Undirected connectivity in log-space

@article{Reingold2008UndirectedCI,
  title={Undirected connectivity in log-space},
  author={Omer Reingold},
  journal={J. ACM},
  year={2008},
  volume={55},
  pages={17:1-17:24}
}
  • O. Reingold
  • Published 1 September 2008
  • Mathematics, Computer Science
  • J. ACM
We present a deterministic, log-space algorithm that solves st-connectivity in undirected graphs. The previous bound on the space complexity of undirected st-connectivity was log4/3(ṡ) obtained by Armoni, Ta-Shma, Wigderson and Zhou (JACM 2000). As undirected st-connectivity is complete for the class of problems solvable by symmetric, nondeterministic, log-space computations (the class SL), this algorithm implies that SL = L (where L is the class of problems solvable by deterministic log-space… 
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References

SHOWING 1-10 OF 97 REFERENCES
Undirected ST-connectivity in log-space
TLDR
A deterministic, log-space algorithm that solves st-connectivity in undirected graphs and implies a way to construct in log- space a fixed sequence of directions that guides a deterministic walk through all of the vertices of any connected graph.
RL⊆SC
TLDR
There exists a deterministic algorithm for the undirected connectivity problem that runs in polynomial time and 0(log2 n) space, obtained by derandomizing the randomized algorithm of [AKL*79], and the de-randomization result is obtained.
An O(log n log log n) space algorithm for undirected st-connectivity
We present a deterministic O(log n log log n) space algorithm for undirected st-connectivity. It is based on the deterministic EREW algorithm of Chong and Lam [6] and uses the universal exploration
S-T Connectivity on Digraphs with a Known Stationary Distribution
TLDR
This work identifies knowledge of the stationary distribution as the gap between the S-T CONNECTIVITY problems the authors know how to solve in logspace and those that capture all of randomized logspace (RL).
S-T Connectivity on Digraphs with a Known Stationary Distribution
TLDR
This work identifies knowledge of the stationary distribution as the gap between the S-T CONNECTIVITY problems the authors know how to solve in logspace and those that capture all of randomized logspace (RL).
Random walks, universal traversal sequences, and the complexity of maze problems
TLDR
Results are derived suggesting that the undirected reachability problem is structurally different from, and easier than, the directed version of NSPACE(logn), an affirmative answer to a question of S. Cook.
Pseudorandom walks on regular digraphs and the RL vs. L problem
TLDR
It is proved that if (2) could be generalized to all regular directed graphs (including ones that are not consistently labelled) then L=RL, and it is shown that such a problem can be solved in deterministic logarithmic space given a log-space pseudorandom walk generator forregular directed graphs.
On traversal sequences, exploration sequences and completeness of kolmogorov random strings
Traversal sequences were defined by Aleliunas et al. (1979) as a tool for the study of undirected s-t-connectivity. In the first part of this thesis we study traversal sequences. We improve on
BP H SPACE(S)⊆DSPACE(S 3/2 )
We prove that any language that can be recognized by a randomized algorithm (with possibly a two-sided error) that runs in spaceO(S) and always terminates can be recognized by deterministic algorithm
Universal Traversal Sequences for Expander Graphs
Graph reachability is a key problem in the study of various logarithmic space complexity classes. Its version for directed graphs is logspace complete for NSPACE(logn), and hence if proved to be in
...
1
2
3
4
5
...