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181
DataStreams and Histograms
, 2001
"... Histograms have been used widely to capture data distribution, to represent the data by a small number of step functions. Dynamic programming algorithms which provide optimal construction of these histograms exist, albeit running in quadratic time and linear space. In this paper we provide linear ti ..."
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Cited by 147 (9 self)
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Histograms have been used widely to capture data distribution, to represent the data by a small number of step functions. Dynamic programming algorithms which provide optimal construction of these histograms exist, albeit running in quadratic time and linear space. In this paper we provide linear time construction of 1 + epsilon approximation of optimal histograms, running in polylogarithmic space. Our results extend to the context of datastreams, and in fact generalize to give 1 + epsilon approximation of several problems in datastreams which require partitioning the index set into intervals. The only assumptions required are that the cost of an interval is monotonic under inclusion (larger interval has larger cost) and that the cost can be computed or approximated in small space. This exhibits a nice class of problems for which we can have near optimal datastream algorithms.
Fast Monte Carlo algorithms for matrices I: Approximating matrix multiplication
 SIAM Journal on Computing
, 2004
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On graph problems in a semistreaming model
 In 31st International Colloquium on Automata, Languages and Programming
, 2004
"... Abstract. We formalize a potentially rich new streaming model, the semistreaming model, that we believe is necessary for the fruitful study of efficient algorithms for solving problems on massive graphs whose edge sets cannot be stored in memory. In this model, the input graph, G = (V, E), is prese ..."
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Cited by 108 (15 self)
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Abstract. We formalize a potentially rich new streaming model, the semistreaming model, that we believe is necessary for the fruitful study of efficient algorithms for solving problems on massive graphs whose edge sets cannot be stored in memory. In this model, the input graph, G = (V, E), is presented as a stream of edges (in adversarial order), and the storage space of an algorithm is bounded by O(n · polylog n), where n = V . We are particularly interested in algorithms that use only one pass over the input, but, for problems where this is provably insufficient, we also look at algorithms using constant or, in some cases, logarithmically many passes. In the course of this general study, we give semistreaming constant approximation algorithms for the unweighted and weighted matching problems, along with a further algorithm improvement for the bipartite case. We also exhibit log n / log log n semistreaming approximations to the diameter and the problem of computing the distance between specified vertices in a weighted graph. These are complemented by Ω(log (1−ɛ) n) lower bounds. 1
Secure multiparty computation of approximations
, 2001
"... Approximation algorithms can sometimes provide efficient solutions when no efficient exact computation is known. In particular, approximations are often useful in a distributed setting where the inputs are held by different parties and may be extremely large. Furthermore, for some applications, the ..."
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Cited by 107 (26 self)
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Approximation algorithms can sometimes provide efficient solutions when no efficient exact computation is known. In particular, approximations are often useful in a distributed setting where the inputs are held by different parties and may be extremely large. Furthermore, for some applications, the parties want to compute a function of their inputs securely, without revealing more information than necessary. In this work we study the question of simultaneously addressing the above efficiency and security concerns via what we call secure approximations. We start by extending standard definitions of secure (exact) computation to the setting of secure approximations. Our definitions guarantee that no additional information is revealed by the approximation beyond what follows from the output of the function being approximated. We then study the complexity of specific secure approximation problems. In particular, we obtain a sublinearcommunication protocol for securely approximating the Hamming distance and a polynomialtime protocol for securely approximating the permanent and related #Phard problems. 1
Characterizing Memory Requirements for Queries over Continuous Data
 In PODS
, 2002
"... This paper deals with continuous conjunctive queries with arithmetic comparisons and optional aggregation over multiple data streams. An algorithm is presented for determining whether or not any given query can be evaluated using a bounded amount of memory for all possible instances of the data stre ..."
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Cited by 99 (11 self)
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This paper deals with continuous conjunctive queries with arithmetic comparisons and optional aggregation over multiple data streams. An algorithm is presented for determining whether or not any given query can be evaluated using a bounded amount of memory for all possible instances of the data streams. For queries that can be evaluated using bounded memory, an execution strategy based on constantsized synopses of the data streams is proposed. For queries that cannot be evaluated using bounded memory, data stream scenarios are identified in which evaluating the queries requires memory linear in the size of the unbounded streams
StreamingData Algorithms for HighQuality Clustering
, 2001
"... As data gathering grows easier, and as researchers discover new ways to interpret data, streamingdata algorithms have become essential in many fields. Data stream computation precludes algorithms that require random access or large memory. In this paper, we consider the problem of clustering data s ..."
