Improved depth lower bounds for small distance connectivity

@article{Beame1998ImprovedDL,
  title={Improved depth lower bounds for small distance connectivity},
  author={Paul Beame and Russell Impagliazzo and Toniann Pitassi},
  journal={computational complexity},
  year={1998},
  volume={7},
  pages={325-345}
}
We consider the problem of determining, given a graph G with specified nodes s and t, whether or not there is a path of at most k edges in G from s to t. We show that solving this problem on polynomialsize unbounded fan-in circuits requires depth $ \Omega{\rm(log\,log}\,k) $ , improving on a depth lower bound of $ \Omega{\rm(log^*}\,k) $ when $ k = {\rm log}^{O(1)}n $ given by Ajtai (1989), Bellantoni et al. (1992). More generally, we obtain an improved size-depth tradeoff lower bound for the… CONTINUE READING

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