Testing Equality in Communication Graphs

Let <inline-formula> <tex-math notation="LaTeX">$G = (V, E)$ </tex-math></inline-formula> be a connected undirected graph with <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> vertices. Suppose that on each vertex of the graph there is a player having an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-bit string. Each player is allowed to communicate with its neighbors according to a (static) agreed communication protocol, and the players must decide, deterministically, if their inputs are all equal. What is the minimum possible total number of bits transmitted in a protocol solving this problem ? We determine this minimum up to a lower order additive term in many cases. In particular, we show that it is <inline-formula> <tex-math notation="LaTeX">$kn/2+o(n)$ </tex-math></inline-formula> for any Hamiltonian <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-vertex graph, and that for any 2-edge connected graph with <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> edges containing no two adjacent vertices of degree exceeding 2 it is <inline-formula> <tex-math notation="LaTeX">$\text {mn}/2+o(n)$ </tex-math></inline-formula>. The proofs combine graph theoretic ideas with tools from additive number theory.

[1]  Noga Alon,et al.  Testing subgraphs in large graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[2]  N. S. Barnett,et al.  Private communication , 1969 .

[3]  Atri Rudra,et al.  The Range of Topological Effects on Communication , 2015, ICALP.

[4]  H. Whitney.  Non-Separable and Planar Graphs. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[5]  H. Whitney.  Non-Separable and Planar Graphs. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Frank Harary,et al.  Graph Theory , 2016 .

[7]  R. Salem,et al.  On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1942, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Richard J. Lipton,et al.  Multi-party protocols , 1983, STOC.

[9]  Jirí Sgall,et al.  An Upper Bound for a Communication Game Related to Time-Space Tradeoffs , 1995, The Mathematics of Paul Erdős I.

[10]  Noga Alon,et al.  Parent-Identifying Codes , 2001, J. Comb. Theory, Ser. A.

[11]  F. Behrend.  On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1946, Proceedings of the National Academy of Sciences of the United States of America.

[12]  Atri Rudra,et al.  Topology Matters in Communication , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[13]  Jens Vygen,et al.  Shorter tours by nicer ears: 7/5-approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs , 2012, ArXiv.

[14]  Nitin H. Vaidya,et al.  Multiparty Equality Function Computation in Networks with Point-to-Point Links , 2011, SIROCCO.

[15]  Noga Alon,et al.  Nearly complete graphs decomposable into large induced matchings and their applications , 2011, STOC '12.

[16]  Aart Blokhuis,et al.  Cocliques in the Kneser graph on the point-hyperplane flags of a projective space , 2014, Comb..