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Solving the induced subgraph problem in the randomized multiparty simultaneous messages model
 Proc 22nd Int Colloq Structural Information and Communication Complexity
, 2015
"... Abstract. We study the message size complexity of recognizing, under the broadcast congested clique model, whether a fixed graph H appears in a given graph G as a minor, as a subgraph or as an induced subgraph. The n nodes of the input graph G are the players, and each player only knows the identiti ..."
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Abstract. We study the message size complexity of recognizing, under the broadcast congested clique model, whether a fixed graph H appears in a given graph G as a minor, as a subgraph or as an induced subgraph. The n nodes of the input graph G are the players, and each player only knows the identities of its immediate neighbors. We are mostly interested in the oneround, simultaneous setup where each player sends a message of sizeO(logn) to a referee that should be able then to determine whether H appears in G. We consider randomized protocols where the players have access to a common random sequence. We completely characterize which graphs H admit such a protocol. For the particular case where H is the path of 4 nodes, we present a new notion called twin ordering, which may be of independent interest. 1
Robust reconstruction of BarabásiAlbert networks in the broadcast congested clique model
"... In the broadcast version of the congested clique model, n nodes communicate in synchronous rounds by writing O(log n)bit messages on a whiteboard, which is visible to all of them. The joint input to the nodes is an undirected nnode graph G, with node i receiving the list of its neighbors in G. Ou ..."
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In the broadcast version of the congested clique model, n nodes communicate in synchronous rounds by writing O(log n)bit messages on a whiteboard, which is visible to all of them. The joint input to the nodes is an undirected nnode graph G, with node i receiving the list of its neighbors in G. Our goal is to design a protocol at the end of which the information contained in the whiteboard is enough for reconstructing G. It has already been shown that there is a oneround protocol for reconstructing graphs with bounded degeneracy. The main drawback of that protocol is that the degeneracy m of the input graph G must be known a priori by the nodes. Moreover, the protocol fails when applied to graphs with degeneracy larger than m. In this paper we address this issue by looking for robust reconstruction protocols, that is, protocols which always give the correct answer and work efficiently when the input is restricted to a certain class. We introduce a very simple, tworound protocol that we call RobustReconstruction. We prove that this protocol is robust for reconstructing the class of BarabásiAlbert trees with (expected) message size O(log n). Moreover, we present computational evidence suggesting that RobustReconstruction also generates logarithmic size messages for arbitrary BarabásiAlbert networks. Finally, we stress the importance of the preferential attachment mechanism (used in the construction of BarabásiAlbert networks) by proving that RobustReconstruction does not generate short messages for random recursive 1 trees.