A unifying look at semigroup computations on meshes with multiple broadcasting

Abstract

Semigroup computations are a fundamental algorithmic tool finding applications in all areas of parallel processing. Given a sequence of m items a ₁, a ₂,..., a _m from a semigroup S with an associative operation ⊕, the semigroup computation problem involves computing a₁ ⊕ a ₂ ⊕ ... ⊕ a _m. We consider the semigroup computation problem involving m (2 ≤ m ≤ n) items on a mesh with multiple broadcasting of size $\sqrt n \times \sqrt n$. Our contribution is to present the first lower bound and the first time-optimal algorithm which apply to the entire range of m (2 ≤ m ≤ n). First, we show that any algorithm that solves the semigroup computation problem must take at least $\Omega (max\{ min\{ log m, log\tfrac{{n^{\tfrac{2}{3}} }}{{m^{\tfrac{1}{3}} }}\} ,\tfrac{{m^{\tfrac{1}{3}} }}{{n^{\tfrac{1}{6}} }}\} )$ time. Second, we show that our bound is tight by designing an algorithm whose running time matches the lower bound. These results unify and generalize all semigroup lower bounds and algorithms known to the authors....