Lower bounds for linear degeneracy testing
Nir Ailon, Bernard Chazelle · Journal of the ACM · 2005
In the late nineties, Erickson proved a remarkable lower bound on the decision tree complexity of one of the central problems of computational geometry: given n numbers, do any r of them add up to 0? His lower bound of Ω( n ⌈ r /2⌉ ), for any fixed r , is optimal if the polynomials at the nodes are linear and at most r -variate. We generalize his bound to s -variate polynomials for s > r . Erickson's bound decays quickly as r grows and never reaches above pseudo-polynomial: we provide an exponential improvement. Our arguments are based on three ideas: (i) a geometrization of Erickson's proof technique; (ii) the use of error-correcting codes; and (iii) a tensor product construction for permutation matrices.