Before revision:  In this paper, we investigate the two aspects of adaptivity and parameterization in sublinear-time algorithms for LIS estimation . We first show that adaptivity helps in LIS estimation, specifically , for every constant \epsilon \in (0,1), every nonadaptive algorithm that outputs an estimate of the length of the LIS in an array of length n to within an additive error of \epsilon \cdot n has to make \log^{\Omega(\log (1/\epsilon))} n) queries.
After revision:  In this paper, we investigate the two aspects of adaptivity and parameterization in sublinear-time algorithms for LIS estimation than the obvious bounds implied by testing monotonicity (for adaptive or nonadaptive algorithms). In this paper, we give the first nontrivial lower bound on the complexity of LIS estimation, specifically , for every constant \epsilon \in (0,1), every nonadaptive algorithm that outputs an estimate of the length of the LIS in an array of length n to within an additive error of \epsilon \cdot n has to make \log^{\Omega(\log (1/\epsilon))} n) queries.