We consider the point-to-point message passing model of communication in which there are $k$ processors
with individual private inputs, each $n$-bit long. Each processor is located at the node of an underlying
undirected graph and has access to private random coins. An edge of the graph is a private channel of
communication between its endpoints. The processors have to compute a given function of all their inputs
by communicating along these channels. While this model has been widely used in distributed computing,
strong lower bounds on the amount of communication needed to compute simple functions have just begun
to appear.
In this work, we prove a tight lower bound of $\Omega(kn)$ on the communication needed for computing the
Tribes function, when the underlying graph is a star of $k + 1$ nodes that has $k$ leaves with inputs and a center
with no input. Lower bound on this topology easily implies comparable bounds for others. Our lower bounds
are obtained by building upon the recent information theoretic techniques of Braverman et.al ([BEO+ 13],
FOCS’13) and combining it with the earlier work of Jayram, Kumar and Sivakumar ([JKS03], STOC’03).
This approach yields information complexity bounds that is of independent interest.

Update on author-list.

We consider the point-to-point message passing model of communication in which there are $k$ processors
with individual private inputs, each $n$-bit long. Each processor is located at the node of an underlying
undirected graph and has access to private random coins. An edge of the graph is a private channel of
communication between its endpoints. The processors have to compute a given function of all their inputs
by communicating along these channels. While this model has been widely used in distributed computing,
strong lower bounds on the amount of communication needed to compute simple functions have just begun
to appear.
In this work, we prove a tight lower bound of $\Omega(kn)$ on the communication needed for computing the
Tribes function, when the underlying graph is a star of $k + 1$ nodes that has $k$ leaves with inputs and a center
with no input. Lower bound on this topology easily implies comparable bounds for others. Our lower bounds
are obtained by building upon the recent information theoretic techniques of Braverman et.al ([BEO+ 13],
FOCS’13) and combining it with the earlier work of Jayram, Kumar and Sivakumar ([JKS03], STOC’03).
This approach yields information complexity bounds that is of independent interest.