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96
Approximating the cutnorm via Grothendieck’s inequality
 Proc. of the 36 th ACM STOC
, 2004
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Limits of dense graph sequences
 J. Combin. Theory Ser. B
"... We show that if a sequence of dense graphs Gn has the property that for every fixed graph F, the density of copies of F in Gn tends to a limit, then there is a natural “limit object”, namely a symmetric measurable function W: [0,1] 2 → [0, 1]. This limit object determines all the limits of subgraph ..."
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Cited by 97 (9 self)
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We show that if a sequence of dense graphs Gn has the property that for every fixed graph F, the density of copies of F in Gn tends to a limit, then there is a natural “limit object”, namely a symmetric measurable function W: [0,1] 2 → [0, 1]. This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a limit object. Along the lines we introduce a rather general model of random graphs, which seems to be interesting on its own right. 1
Testing that distributions are close
 In IEEE Symposium on Foundations of Computer Science
, 2000
"... Given two distributions over an n element set, we wish to check whether these distributions are statistically close by only sampling. We give a sublinear algorithm which uses O(n 2/3 ɛ −4 log n) independent samples from each distribution, runs in time linear in the sample size, makes no assumptions ..."
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Cited by 77 (16 self)
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Given two distributions over an n element set, we wish to check whether these distributions are statistically close by only sampling. We give a sublinear algorithm which uses O(n 2/3 ɛ −4 log n) independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases ɛ when the distance between the distributions is small (less than max ( 2 32 3 √ n, ɛ 4 √)) or large (more n than ɛ) in L1distance. We also give an Ω(n 2/3 ɛ −2/3) lower bound. Our algorithm has applications to the problem of checking whether a given Markov process is rapidly mixing. We develop sublinear algorithms for this problem as well.
Property Testing
 Handbook of Randomized Computing, Vol. II
, 2000
"... this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well). ..."
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Cited by 76 (10 self)
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this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well).
Efficient sampling algorithm for estimating subgraph concentrations and detecting network motifs
 Bioinformatics
, 2004
"... Biological and engineered networks have recently been shown to display network motifs: a small set of characteristic patterns which occur much more frequently than in randomized networks with the same degree sequence. Network motifs were demonstrated to play key information processing roles in biolo ..."
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Cited by 70 (0 self)
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Biological and engineered networks have recently been shown to display network motifs: a small set of characteristic patterns which occur much more frequently than in randomized networks with the same degree sequence. Network motifs were demonstrated to play key information processing roles in biological regulation networks. Existing algorithms for detecting network motifs act by exhaustively enumerating all subgraphs with a given number of nodes in the network. The runtime of such full enumeration algorithms increases strongly with network size. Here we present a novel algorithm that allows estimation of subgraph concentrations and detection of network motifs at a run time that is asymptotically independent of the network size. This algorithm is based on random sampling of subgraphs. Network motifs are detected with a surprisingly small number of samples in a wide variety of networks. Our method can be applied to estimate the concentrations of larger subgraphs in larger networks than was previously possible with full enumeration algorithms. We present results for highorder motifs in several biological networks and discuss their possible functions. Availability: A software tool for estimating subgraph concentrations and detecting network motifs (mfinder 2.0) and further information is available at:
The counting lemma for regular kuniform hypergraphs. Random Structures and Algorithms
"... Abstract. Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εr ..."
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Cited by 70 (12 self)
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Abstract. Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓpartite graph with V (G) = V1 ∪ · · · ∪ Vℓ and Vi  = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are εregular of density d for ℓ 1 ≤ i < j ≤ ℓ, then G contains (1 ± fℓ(ε))d
Testing of Clustering
 In Proc. 41th Annu. IEEE Sympos. Found. Comput. Sci
, 2000
"... A set X of points in ! d is (k; b)clusterable if X can be partitioned into k subsets (clusters) so that the diameter (alternatively, the radius) of each cluster is at most b. We present algorithms that by sampling from a set X , distinguish between the case that X is (k; b)clusterable and the ca ..."
