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 A_n,d, a monotone increasing d-dimensional array with n entries along each dimension, and a real number x, we want to decide whether x∈A_n,d, by performing a sequence of comparisons between x and some entries of A_n,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?4. For d=4, our new algorithm is optimal up to the lower order terms.