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Cited by 95 (1 self)
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As data gathering grows easier, and as researchers discover new ways to interpret data, streamingdata algorithms have become essential in many fields. Data stream computation precludes algorithms that require random access or large memory. In this paper, we consider the problem of clustering data streams, which is important in the analysis a variety of sources of data streams, such as routing data, telephone records, web documents, and clickstreams. We provide a new clustering algorithms with theoretical guarantees on its performance. We give empirical evidence of its superiority over the commonlyused kMeans algorithm. We then adapt our algorithm to be able to operate on data streams and experimentally demonstrate its superior performance in this context.
Maintaining Variance and kMedians over Data Stream Windows
 In PODS
, 2003
"... The sliding window model is useful for discounting stale data in data stream applications. In this model, data elements arrive continually and only the most recent N elements are used when answering queries. We present a novel technique for solving two important and related problems in the sliding w ..."
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Cited by 93 (1 self)
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The sliding window model is useful for discounting stale data in data stream applications. In this model, data elements arrive continually and only the most recent N elements are used when answering queries. We present a novel technique for solving two important and related problems in the sliding window model  maintaining variance and maintaining a k median clustering. Our solution to the problem of maintaining variance provides a continually updated estimate of the variance of the last N values in a data stream with relative error of at most # using O( # 2 log N) memory. We present a constantfactor approximation algorithm which maintains an approximate kmedian solution for the last N data points using O( N) memory, where # < 1/2 is a parameter which trades o# the space bound with the approximation factor of O(2 ).
Faster CoreSet Constructions and Data Stream Algorithms in Fixed Dimensions
 Comput. Geom. Theory Appl
, 2003
"... We speed up previous (1 + ")factor approximation algorithms for a number of geometric optimization problems in xed dimensions: diameter, width, minimumradius enclosing cylinder, minimumwidth annulus, minimumvolume bounding box, minimumwidth cylindrical shell, etc. ..."
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Cited by 86 (6 self)
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We speed up previous (1 + ")factor approximation algorithms for a number of geometric optimization problems in xed dimensions: diameter, width, minimumradius enclosing cylinder, minimumwidth annulus, minimumvolume bounding box, minimumwidth cylindrical shell, etc.
The String Edit Distance Matching Problems with Moves
, 2006
"... The edit distance between two strings S and R is defined to be the minimum number of character inserts, deletes and changes needed to convert R to S. Given a text string t of length n, and a pattern string p of length m, informally, the string edit distance matching problem is to compute the smalles ..."
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Cited by 72 (3 self)
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The edit distance between two strings S and R is defined to be the minimum number of character inserts, deletes and changes needed to convert R to S. Given a text string t of length n, and a pattern string p of length m, informally, the string edit distance matching problem is to compute the smallest edit distance between p and substrings of t. We relax the problem so that (a) we allow an additional operation, namely, substring moves, and (b) we allow approximation of this string edit distance. Our result is a near linear time deterministic algorithm to produce a factor of O(log n log ∗ n) approximation to the string edit distance with moves. This is the first known significantly subquadratic algorithm for a string edit distance problem in which the distance involves nontrivial alignments. Our results are obtained by embedding strings into L1 vector space using a simplified parsing technique we call Edit
Efficient semistreaming algorithms for local triangle counting in massive graphs
 in KDD’08, 2008
"... In this paper we study the problem of local triangle counting in large graphs. Namely, given a large graph G = (V, E) we want to estimate as accurately as possible the number of triangles incident to every node v ∈ V in the graph. The problem of computing the global number of triangles in a graph ha ..."
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Cited by 70 (4 self)
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In this paper we study the problem of local triangle counting in large graphs. Namely, given a large graph G = (V, E) we want to estimate as accurately as possible the number of triangles incident to every node v ∈ V in the graph. The problem of computing the global number of triangles in a graph has been considered before, but to our knowledge this is the first paper that addresses the problem of local triangle counting with a focus on the efficiency issues arising in massive graphs. The distribution of the local number of triangles and the related local clustering coefficient can be used in many interesting applications. For example, we show that the measures we compute can help to detect the presence of spamming activity in largescale Web graphs, as well as to provide useful features to assess content quality in social networks. For computing the local number of triangles we propose two approximation algorithms, which are based on the idea of minwise independent permutations (Broder et al. 1998). Our algorithms operate in a semistreaming fashion, using O(V ) space in main memory and performing O(log V ) sequential scans over the edges of the graph. The first algorithm we describe in this paper also uses O(E) space in external memory during computation, while the second algorithm uses only main memory. We present the theoretical analysis as well as experimental results in massive graphs demonstrating the practical efficiency of our approach. Luca Becchetti was partially supported by EU Integrated