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تسجيل مستخدم جديدA Gaussian fixed point random walk
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In this note, we design a discrete random walk on the real line which takes steps $0, pm 1$ (and one with steps in ${pm 1, 2}$) where at least $96%$ of the signs are $pm 1$ in expectation, and which has $mathcal{N}(0,1)$ as a stationary distribution. As an immediate corollary, we obtain an online version of Banaszczyks discrepancy result for partial colorings and $pm 1, 2$ signings. Additionally, we recover linear time algorithms for logarithmic bounds for the Koml{o}s conjecture in an oblivious online setting.
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Motivated by storage applications, we study the following data structure problem: An encoder wishes to store a collection of jointly-distributed files $overline{X}:=(X_1,X_2,ldots, X_n) sim mu$ which are emph{correlated} ($H_mu(overline{X}) ll sum_i
H_mu(X_i)$), using as little (expected) memory as possible, such that each individual file $X_i$ can be recovered quickly with few (ideally constant) memory accesses. In the case of independent random files, a dramatic result by Pat (FOCS08) and subsequently by Dodis, Pat and Thorup (STOC10) shows that it is possible to store $overline{X}$ using just a emph{constant} number of extra bits beyond the information-theoretic minimum space, while at the same time decoding each $X_i$ in constant time. However, in the (realistic) case where the files are correlated, much weaker results are known, requiring at least $Omega(n/polylg n)$ extra bits for constant decoding time, even for simple joint distributions $mu$. We focus on the natural case of compressingemph{Markov chains}, i.e., storing a length-$n$ random walk on any (possibly directed) graph $G$. Denoting by $kappa(G,n)$ the number of length-$n$ walks on $G$, we show that there is a succinct data structure storing a random walk using $lg_2 kappa(G,n) + O(lg n)$ bits of space, such that any vertex along the walk can be decoded in $O(1)$ time on a word-RAM. For the harder task of matching the emph{point-wise} optimal space of the walk, i.e., the empirical entropy $sum_{i=1}^{n-1} lg (deg(v_i))$, we present a data structure with $O(1)$ extra bits at the price of $O(lg n)$ decoding time, and show that any improvement on this would lead to an improved solution on the long-standing Dictionary problem. All of our data structures support the emph{online} version of the problem with constant update and query time.
One of the most studied models of SAT is random SAT. In this model, instances are composed from clauses chosen uniformly randomly and independently of each other. This model may be unsatisfactory in that it fails to describe various features of SAT i
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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,
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