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A sublinear time algorithm for pagerank computations
 In WAW
, 2012
"... Abstract. In a network, identifying all vertices whose PageRank is more than a given threshold value ∆ is a basic problem that has arisen in Web and social network analyses. In this paper, we develop a nearly optimal, sublinear time, randomized algorithm for a close variant of this problem. When giv ..."
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Cited by 9 (3 self)
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Abstract. In a network, identifying all vertices whose PageRank is more than a given threshold value ∆ is a basic problem that has arisen in Web and social network analyses. In this paper, we develop a nearly optimal, sublinear time, randomized algorithm for a close variant of this problem. When given a directed network G = (V, E), a threshold value ∆, and a positive constant c> 3, with probability 1 − o(1), our algorithm will return a subset S ⊆ V with the property that S contains all vertices of PageRank at least ∆ and no vertex with PageRank less than ∆/c. The running time of our algorithm is always). In addition, our algorithm can be efficiently implemented in various network access models including the Jump and Crawl query model recently studied by [6], making it suitable for dealing with large social and information networks. Õ( n As part of our analysis, we show that any algorithm for solving this problem must have expected time complexity of Ω ( n). Thus, our algorithm is optimal up to logarithmic factors. Our algorithm (for identifying vertices with significant PageRank) applies a multiscale sampling scheme that uses a fast personalized PageRank estimator as its main subroutine. For that, we develop a new local randomized algorithm for approximating personalized PageRank which is more robust than the earlier ones developed by Jeh and Widom [9] and by Andersen, Chung, and Lang [2]. 1
Multiscale matrix sampling and sublineartime PageRank computation
 CoRR
"... Abstract A fundamental problem arising in many applications in Web science and social network analysis is the problem of identifying all nodes in a network whose PageRank exceeds a given threshold ∆. In this paper, we study the probabilistic version of the problem where given an arbitrary approxima ..."
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Abstract A fundamental problem arising in many applications in Web science and social network analysis is the problem of identifying all nodes in a network whose PageRank exceeds a given threshold ∆. In this paper, we study the probabilistic version of the problem where given an arbitrary approximation factor c > 1, we are asked to output a set S of nodes such that with high probability, S contains all nodes of PageRank at least ∆, and no node of PageRank smaller than ∆/c. We call this problem SignificantPageRanks. We develop a nearly optimal, local algorithm for the problem with runtime complexitỹ O(n/∆) on networks with n nodes, where the tilde hides a polylogarithmic factor. We show that any algorithm for solving this problem must have runtime of Ω(n/∆), rendering our algorithm optimal up to logarithmic factors. Our algorithm has sublinear time complexity for applications including Web crawling and Web search that require efficient identification of nodes whose PageRanks are above a threshold ∆ = n δ , for some constant 0 < δ < 1. Our algorithm comes with two main technical contributions. The first is a multiscale sampling scheme for a basic matrix problem that could be of interest on its own. For us, it appears as an abstraction of a subproblem we need to tackle in order to solve the SignificantPageRanks problem, but we hope that this abstraction will be useful in designing fast algorithms for identifying nodes that are significant beyond PageRank measurements. In the abstract matrix problem it is assumed that one can access an unknown rightstochastic matrix by querying its rows, where the cost of a query and the accuracy of the answers depend on a precision parameter ǫ. At a cost propositional to 1/ǫ, the query will return a list of O(1/ǫ) entries and their indices that provide an ǫprecision approximation of the row. Our task is to find a set that contains all columns whose sum is at least ∆, and omits any column whose sum is less than ∆/c. Our multiscale sampling scheme solves this problem with costÕ(n/∆), while traditional sampling algorithms would take time Θ((n/∆) 2 ). Our second main technical contribution is a new local algorithm for approximating personalized PageRank, which is more robust than the earlier ones developed in Together with our multiscale sampling scheme we are able to optimally solve the SignificantPageRanks problem.
Optimal cuts and bisections on the real line in polynomial time
 In CoRR abs/1207.0933, 2012. 8 S. Khot. Ruling out PTAS for Graph MinBisection, Densest Subgraph and Bipartite Clique. In Proceedings of the 45th Annual Symposium on Foundations of Computer Science
, 2004
"... The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time algorithm. Our results depend on a new technique of dealing with met ..."
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The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time algorithm. Our results depend on a new technique of dealing with metric equalities and their connection to dynamic programming. The method of our solution could be also of independent interest.
REGULAR PAPER Approximation Algorithms for Discrete Polynomial Optimization
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Optimal Cuts and Partitions in Tree Metrics in Polynomial Time
, 2012
"... We present a polynomial time dynamic programming algorithm for optimal partitions in the shortest path metric induced by a tree. This resolves, among other things, the exact complexity status of the optimal partition problems in one dimensional geometric metric settings. Our method of solution could ..."
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We present a polynomial time dynamic programming algorithm for optimal partitions in the shortest path metric induced by a tree. This resolves, among other things, the exact complexity status of the optimal partition problems in one dimensional geometric metric settings. Our method of solution could be also of independent interest in other applications. We discuss also an extension of our method to the class of metrics induced by the bounded treewidth graphs.