Conflict-free access to templates of trees and hypercubes in parallel memory systems

Abstract

In this paper, we deal with the problem of mapping data structures, called hosts, into as few distinct memory modules as possible to guarantee that sets of distinct host nodes, called templates, can be accessed in parallel and without memory conflicts. An efficient solution to this important problem leads to a higher memory bandwidth and a better overall performance of a multiprocessor system.

Considering a binomial tree as the host, we devise for the first time a recursive mapping of its nodes which allows conflict-free access to any binomial subtree. Since the overlappings among various template instances intricate the problem, thus requiring more memory modules than the template size, we define what are called the oriented templates (subtrees) for which the conflict-freeness is guaranteed using the number of memory modules equal to the template size.

We also investigate the conflict-free access to d-dimensional subcubes of n-dimensional hypercubes. In this context, hypercubes model sets of items indexed with n-digit (binary or non-binary) in which parallel accesses will be made to sets of items differing in an arbitrary collection of d-digit positions. With the help of the coding theory, we propose a novel approach to solve the subcube access problem. Codes with minimum distance d ≥ 2 play a crucial role in our applications. In fact, we prove that any occurrence of a subcube Q _s ⊂ Q _n for 0 ≤ s ≤ d −1, can be accessed without conflicts using [2ⁿ/M] memory modules, by associating an n-dimensional hypercube, Q _n , with a linear code C of length n, size M and minimum distance d. Associating the hypercube nodes with maximum distance separable (MDS) codes, our problem is solved optimally both in terms of the number of memory modules required and the amount of load per module. These codes can be easily modified (without node relocation) when the size of the host or the number of available memory modules change.

This work is partially supported by Texas Advanced Technology Program grant TATP-003594031, and by Progetto 40% “Efficienza di Algoritmi e Progetto di Strutture Informative.