A strong direct product theorem for the tribes function via the smooth-rectangle bound

The main result of this paper is an optimal strong direct product result for the two-party public-coin randomized communication complexity of the Tribes function. This is proved by providing an alternate proof of the optimal lower bound of \Omega(n) for the randomised communication complexity of the Tribes function using the so-called smooth-rectangle bound, introduced by Jain and Klauck [JK10]. The optimal \Omega(n) lower bound for Tribes was originally proved by Jayram, Kumar and Sivakumar [JKS03], using a more powerful lower bound technique, namely the information complexity bound. The information complexity bound is known to be at least as strong a lower bound method as the smooth-rectangle bound [KLL+12]. On the other hand, we are not aware of any function or relation for which the smooth-rectangle bound is (asymptotically) smaller than its public-coin randomized communication complexity. The optimal direct product for Tribes is obtained by combining our smooth-rectangle bound for tribes with the strong direct product result of Jain and Yao [JY12] in terms of smooth-rectangle bound.