239 research outputs found

    Decremental Single-Source Reachability in Planar Digraphs

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    In this paper we show a new algorithm for the decremental single-source reachability problem in directed planar graphs. It processes any sequence of edge deletions in O(nlog2nloglogn)O(n\log^2{n}\log\log{n}) total time and explicitly maintains the set of vertices reachable from a fixed source vertex. Hence, if all edges are eventually deleted, the amortized time of processing each edge deletion is only O(log2nloglogn)O(\log^2 n \log \log n), which improves upon a previously known O(n)O(\sqrt{n}) solution. We also show an algorithm for decremental maintenance of strongly connected components in directed planar graphs with the same total update time. These results constitute the first almost optimal (up to polylogarithmic factors) algorithms for both problems. To the best of our knowledge, these are the first dynamic algorithms with polylogarithmic update times on general directed planar graphs for non-trivial reachability-type problems, for which only polynomial bounds are known in general graphs

    On Semi-Algebraic Proofs and Algorithms

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    Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs

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    As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems, such as maximum matching and minimum vertex cover, on large datasets. For massive inputs, several different computational models have been introduced, including the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In each model, algorithms are analyzed in terms of resources such as space used or rounds of communication needed, in addition to the more traditional approximation ratio. In this paper, we give a single unified approach that yields better approximation algorithms for matching and vertex cover in all these models. The highlights include: * The first one pass, significantly-better-than-2-approximation for matching in random arrival streams that uses subquadratic space, namely a (1.5+ϵ)(1.5+\epsilon)-approximation streaming algorithm that uses O(n1.5)O(n^{1.5}) space for constant ϵ>0\epsilon > 0. * The first 2-round, better-than-2-approximation for matching in the MPC model that uses subquadratic space per machine, namely a (1.5+ϵ)(1.5+\epsilon)-approximation algorithm with O(mn+n)O(\sqrt{mn} + n) memory per machine for constant ϵ>0\epsilon > 0. By building on our unified approach, we further develop parallel algorithms in the MPC model that give a (1+ϵ)(1 + \epsilon)-approximation to matching and an O(1)O(1)-approximation to vertex cover in only O(loglogn)O(\log\log{n}) MPC rounds and O(n/polylog(n))O(n/poly\log{(n)}) memory per machine. These results settle multiple open questions posed in the recent paper of Czumaj~et.al. [STOC 2018]

    Efficient Statistics, in High Dimensions, from Truncated Samples

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    We provide an efficient algorithm for the classical problem, going back to Galton, Pearson, and Fisher, of estimating, with arbitrary accuracy the parameters of a multivariate normal distribution from truncated samples. Truncated samples from a dd-variate normal N(μ,Σ){\cal N}(\mathbf{\mu},\mathbf{\Sigma}) means a samples is only revealed if it falls in some subset SRdS \subseteq \mathbb{R}^d; otherwise the samples are hidden and their count in proportion to the revealed samples is also hidden. We show that the mean μ\mathbf{\mu} and covariance matrix Σ\mathbf{\Sigma} can be estimated with arbitrary accuracy in polynomial-time, as long as we have oracle access to SS, and SS has non-trivial measure under the unknown dd-variate normal distribution. Additionally we show that without oracle access to SS, any non-trivial estimation is impossible.Comment: to appear at 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 201

    On the Symmetries of and Equivalence Test for Design Polynomials

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    In a Nisan-Wigderson design polynomial (in short, a design polynomial), every pair of monomials share a few common variables. A useful example of such a polynomial, introduced in [Neeraj Kayal et al., 2014], is the following: NW_{d,k}({x}) = sum_{h in F_d[z], deg(h) <= k}{ prod_{i=0}^{d-1}{x_{i, h(i)}}}, where d is a prime, F_d is the finite field with d elements, and k << d. The degree of the gcd of every pair of monomials in NW_{d,k} is at most k. For concreteness, we fix k = ceil[sqrt{d}]. The family of polynomials NW := {NW_{d,k} : d is a prime} and close variants of it have been used as hard explicit polynomial families in several recent arithmetic circuit lower bound proofs. But, unlike the permanent, very little is known about the various structural and algorithmic/complexity aspects of NW beyond the fact that NW in VNP. Is NW_{d,k} characterized by its symmetries? Is it circuit-testable, i.e., given a circuit C can we check efficiently if C computes NW_{d,k}? What is the complexity of equivalence test for NW, i.e., given black-box access to a f in F[{x}], can we check efficiently if there exists an invertible linear transformation A such that f = NW_{d,k}(A * {x})? Characterization of polynomials by their symmetries plays a central role in the geometric complexity theory program. Here, we answer the first two questions and partially answer the third. We show that NW_{d,k} is characterized by its group of symmetries over C, but not over R. We also show that NW_{d,k} is characterized by circuit identities which implies that NW_{d,k} is circuit-testable in randomized polynomial time. As another application of this characterization, we obtain the "flip theorem" for NW. We give an efficient equivalence test for NW in the case where the transformation A is a block-diagonal permutation-scaling matrix. The design of this algorithm is facilitated by an almost complete understanding of the group of symmetries of NW_{d,k}: We show that if A is in the group of symmetries of NW_{d,k} then A = D * P, where D and P are diagonal and permutation matrices respectively. This is proved by completely characterizing the Lie algebra of NW_{d,k}, and using an interplay between the Hessian of NW_{d,k} and the evaluation dimension
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