Searching monotone multi-dimensional arrays

A d-dimensional array of real numbers is called monotone increasing   if its entries are increasing along each dimension. Given An,dAn,d, a monotone increasing d-dimensional array with n entries along each dimension, and a real number x  , we want to decide whether x∈An,dx∈An,d, by performing a sequence of comparisons between x   and some entries of An,dAn,d. We want to minimize the number of comparisons used. In this paper we investigate this search problem, we generalize Linial and Saks’ search algorithm [N. Linial, M. Saks, Searching ordered structures, J. Algorithms 6 (1985) 86–103] for monotone three-dimensional arrays to d  -dimensions for d⩾4d⩾4. For d=4d=4, our new algorithm is optimal up to the lower order terms.