Universal Communication, Universal Graphs, and Graph Labeling

TL;DR

This paper introduces a universal SMP communication model linking communication complexity to graph labeling, providing new protocols for distance and adjacency problems in various graph classes, with implications for efficient labeling schemes.

Contribution

It presents a universal SMP protocol for distributive lattices, relates it to graph labeling, and establishes bounds and protocols for various graph families, including trees and planar graphs.

Findings

O(k^2) communication protocol for distributive lattices distance problem

Constant-cost protocols for adjacency in trees, low-arboricity, and planar graphs

O(log n) labeling scheme for distance in planar graphs

Abstract

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 $\Omega(n^{1/4})$ 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.