Recently Rubinfeld et al. (ICS 2011, pp. 223-238) proposed a new model of sublinear algorithms called local computation algorithms. In this model, a computation problem F may have more than one legal solution and each of them consists of many bits. The local computation algorithm for F should answer in an online fashion, for any index i, the i th bit of some legal solution of F . Further, all the answers given by the algorithm should be consistent with at least one solution of F . In this work, we continue the study of local computation algorithms. In particular, we develop a technique which under certain conditions can be applied to construct local computation algorithms that run not only in polylogarithmic time but also in polylogarithmic space. Moreover, these local computation algorithms are easily parallelizable and can answer all parallel queries consistently. Our main technical tools are pseudorandom numbers with bounded independence and the theory of branching processes.
In this work, we consider the problems of testing whether a distribution over {0, 1} n is k-wise (resp. ( , k)-wise) independent using samples drawn from that distribution.For the problem of distinguishing k-wise independent distributions from those that are δ-far from k-wise independence in statistical distance, we upper bound the number of required samples byÕ(n k /δ 2 ) and lower bound it by Ω(n k−1 2 /δ) (these bounds hold for constant k, and essentially the same bounds hold for general k). To achieve these bounds, we use Fourier analysis to relate a distribution's distance from k-wise independence to its biases, a measure of the parity imbalance it induces on a set of variables. The relationships we derive are tighter than previously known, and may be of independent interest.To distinguish ( , k)-wise independent distributions from those that are δ-far from ( , k)-wise independence in statistical distance, we upper bound the number of required samples by O`k log n δ 2 2´a nd lower bound it by Ω " √ k log n 2 k ( +δ) √ log 1/2 k ( +δ) «. Although these bounds are an exponential improvement (in terms of n and k) over the corresponding bounds for testing k-wise independence, we give evidence that the time complexity of testing ( , k)-wise independence is unlikely to be poly(n, 1/ , 1/δ) for k = Θ(log n), since this would disprove a plausible conjecture concerning the hardness of finding hidden cliques in random graphs. Under the conjecture, our result implies that for, say, k = log n and = 1/n 0.99 , there is a set of ( , k)-wise indepen-
Posttranslational modifications (PTMs) are important strategies used by eukaryotic organisms to modulate their phenotypes. One of the well studied PTMs, arginine methylation, is catalyzed by protein arginine methyltransferases (PRMTs) with SAM as the methyl donor. The functions of PRMTs have been broadly studied in different biological processes and diseased states, but the molecular basis for arginine methylation is not well defined. In this study, we report the transient-state kinetic analysis of PRMT1 catalysis. The fast association and dissociation rates suggest that PRMT1 catalysis of histone H4 methylation follows a rapid equilibrium sequential kinetic mechanism. The data give direct evidence that the chemistry of methyl transfer is the major rate-limiting step, and that binding of the cofactor SAM or SAH affects the association and dissociation of H4 with PRMT1. Importantly, from the stopped-flow fluorescence measurements, we have identified a critical kinetic step suggesting a precatalytic conformational transition induced by substrate binding. These results provide new insights into the mechanism of arginine methylation and the rational design of PRMT inhibitors.
Abstract. We propose a general method for converting online algorithms to local computation algorithms, 3 by selecting a random permutation of the input, and simulating running the online algorithm. We bound the number of steps of the algorithm using a query tree, which models the dependencies between queries. We improve previous analyses of query trees on graphs of bounded degree, and extend this improved analysis to the cases where the degrees are distributed binomially, and to a special case of bipartite graphs. Using this method, we give a local computation algorithm for maximal matching in graphs of bounded degree, which runs in time and space O(log 3 n). We also show how to convert a large family of load balancing algorithms (related to balls and bins problems) to local computation algorithms. This gives several local load balancing algorithms which achieve the same approximation ratios as the online algorithms, but run in O(log n) time and space. Finally, we modify existing local computation algorithms for hypergraph 2-coloring and k-CNF and use our improved analysis to obtain better time and space bounds, of O(log 4 n), removing the dependency on the maximal degree of the graph from the exponent.
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