Problem 65: Communication Complexity of Connectivity

From Open Problems in Sublinear Algorithms
Revision as of 16:28, 16 June 2022 by 98.110.124.189 (talk)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
Suggested by Andrew McGregor
Source Bertinoro 2014
Short link https://sublinear.info/65

A recent result in graph sketching [AhnGM-12] can be rephrased in terms of a simultaneous message communication protocol with public coins. Specifically, suppose that $n$ players are each given a row of the adjacency matrix of some graph. The players simultaneously send a message to a central player who must then determine whether the graph is connected. Existing work shows that this is possible with $O(\log^3 n)$ bit messages from each player. Are $O(\log^2 n)$ or $O(\log n)$ bits sufficient? Also, is there a non-trivial lower bound if the players must use private coins?

Update

The conjecture has been resolved by Yu [Yu-21].