# Recursively divisible problems

## Abstract

We introduce the concept of a (*p, d*)-*divisible* problem, where *p* reflects the number of processors and *d* the number of communication phases of a parallel algorithm solving the problem. We call problems that are *recursively* (*n*^{∈}, *O*(1))-divisible in a work-optimal way with 0 <ε < **1***ideally divisible* and give motivation drawn from parallel computing for the relevance of that concept. We show that several important problems are ideally divisible. For example, sorting is recursively (*n*^{1/3}, 4)-divisible in a work-optimal way. On the other hand, we also provide some results of lower bound type. For example, ideally divisible problems appear to be a proper subclass of the functional complexity class *FP* of sequentially feasible problems. Finally, we also give some extensions and variations of the concept of (*p, d*)-divisibility.

## Keywords

Parallel Algorithm Communication Phase Sorting Problem Proper Subclass Parallel Random Access Machine## Preview

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