186 research outputs found

    Probabilistic Spectral Sparsification In Sublinear Time

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    In this paper, we introduce a variant of spectral sparsification, called probabilistic (ε,δ)(\varepsilon,\delta)-spectral sparsification. Roughly speaking, it preserves the cut value of any cut (S,Sc)(S,S^{c}) with an 1±ε1\pm\varepsilon multiplicative error and a δS\delta\left|S\right| additive error. We show how to produce a probabilistic (ε,δ)(\varepsilon,\delta)-spectral sparsifier with O(nlogn/ε2)O(n\log n/\varepsilon^{2}) edges in time O~(n/ε2δ)\tilde{O}(n/\varepsilon^{2}\delta) time for unweighted undirected graph. This gives fastest known sub-linear time algorithms for different cut problems on unweighted undirected graph such as - An O~(n/OPT+n3/2+t)\tilde{O}(n/OPT+n^{3/2+t}) time O(logn/t)O(\sqrt{\log n/t})-approximation algorithm for the sparsest cut problem and the balanced separator problem. - A n1+o(1)/ε4n^{1+o(1)}/\varepsilon^{4} time approximation minimum s-t cut algorithm with an εn\varepsilon n additive error

    Effective Resistances in Non-Expander Graphs

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    Densest Subgraph in Dynamic Graph Streams

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    In this paper, we consider the problem of approximating the densest subgraph in the dynamic graph stream model. In this model of computation, the input graph is defined by an arbitrary sequence of edge insertions and deletions and the goal is to analyze properties of the resulting graph given memory that is sub-linear in the size of the stream. We present a single-pass algorithm that returns a (1+ϵ)(1+\epsilon) approximation of the maximum density with high probability; the algorithm uses O(\epsilon^{-2} n \polylog n) space, processes each stream update in \polylog (n) time, and uses \poly(n) post-processing time where nn is the number of nodes. The space used by our algorithm matches the lower bound of Bahmani et al.~(PVLDB 2012) up to a poly-logarithmic factor for constant ϵ\epsilon. The best existing results for this problem were established recently by Bhattacharya et al.~(STOC 2015). They presented a (2+ϵ)(2+\epsilon) approximation algorithm using similar space and another algorithm that both processed each update and maintained a (4+ϵ)(4+\epsilon) approximation of the current maximum density in \polylog (n) time per-update.Comment: To appear in MFCS 201

    Effective Resistances in Non-Expander Graphs

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    Effective resistances are ubiquitous in graph algorithms and network analysis. In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair ss and tt. We consider the classical adjacency list model for local algorithms. While recent works have provided sublinear time algorithms for expander graphs, we prove several lower bounds for general graphs of nn vertices and mm edges: 1.It needs Ω(n)\Omega(n) queries to obtain 1.011.01-approximations of the effective resistance of an adjacent pair ss and tt, even for graphs of degree at most 3 except ss and tt. 2.For graphs of degree at most dd and any parameter \ell, it needs Ω(m/)\Omega(m/\ell) queries to obtain cmin{d,}c \cdot \min\{d, \ell\}-approximations where c>0c>0 is a universal constant. Moreover, we supplement the first lower bound by providing a sublinear time (1+ϵ)(1+\epsilon)-approximation algorithm for graphs of degree 2 except the pair ss and tt. One of our technical ingredients is to bound the expansion of a graph in terms of the smallest non-trivial eigenvalue of its Laplacian matrix after removing edges. We discover a new lower bound on the eigenvalues of perturbed graphs (resp. perturbed matrices) by incorporating the effective resistance of the removed edge (resp. the leverage scores of the removed rows), which may be of independent interest

