Before revision:  We derive lower-bound results showing that, for \lambda =\Omega(\sqrt{n \log n}), the upper bound on the running time of this randomized algorithm matches the lower bound that we derive on the time complexity of  Rich Lines in the algebraic computation trees model.
After revision:  We derive lower-bound results showing that, for \lambda =\Omega(\sqrt{n \log n}), the upper bound on the running time of this randomized algorithm matches the lower bound that we derive on the time complexity of  Rich Lines in the algebraic computation trees model.