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Cited by 60 (13 self)
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A set X of points in ! d is (k; b)clusterable if X can be partitioned into k subsets (clusters) so that the diameter (alternatively, the radius) of each cluster is at most b. We present algorithms that by sampling from a set X , distinguish between the case that X is (k; b)clusterable and the case that X is fflfar from being (k; b 0 )clusterable for any given 0 ! ffl 1 and for b 0 b. In fflfar from being (k; b 0 )clusterable we mean that more than ffl \Delta jX j points should be removed from X so that it becomes (k; b 0 )clusterable. We give algorithms for a variety of cost measures that use a sample of size independent of jX j, and polynomial in k and 1=ffl. Our algorithms can also be used to find approximately good clusterings. Namely, these are clusterings of all but an fflfraction of the points in X that have optimal (or close to optimal) cost. The benefit of our algorithms is that they construct an implicit representation of such clusterings in time independ...
Energy Minimization via Graph Cuts: Settling What is Possible
 IN IEEE CONFERENCE ON COMPUTER VISION AND PATTERN RECOGNITION
, 2005
"... The recent explosion of interest in graph cut methods in computer vision naturally spawns the question: what energy functions can be minimized via graph cuts? This question was first attacked by two papers of Kolmogorov and Zabih [23, 24], in which they dealt with functions with pairwise and triplew ..."
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Cited by 44 (0 self)
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The recent explosion of interest in graph cut methods in computer vision naturally spawns the question: what energy functions can be minimized via graph cuts? This question was first attacked by two papers of Kolmogorov and Zabih [23, 24], in which they dealt with functions with pairwise and triplewise pixel interactions. In this work, we extend their results in two directions. First, we examine the case of kwise pixel interactions; the results are derived from a purely algebraic approach. Second, we discuss the applicability of provably approximate algorithms. Both of these developments should help researchers best understand what can and cannot be achieved when designing graph cut based algorithms.
Approximating the Minimum Spanning Tree Weight in Sublinear Time
 In Proceedings of the 28th Annual International Colloquium on Automata, Languages and Programming (ICALP
, 2001
"... We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set {1,...,w}, and given a parameter 0 < ε < 1/2, estimates in time O(dwε−2 log dw ε) the weight of the minimum spanning tree of G with a relative erro ..."
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Cited by 38 (6 self)
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We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set {1,...,w}, and given a parameter 0 < ε < 1/2, estimates in time O(dwε−2 log dw ε) the weight of the minimum spanning tree of G with a relative error of at most ε. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of Ω(dwε−2) on the probe and time complexity of any approximation algorithm for MST weight. The essential component of our algorithm is a procedure for estimating in time O(dε−2 log d ε) the number of connected components of an unweighted graph to within an additive error of εn. (This becomes O(ε−2 log 1 ε) for d = O(1).) The time bound is shown to be tight up to within the log d ε factor. Our connectedcomponents algorithm picks O(1/ε2) vertices in the graph and then grows “local spanning trees” whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST. 1
Gaussian Bounds for Noise Correlation of Functions and Tight Analysis of Long Codes
 In IEEE Symposium on Foundations of Computer Science (FOCS
, 2008
"... In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recentl ..."
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Cited by 37 (5 self)
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In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recently obtained with O’Donnell and Oleszkiewicz for multilinear polynomials with low influences and bounded degree and on properties of multidimensional Gaussian distributions. We present two applications of the new bounds to the theory of social choice. We show that Majority is asymptotically the most predictable function among all low influence functions given a random sample of the voters. Moreover, we derive an almost tight bound in the context of Condorcet aggregation and low influence voting schemes on a large number of candidates. In particular, we show that for every low influence aggregation function, the probability that Condorcet voting on k candidates will result in a unique candidate that is preferable to all others is k−1+o(1). This matches the asymptotic behavior of the majority function for which the probability is k−1−o(1). A number of applications in hardness of approximation in theoretical computer science were