Universal Communication, Universal Graphs, and Graph Labeling

We introduce a communication model called universal SMP, in which Alice and Bob receive a function $f$ belonging to a family $\mathcal{F}$, and inputs $x$ and $y$. Alice and Bob use shared randomness to send a message to a third party who cannot see $f, x, y$, or the shared randomness, and must decide $f(x,y)$. Our main application of universal SMP is to relate communication complexity to graph labeling, where the goal is to give a short label to each vertex in a graph, so that adjacency or other functions of two vertices $x$ and $y$ can be determined from the labels $\ell(x),\ell(y)$. We give a universal SMP protocol using $O(k^2)$ bits of communication for deciding whether two vertices have distance at most $k$ on distributive lattices (generalizing the $k$-Hamming Distance problem in communication complexity), and explain how this implies an $O(k^2\log n)$ labeling scheme for determining $\mathrm{dist}(x,y) \leq k$ on distributive lattices with size $n$; in contrast, we show that a universal SMP protocol for determining $\mathrm{dist}(x,y) \leq 2$ in modular lattices (a superset of distributive lattices) has super-constant $Ω(n^{1/4})$ communication cost. On the other hand, we demonstrate that many graph families known to have efficient adjacency labeling schemes, such as trees, low-arboricity graphs, and planar graphs, admit constant-cost communication protocols for adjacency. Trees also have an $O(k)$ protocol for deciding $\mathrm{dist}(x,y) \leq k$ and planar graphs have an $O(1)$ protocol for $\mathrm{dist}(x,y) \leq 2$, which implies a new $O(\log n)$ labeling scheme for the same problem on planar graphs.