On high‐speed parallel algorithms using redundant coding

We introduce a concept of local computability for designing high‐speed parallel algorithms on fan‐in restricted models. A function is k‐locally computable if each subfunction sepends on only at most k input variables. If k is a constant independent of n, the number of input variables, we can construct an O(1) time parallel algorithm for F on a fan‐in restricted computation model. In order to realize the local computability, we use a redundant coding scheme. We show that a binary operation of any finite Abelian group is k‐locally computable under a redundant coding scheme, where k is a constant independent of the order of the group. We also show that we can design a redundant coding scheme for a residue ring Z_m of integers under which addition and multiplication can be performed in O(1) and O(log log log m) time, respectively, in parallel, when m is the product of the smallest r primes.