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Counting Stars and Other Small Subgraphs in Sublinear Time
"... Detecting and counting the number of copies of certain subgraphs (also known as network motifs or graphlets), is motivated by applications in a variety of areas ranging from Biology to the study of the WorldWideWeb. Several polynomialtime algorithms have been suggested for counting or detecting t ..."
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Detecting and counting the number of copies of certain subgraphs (also known as network motifs or graphlets), is motivated by applications in a variety of areas ranging from Biology to the study of the WorldWideWeb. Several polynomialtime algorithms have been suggested for counting or detecting the number of occurrences of certain network motifs. However, a need for more efficient algorithms arises when the input graph is very large, as is indeed the case in many applications of motif counting. In this paper we design sublineartime algorithms for approximating the number of copies of certain constantsize subgraphs in a graph G. That is, our algorithms do not read the whole graph, but rather query parts of the graph. Specifically, we consider algorithms that may query the degree of any vertex of their choice and may ask for any neighbor of any vertex of their choice. The main focus of this work is on the basic problem of counting the number of length2 paths and more generally on counting the number of stars of a certain size. Specifically, we design an algorithm that, given an approximation parameter 0 < ɛ < 1 and query access to a graph G, outputs an estimate ˆνs such that with high constant probability, (1−ɛ)νs(G) ≤ ˆνs ≤ (1+ɛ)νs(G), where νs(G) denotes the number of stars of size s + 1 in the graph. The expected query ( complexity and { running time of}) the algorithm are O
Introduction to testing graph properties
 In Property Testing
, 2010
"... Abstract. The aim of this article is to introduce the reader to the study of testing graph properties, while focusing on the main models and issues involved. No attempt is made to provide a comprehensive survey of this ..."
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Abstract. The aim of this article is to introduce the reader to the study of testing graph properties, while focusing on the main models and issues involved. No attempt is made to provide a comprehensive survey of this
Facility location in sublinear time
 In 32nd International Colloquium on Automata, Languages, and Programming
, 2005
"... Abstract. In this paper we present a randomized constant factor approximation algorithm for the problem of computing the optimal cost of the metric Minimum Facility Location problem, in the case of uniform costs and uniform demands, and in which every point can open a facility. By exploiting the fac ..."
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Abstract. In this paper we present a randomized constant factor approximation algorithm for the problem of computing the optimal cost of the metric Minimum Facility Location problem, in the case of uniform costs and uniform demands, and in which every point can open a facility. By exploiting the fact that we are approximating the optimal cost without computing an actual solution, we give the first algorithm for this problem with running time O(n log 2 n), where n is the number of metric space points. Since the size of the representation of an npoint metric space is Θ(n 2), the complexity of our algorithm is sublinear with respect to the input size. We consider also the general version of the metric Minimum Facility Location problem and we show that there is no o(n 2)time algorithm, even a randomized one, that approximates the optimal solution to within any factor. This result can be generalized to some related problems, and in particular, the cost of minimumcost matching, the cost of bichromatic matching, or the cost of n/2median cannot be approximated in o(n 2)time. 1
Approximating the Weight of the Euclidean Minimum Spanning Tree in Sublinear Time
"... We consider the problem of computing the weight of a Euclidean minimum spanning tree for a set of n points in Rd. We focus on the setting where the input point set is supported by certain basic (and commonly used) geometric data structures that can provide efficient access to the input in a struct ..."
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We consider the problem of computing the weight of a Euclidean minimum spanning tree for a set of n points in Rd. We focus on the setting where the input point set is supported by certain basic (and commonly used) geometric data structures that can provide efficient access to the input in a structured way. We present an algorithm that estimates with high probability the weight of a Euclidean minimum spanning tree of a set of points to within 1 + " using only eO(pn poly(1=")) queries for constant d. The algorithm assumes that the input is supported by a minimal bounding cube enclosing it, by orthogonal range queries, and by cone approximate nearest neighbors queries.
Approximate Testing of Visual Properties
 Proc. Sixth Int’l Workshop Approximation Algorithms for Combinatorial Optimization Problems
, 2003
"... Abstract. We initiate a study of property testing as applied to visual properties of images. Property testing is a rapidly developing area investigating algorithms that, with a small number of local checks, distinguish objects satisfying a given property from objects which need to be modified signif ..."
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Abstract. We initiate a study of property testing as applied to visual properties of images. Property testing is a rapidly developing area investigating algorithms that, with a small number of local checks, distinguish objects satisfying a given property from objects which need to be modified significantly to satisfy the property. We study visual properties of discretized images represented by n × n matrices of binary pixel values. We obtain algorithms with query complexity independent of n for several basic properties: being a halfplane, connectedness and convexity. 1
On derandomizing probabilistic sublineartime algorithms
 IN PROCEEDINGS OF THE 22TH ANNUAL IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2007
"... There exists a positive constant α < 1 such that for any function T (n) ≤ n α and for any problem L ∈ BPTIME(T (n)), there exists a deterministic algorithm running in poly(T (n)) time which decides L, except for at most a 2 −Ω(T (n) log T (n)) fraction of inputs of length n. The proof uses a nove ..."
