Exploiting the structure of the 2-dimensional sorting problem associated with the polynomial product has been the strategy in the design of certain algorithms which are faster for a large class of problems than those found in the literature. First a parallel is drawn between GEN–MULT and Horowitz’s SORT–MULT algorithm [A sorting algorithm for polynomial multiplication, J. Assoc. Comput. Mach., 22 (1975), pp. 450–462]. The former owes its better performance to the global incorporation of the structure. A geometric picture of the relative location of the product exponents enabled us to present a new approach, implemented by the SLICE algorithms, in which the product exponents are sorted in separate parallel slices. For a class of problems of moderate size and sparsity, which was investigated by Johnson [Sparse polynomial arithmetic, Proc. Eurosam Conf., SIGSAM Bull., 8 (1974), pp. 63–71] with the ALTRAN system, the simple SLICE–S algorithm gave a 1:4 improvement, timewise and relative to sorting effort, as ...