Decidability classes for mobile agents computing
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Collaborative search on the plane without communication
PODC '12 Proceedings of the 2012 ACM symposium on Principles of distributed computing
Brief announcement: what can be computed without communication?
PODC '12 Proceedings of the 2012 ACM symposium on Principles of distributed computing
On the locality of some NP-complete problems
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
What can be computed without communications?
SIROCCO'12 Proceedings of the 19th international conference on Structural Information and Communication Complexity
Memory lower bounds for randomized collaborative search and implications for biology
DISC'12 Proceedings of the 26th international conference on Distributed Computing
Randomized distributed decision
DISC'12 Proceedings of the 26th international conference on Distributed Computing
ACM Computing Surveys (CSUR)
What can be decided locally without identifiers?
Proceedings of the 2013 ACM symposium on Principles of distributed computing
Towards a complexity theory for local distributed computing
Journal of the ACM (JACM)
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A central theme in distributed network algorithms concerns understanding and coping with the issue of {\em locality}. Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for \emph{distributed decision problems}. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard $\cal{LOCAL}$ model of computation and define $LD(t)$ (for {\em local decision}) as the class of decision problems that can be solved in $t$ communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class $BPLD(t,p,q)$, containing all languages for which there exists a randomized algorithm that runs in $t$ rounds, accepts correct instances with probability at least $p$ and rejects incorrect ones with probability at least $q$. We show that $p^2+q = 1$ is a threshold for the containment of $LD(t)$ in $BPLD(t,p,q)$. More precisely, we show that there exists a language that does not belong to $LD(t)$ for any $t=o(n)$ but does belong to $BPLD(0,p,q)$ for any $p,q\in (0,1]$ such that $p^2+q\leq 1$. On the other hand, we show that, restricted to hereditary languages, $BPLD(t,p,q) = LD(O(t))$, for any function $t$ and any $p,q\in (0,1]$ such that $p^2+q>, 1$. In addition, we investigate the impact of non-determinism on local decision, and establish some structural results inspired by classical computational complexity theory. Specifically, we show that non-determinism does help, but that this help is limited, as there exist languages that cannot be decided non-deterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with non-determinism that enables to decide \emph{all} languages \emph{in constant time}. Finally, we introduce the notion of local reduction, and establish some completeness results.