# An O(√n)-worst-case-time solution to the granularity problem

## Abstract

In this paper we deal with the granularity problem, that is, the problem of implementing a shared memory in a distributed system where *n* processors are connected to *n* memory modules through a complete network (Module Parallel Computer). We present a memory organization scheme where *m* ∈ *⊗*(*n*^{2}) variables, each replicated into a 2*c* — 1 copies (for constant *c*), are evenly distributed among the *n* modules, so that a suitable access protocol allows any set of at most *n* distinct read/write operations to be performed by the processors in *O*(*√n*) parallel steps in the worst case. The well known strategy based on multiple copies is needed to avoid the worst-case *O*(*n*)-time, since only a majority of the copies of each variable need be accessed for any operation. The memory organization scheme can be extended to deal with *m* ∈ *⊗*(*n*^{3}) variables attaining an *O*(*n*^{2/3})-time complexity in the worst case.

## Key Words

Algorithms and Data Structures Theory of Parallel and Distributed Computing P-RAM Simulation## Preview

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