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Universal Communication, Universal Graphs, and Graph Labeling

Author Nathaniel Harms



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Nathaniel Harms
  • University of Waterloo, Canada

Acknowledgements

Thanks to Eric Blais for comments on the structure of this paper; Amit Levi for helpful discussions and comments on the presentation; Anna Lubiw for an introduction to planar graphs and graph labeling; Corwin Sinnamon for comments on distributive lattices; and Sajin Sasy for observing the possible applications to privacy.

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Nathaniel Harms. Universal Communication, Universal Graphs, and Graph Labeling. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 33:1-33:27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ITCS.2020.33

Abstract

We introduce a communication model called universal SMP, in which Alice and Bob receive a function f belonging to a family 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 ℓ(x), ℓ(y). We give a universal SMP protocol using O(k^2) bits of communication for deciding whether two vertices have distance at most k in distributive lattices (generalizing the k-Hamming Distance problem in communication complexity), and explain how this implies a O(k^2 log n) labeling scheme for deciding dist(x,y) ≤ k on distributive lattices with size n; in contrast, we show that a universal SMP protocol for determining dist(x,y) ≤ 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 dist(x,y) ≤ k and planar graphs have an O(1) protocol for dist(x,y) ≤ 2, which implies a new O(log n) labeling scheme for the same problem on planar graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
  • Mathematics of computing → Graph coloring
  • Mathematics of computing → Trees
  • Theory of computation → Models of computation
  • Theory of computation → Communication complexity
  • Theory of computation → Computational geometry
  • Theory of computation → Generating random combinatorial structures
Keywords
  • Universal graphs
  • graph labeling
  • distance labeling
  • planar graphs
  • lattices
  • hamming distance

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