Online Algorithms with Lookaround

In this work, we give a unifying view of locality in four settings: distributed algorithms, sequential greedy algorithms, dynamic algorithms, and online algorithms. We introduce a new model of computing, called the online-LOCAL model: the adversary reveals the nodes of the input graph one by one, in the same way as in classical online algorithms, but for each new node the algorithm can also inspect its radius-$T$ neighborhood before choosing the output. Instead of looking ahead in time, we have the power of looking around in space. We compare the online-LOCAL model with three other models: the LOCAL model of distributed computing, where each node produces its output based on its radius-$T$ neighborhood, its sequential counterpart SLOCAL, and the dynamic-LOCAL model, where changes in the dynamic input graph only influence the radius-$T$ neighborhood of the point of change. SLOCAL and dynamic-LOCAL models are sandwiched between LOCAL and online-LOCAL models, with LOCAL being the weakest and online-LOCAL the strongest model. In this work, we seek to answer the following question: is the online-LOCAL model strictly stronger than the LOCAL model when we look at graph algorithms for solving locally checkable labeling problems (LCLs)? First, we show that for LCL problems in paths, cycles, and rooted trees, all four models are roughly equivalent: the locality of any LCL problem falls in the same broad class - $O(\log^* n)$, $\Theta(\log n)$, or $n^{\Theta(1)}$ - in all four models. In particular, prior work on the LOCAL model directly generalizes to all four models. Second, we show that this equivalence does not hold in two-dimensional grids. We show that the locality of the $3$-coloring problem is $O(\log n)$ in the online-LOCAL model, while it is known to be $\Omega(\sqrt{n})$ in the LOCAL model.