In distributed computing, questions of computability are exquisitely sensitive to minute details of the model assumptions, and there is no universally agreed upon model of network computing. Here, we study which functions are computable by deterministic and anonymous agents in either static or dynamic networks. We consider various communication assumptions common in the literature, and in each case we strive to characterize the set of computable functions, organizing existing results as well as offering new ones, alongside new proofs which bring new understanding of this computability landscape.

We revisit asynchronous computing in networks of crash-prone processes, under the asynchronous variant of the standard LOCAL model, recently introduced by Fraigniaud et al. [DISC 2022]. We focus on the vertex coloring problem, and our contributions concern both lower and upper bounds for this problem. On the upper bound side, we design an algorithm tolerating an arbitrarily large number of crash failures that computes an $O(Δ^2)$-coloring of any $n$-node graph of maximum degree $Δ$, in $O(\log^\star n)$ rounds. This extends Linial's seminal result from the (synchronous failure-free) LOCAL model to its asynchronous crash-prone variant. Then, by allowing a dependency on $Δ$ on the runtime, we show that we can reduce the colors to $\big(\frac12(Δ+1)(Δ+2)-1 \big)$. For cycles (i.e., for $Δ=2$), our algorithm achieves a 5-coloring of any $n$-node cycle, in $O(\log^\star n)$ rounds. This improves the known 6-coloring algorithm by Fraigniaud et al., and fixes a bug in their algorithm, which was erroneously claimed to produce a 5-coloring. On the lower bound side, we show that, for $k<5$, and for every prime integer~$n$, no algorithm can $k$-color the $n$-node cycle in the asynchronous crash-prone variant of LOCAL, independently from the round-complexities of the algorithms. This lower bound is obtained by reduction from an original extension of the impossibility of solving weak symmetry-breaking in the wait-free shared-memory model. We show that this impossibility still holds even if the processes are provided with inputs susceptible to help breaking symmetry.

We present a wait-free algorithm for proper coloring the n ≥ 3 nodes of the asynchronous cycle Cn, where each crash-prone node starts with its (unique) identifier as input. The algorithm is independent of n and uses up to five colors, and each node terminates upon completing at most O(log*n) write-read-compute steps.

We present a wait-free algorithm for proper coloring the n nodes of the asynchronous cycle C_n, where each crash-prone node starts with its (unique) identifier as input. The algorithm is independent of n ≥ 3, and runs in O(log^*n) rounds in C_n. This round-complexity is optimal thanks to a known matching lower bound, which applies even to synchronous (failure-free) executions. The range of colors used by our algorithm, namely {0,…,4}, is optimal too, thanks to a known lower bound on the minimum number of names for which renaming is solvable wait-free in shared-memory systems, whenever n is a power of a prime. Indeed, our model coincides with the shared-memory model whenever n = 3, and the minimum number of names for which renaming is possible in 3-process shared-memory systems is 5.