    On Solving Linear Systems in Sublinear Time

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    We study sublinear algorithms that solve linear systems locally. In the classical version of this problem the input is a matrix S in R^{n x n} and a vector b in R^n in the range of S, and the goal is to output x in R^n satisfying Sx=b. For the case when the matrix S is symmetric diagonally dominant (SDD), the breakthrough algorithm of Spielman and Teng [STOC 2004] approximately solves this problem in near-linear time (in the input size which is the number of non-zeros in S), and subsequent papers have further simplified, improved, and generalized the algorithms for this setting. Here we focus on computing one (or a few) coordinates of x, which potentially allows for sublinear algorithms. Formally, given an index u in [n] together with S and b as above, the goal is to output an approximation x^_u for x^*_u, where x^* is a fixed solution to Sx=b. Our results show that there is a qualitative gap between SDD matrices and the more general class of positive semidefinite (PSD) matrices. For SDD matrices, we develop an algorithm that approximates a single coordinate x_{u} in time that is polylogarithmic in n, provided that S is sparse and has a small condition number (e.g., Laplacian of an expander graph). The approximation guarantee is additive | x^_u-x^*_u | 0. We further prove that the condition-number assumption is necessary and tight. In contrast to the SDD matrices, we prove that for certain PSD matrices S, the running time must be at least polynomial in n (for the same additive approximation), even if S has bounded sparsity and condition number

    Quantum Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving

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    Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, "spectral sparsification" reduces the number of edges to near-linear in the number of nodes, while approximately preserving the cut and spectral structure of the graph. In this work we demonstrate a polynomial quantum speedup for spectral sparsification and many of its applications. In particular, we give a quantum algorithm that, given a weighted graph with nn nodes and mm edges, outputs a classical description of an ϵ\epsilon-spectral sparsifier in sublinear time O~(mn/ϵ)\tilde{O}(\sqrt{mn}/\epsilon). This contrasts with the optimal classical complexity O~(m)\tilde{O}(m). We also prove that our quantum algorithm is optimal up to polylog-factors. The algorithm builds on a string of existing results on sparsification, graph spanners, quantum algorithms for shortest paths, and efficient constructions for kk-wise independent random strings. Our algorithm implies a quantum speedup for solving Laplacian systems and for approximating a range of cut problems such as min cut and sparsest cut.Comment: v2: several small improvements to the text. An extended abstract will appear in FOCS'20; v3: corrected a minor mistake in Appendix

    Multi-Scale Matrix Sampling and Sublinear-Time PageRank Computation

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    A fundamental problem arising in many applications in Web science and social network analysis is, given an arbitrary approximation factor c>1c>1, to output a set SS of nodes that with high probability contains all nodes of PageRank at least Δ\Delta, and no node of PageRank smaller than Δ/c\Delta/c. We call this problem {\sc SignificantPageRanks}. We develop a nearly optimal, local algorithm for the problem with runtime complexity O~(n/Δ)\tilde{O}(n/\Delta) on networks with nn nodes. We show that any algorithm for solving this problem must have runtime of Ω(n/Δ){\Omega}(n/\Delta), rendering our algorithm optimal up to logarithmic factors. Our algorithm comes with two main technical contributions. The first is a multi-scale sampling scheme for a basic matrix problem that could be of interest on its own. In the abstract matrix problem it is assumed that one can access an unknown {\em right-stochastic matrix} by querying its rows, where the cost of a query and the accuracy of the answers depend on a precision parameter ϵ\epsilon. At a cost propositional to 1/ϵ1/\epsilon, the query will return a list of O(1/ϵ)O(1/\epsilon) entries and their indices that provide an ϵ\epsilon-precision approximation of the row. Our task is to find a set that contains all columns whose sum is at least Δ\Delta, and omits any column whose sum is less than Δ/c\Delta/c. Our multi-scale sampling scheme solves this problem with cost O~(n/Δ)\tilde{O}(n/\Delta), while traditional sampling algorithms would take time Θ((n/Δ)2)\Theta((n/\Delta)^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 \cite{JehW03,AndersenCL06} and is highly efficient particularly for networks with large in-degrees or out-degrees. Together with our multiscale sampling scheme we are able to optimally solve the {\sc SignificantPageRanks} problem.Comment: Accepted to Internet Mathematics journal for publication. An extended abstract of this paper appeared in WAW 2012 under the title "A Sublinear Time Algorithm for PageRank Computations
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