Computing Downwards Accumulations on Trees Quickly

Abstract

\em Downwards passes\/ on binary trees are essentially functions which pass information down a tree, from the root towards the leaves. Under certain conditions, a downwards pass is both `efficient' (computable in a functional style in parallel time proportional to the depth of the tree) and `manipulable' (enjoying a number of distributivity properties useful in program construction); we call a downwards pass satisfying these conditions a \em downwards accumulation. In this paper, we show that these conditions do in fact yield a stronger conclusion: the accumulation can be computed in parallel time proportional to the \em logarithm\/ of the depth of the tree, on a CREW PRAM machine.