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There exists a positive constant α < 1 such that for any function T (n) ≤ n α and for any problem L ∈ BPTIME(T (n)), there exists a deterministic algorithm running in poly(T (n)) time which decides L, except for at most a 2 −Ω(T (n) log T (n)) fraction of inputs of length n. The proof uses a novel derandomization technique based on a new type of randomness extractors, called exposureresilient extractors. An exposureresilient extractor is an efficient procedure that, from a random variable with imperfect minentropy, produces randomness that passes all statistical tests including those that have bounded access to the random variable, with adaptive queries that can depend on the string being tested. More precisely, EXT: {0, 1} n × {0, 1} d → {0, 1} m is a (k, ɛ)exposure resilient extractor resistant to q queries if, when the minentropy of x is at least k and y is random, EXT(x, y) looks ɛrandom to all statistical tests modeled by oracle circuits of unbounded complexity that can query q bits of x. Besides the extractor that is needed for the above derandomization (whose parameters are tailored for this application), we construct, for any δ < 1, a (k, ɛ)exposure resilient extractor with query resistance n δ, k = n − n Ω(1) , ɛ = n −Ω(1) , m = n Ω(1) and d = O(log n).
New Sublinear Methods in the Struggle against Classical Problems
, 2010
"... We study the time and query complexity of approximation algorithms that access only a minuscule fraction of the input, focusing on two classical sources of problems: combinatorial graph optimization and manipulation of strings. The tools we develop find applications outside of the area of sublinear ..."
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We study the time and query complexity of approximation algorithms that access only a minuscule fraction of the input, focusing on two classical sources of problems: combinatorial graph optimization and manipulation of strings. The tools we develop find applications outside of the area of sublinear algorithms. For instance, we obtain a more efficient approximation algorithm for edit distance and distributed algorithms for combinatorial problems on graphs that run in a constant number of communication rounds.
Testing Euclidean minimum spanning trees in the plane
 ACM Transactions on Algorithms
, 2007
"... Given a Euclidean graph G over a set P of n points in the plane, we are interested in verifying whether G is a Euclidean minimum spanning tree (EMST) of P or G differs from it in more than ǫn edges. We assume that G is given in adjacency list representation and the point/vertex set P is given in an ..."
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Given a Euclidean graph G over a set P of n points in the plane, we are interested in verifying whether G is a Euclidean minimum spanning tree (EMST) of P or G differs from it in more than ǫn edges. We assume that G is given in adjacency list representation and the point/vertex set P is given in an array. We present a property testing algorithm that accepts graph G if it is an EMST of P and that rejects with probability at least 2 3 if G differs from every EMST of P in more than ǫn edges. Our algorithm runs in O ( � n/ǫ · log2 (n/ǫ)) time and has a query complexity of O ( � n/ǫ · log(n/ǫ)).
EXPOSURERESILIENT EXTRACTORS AND THE DERANDOMIZATION OF PROBABILISTIC SUBLINEAR TIME ALGORITHMS
"... Abstract. There exists a positive constant α < 1 such that for any function T (n) ≤ n α and for any problem L ∈ BPTIME(T (n)), there exists a deterministic algorithm running in poly(T (n)) time which decides L, except for at most a 2 −Ω(T (n) log T (n)) fraction of inputs of length n. The proof use ..."
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Abstract. There exists a positive constant α < 1 such that for any function T (n) ≤ n α and for any problem L ∈ BPTIME(T (n)), there exists a deterministic algorithm running in poly(T (n)) time which decides L, except for at most a 2 −Ω(T (n) log T (n)) fraction of inputs of length n. The proof uses a novel derandomization technique based on a new type of randomness extractors, called exposureresilient extractors. An exposureresilient extractor is an efficient procedure that, from a random variable with imperfect randomness, produces randomness that passes all statistical tests including those that have bounded access to the random variable, with adaptive queries that can depend on the string being tested. More precisely, EXT: {0, 1} n × {0, 1} d → {0, 1} m is a (k, ɛ)exposureresilient extractor resistant to q queries if, when the minentropy of the random variable x is at least k and the random variable y is uniformly distributed, EXT(x, y) looks ɛrandom to all statistical tests modeled by oracle circuits of unbounded size that can query q bits of x. Besides the extractor that is needed for the proof of the main result (whose parameters are tailored for this application), we construct, for any δ < 1, a polynomialtime computable (k, ɛ)exposureresilient extractor with query resistance n δ, k = n − n Ω(1) , ɛ = n −Ω(1) , m = n Ω(1) and d = O(log n).
A NearOptimal SublinearTime Algorithm for Approximating the Minimum Vertex Cover Size
, 2011
"... We give a nearly optimal sublineartime algorithm for approximating the size of a minimum vertex cover in a graph G. The algorithm may query the degree deg(v) of any vertex v of its choice, and for each 1 ≤ i ≤ deg(v), it may ask for the ith neighbor of v. Letting VCopt(G) denote the minimum size of ..."
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We give a nearly optimal sublineartime algorithm for approximating the size of a minimum vertex cover in a graph G. The algorithm may query the degree deg(v) of any vertex v of its choice, and for each 1 ≤ i ≤ deg(v), it may ask for the ith neighbor of v. Letting VCopt(G) denote the minimum size of vertex cover in G, the algorithm outputs, with high constant success probability, an estimate ̂VC(G) such that VCopt(G) ≤ ̂ VC(G) ≤ 2VCopt(G) + ǫn, where ǫ is a given additive approximation parameter. We refer to such an estimate as a (2, ǫ)estimate. The query complexity and running time of the algorithm are Õ ( ¯ d · poly(1/ǫ)), where ¯ d denotes the average vertex degree in the graph. The best previously known sublinear algorithm, of Yoshida et al. (STOC 2009), has query complexity and running time O(d4 /ǫ2), where d is the maximum degree in the graph. Given the lower bound of Ω ( ¯ d) (for constant ǫ) for obtaining such an estimate (with any constant multiplicative factor) due to Parnas and Ron (TCS 2007), our result is nearly optimal. In the case that the graph is dense, that is, the number of edges is Θ(n2), we consider another model, in which the algorithm may ask, for any pair of vertices u and v, whether there is an edge between u and v. We show how to adapt the algorithm that uses neighbor queries to this model and obtain an