Local Distributed Decision

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.