Universal Communication, Universal Graphs, and Graph Labeling
Nathaniel Harms

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 belonging to a family , and inputs and . Alice and Bob use shared randomness to send a message to a third party who cannot see , or the shared randomness, and must decide . 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 and can be determined from the labels . We give a universal SMP protocol using bits of communication for deciding whether two vertices have distance at most on distributive lattices (generalizing the -Hamming Distance problem in communication complexity), and explain how this implies an labeling scheme for…
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