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تسجيل مستخدم جديدFinite Volume Spaces and Sparsification
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We introduce and study finite $d$-volumes - the high dimensional generalization of finite metric spaces. Having developed a suitable combinatorial machinery, we define $ell_1$-volumes and show that they contain Euclidean volumes and hypertree volumes. We show that they can approximate any $d$-volume with $O(n^d)$ multiplicative distortion. On the other hand, contrary to Bourgains theorem for $d=1$, there exists a $2$-volume that on $n$ vertices that cannot be approximated by any $ell_1$-volume with distortion smaller than $tilde{Omega}(n^{1/5})$. We further address the problem of $ell_1$-dimension reduction in the context of $ell_1$ volumes, and show that this phenomenon does occur, although not to the same striking degree as it does for Euclidean metrics and volumes. In particular, we show that any $ell_1$ metric on $n$ points can be $(1+ epsilon)$-approximated by a sum of $O(n/epsilon^2)$ cut metrics, improving over the best previously known bound of $O(n log n)$ due to Schechtman. In order to deal with dimension reduction, we extend the techniques and ideas introduced by Karger and Bencz{u}r, and Spielman et al.~in the context of graph Sparsification, and develop general methods with a wide range of applications.
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We consider a fundamental algorithmic question in spectral graph theory: Compute a spectral sparsifier of random-walk matrix-polynomial $$L_alpha(G)=D-sum_{r=1}^dalpha_rD(D^{-1}A)^r$$ where $A$ is the adjacency matrix of a weighted, undirected graph,
$D$ is the diagonal matrix of weighted degrees, and $alpha=(alpha_1...alpha_d)$ are nonnegative coefficients with $sum_{r=1}^dalpha_r=1$. Recall that $D^{-1}A$ is the transition matrix of random walks on the graph. The sparsification of $L_alpha(G)$ appears to be algorithmically challenging as the matrix power $(D^{-1}A)^r$ is defined by all paths of length $r$, whose precise calculation would be prohibitively expensive. In this paper, we develop the first nearly linear time algorithm for this sparsification problem: For any $G$ with $n$ vertices and $m$ edges, $d$ coefficients $alpha$, and $epsilon > 0$, our algorithm runs in time $O(d^2mlog^2n/epsilon^{2})$ to construct a Laplacian matrix $tilde{L}=D-tilde{A}$ with $O(nlog n/epsilon^{2})$ non-zeros such that $tilde{L}approx_{epsilon}L_alpha(G)$. Matrix polynomials arise in mathematical analysis of matrix functions as well as numerical solutions of matrix equations. Our work is particularly motivated by the algorithmic problems for speeding up the classic Newtons method in applications such as computing the inverse square-root of the precision matrix of a Gaussian random field, as well as computing the $q$th-root transition (for $qgeq1$) in a time-reversible Markov model. The key algorithmic step for both applications is the construction of a spectral sparsifier of a constant degree random-walk matrix-polynomials introduced by Newtons method. Our algorithm can also be used to build efficient data structures for effective resistances for multi-step time-reversible Markov models, and we anticipate that it could be useful for other tasks in network analysis.
A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA19] states that in every $n$-vertex graph $G$ with maximum degree $Delta$, sampling $O(log{n})$ colors per each vertex independently from $Delta+1$ colors almost certainly allow
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