Lower bounds for linear degeneracy testing

Abstract. 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 (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. Categories and Subject Descriptors: F.2.0 [Analysis of Algorithms and Problem Complexity]: General General Terms: Theory Additional Key Words and Phrases: Computational geometry, linear decision trees, lower bounds

• Type: Conference
• Year: 2004
• Where: STOC
• Authors: Nir Ailon, Bernard Chazelle
• Keywords: Additional Key Words, Algorithms, Computational Geometry, STOC 2004, Tensor Product Construction