he noisy broadcast model was first studied by [Gallager, 1988] where an n-character input is distributed among n processors, so that each processor receives one input bit. Computation proceeds in rounds, where in each round each processor broadcasts a single character, and each reception is corrupted independently at random with some probability p. [Gallager, 1988] gave an algorithm for all processors to learn the input in O(log log n) rounds with high probability. Later, a matching lower bound of Omega(log log n) was given by [Goyal et al., 2008]. We study a relaxed version of this model where each reception is erased and replaced with a `?' independently with probability p, so the processors have knowledge of whether a bit has been corrupted. In this relaxed model, we break past the lower bound of [Goyal et al., 2008] and obtain an O(log^* n)-round algorithm for all processors to learn the input with high probability. We also show an O(1)-round algorithm for the same problem when the alphabet size is Omega(poly(n)).
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Enhanced diffusivity in perturbed senile reinforced random walk models
We consider diffusivity of random walks with transition probabilities depending on the number of consecutive traversals of the last traversed edge, the so called senile reinforced random walk (SeRW). In one dimension, the walk is known to be sub-diffusive with identity reinforcement function. We perturb the model by introducing a small probability δ of escaping the last traversed edge at each step. The perturbed SeRW model is diffusive for any δ > 0 , with enhanced diffusivity (≫ O ( δ^2 ) ) in the small δ regime. We further study stochastically perturbed SeRW models by having the last edge escape probability of the form δ ξ n with ξ n ’s being independent random variables. Enhanced diffusivity in such models are logarithmically close to the so called residual diffusivity (positive in the zero δ limit), with diffusivity between O ( 1/| log δ | ) and O ( 1/ log | log δ | ) . Finally, we generalize our results to higher dimensions where the unperturbed model is already diffusive. The enhanced diffusivity can be as much as O ( 1/log ^2 δ )
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- Award ID(s):
- 1632935
- NSF-PAR ID:
- 10158977
- Date Published:
- Journal Name:
- Asymptotic analysis
- ISSN:
- 0921-7134
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/DOI: 10.3233/ASY-201611
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