplicates and heavy hitters problems. We obtain these lower bounds using reductions from the communication complexity problem augmented indexing. Copyright © 2011 ACM.

Furedi Z.,Renyi Institute of Mathematics | Jiang T.,Miami University Ohio
Journal of Combinatorial Theory. Series A | Year: 2014

A k-uniform linear cycle of length ℓ, denoted by Cℓ(k), is a cyclic list of k-sets A1, . . . , Aℓ such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k ≥ 5 and ℓ ≥ 3 and sufficiently large n we determine the largest size of a k-uniform set family on [n] not containing a linear cycle of length ℓ. For odd ℓ = 2t + 1 the unique extremal family FS consists of all k-sets in [n] intersecting a fixed t-set S in [n]. For even ℓ = 2t + 2, the unique extremal family consists of FS plus all the k-sets outside S containing some fixed two elements. For k ≥ 4 and large n we also establish an exact result for so-called minimal cycles. For all k ≥ 4 our results substantially extend Erdos's result on largest k-uniform families without t + 1 pairwise disjoint members and confirm, in a stronger form, a conjecture of Mubayi and Verstraëte. Our main method is the delta system method. © 2014 Elsevier Inc.

Pach J.,Ecole Polytechnique Federale de Lausanne | Radoicic R.,City University of New York | Toth G.,Renyi Institute of Mathematics
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2012

A tangle is a graph drawn in the plane so that any pair of edges have precisely one point in common, and this point is either an endpoint or a point of tangency. If we allow a third option: the common point may be a proper crossing between the two edges, then the graph is called a tangled thrackle. We establish the following analogues of Conway's thrackle conjecture: The number of edges of a tangle cannot exceed its number of vertices, n. We also prove that the number of edges of an x-monotone tangled thrackle with n vertices is at most n + 1. Both results are tight for n > 3. For not necessarily x-monotone tangled thrackles, we have a somewhat weaker, but nearly linear, upper bound. © 2012 Springer-Verlag.

Frankl P.,Renyi Institute of Mathematics
Journal of Combinatorial Theory. Series A | Year: 2013

The main result is the following. Let F be a family of k-subsets of an n-set, containing no s + 1 pairwise disjoint edges. Then for n ≥ (2. s + 1). k - s one has |F|≤(n k)-(n-s k). This upper bound is the best possible and confirms a conjecture of Erdo{double acute}s dating back to 1965. The proof is surprisingly compact. It applies a generalization of Katona's Intersection Shadow Theorem. © 2013 Elsevier Inc..

Csirmaz L.,Central European University | Tardos G.,Simon Fraser University | Tardos G.,Renyi Institute of Mathematics
IEEE Transactions on Information Theory | Year: 2013

The information rate for an access structure is the reciprocal of the load of the optimal secret sharing scheme for this structure. We determine this value for all trees: it is , where is the size of the largest core of the tree. A subset of the vertices of a tree is a core if it induces a connected subgraph and for each vertex in the subset one finds a neighbor outside the subset. Our result follows from a lower and an upper bound on the information rate that applies for any graph and happen to coincide for trees because of a correspondence between the size of the largest core and a quantity related to a fractional cover of the tree with stars.© 2012 IEEE.

Furedi Z.,Renyi Institute of Mathematics
European Journal of Combinatorics | Year: 2013

Given a tree T on v vertices and an integer k ≥ 2 one can define the k-expansion T (k) as a k-uniform linear hypergraph by enlarging each edge with a new, distinct set of k - 2 vertices. T (k) has v+(v-1)(k-2) vertices. The aim of this paper is to show that using the delta-system method one can easily determine asymptotically the size of the largest T (k)-free n-vertex hypergraph, i.e., the Turán number of T (k). © 2013 Elsevier Ltd.

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