On resource placements in 3D tori

doi:10.1016/S0743-7315(03)00122-9

Journal of Parallel and Distributed Computing

Volume 63, Issue 9, September 2003, Pages 838-845

https://doi.org/10.1016/S0743-7315(03)00122-9 Get rights and content

Abstract

Perfect distance-d placements in 3D tori is investigated. It is proved that there exists no linear perfect distance-1 placement other than the known ones so far. Perfect distance-d placements in 3D tori, where at least one of the dimensions is 2, are presented. Furthermore, quasi-perfect placements for 3D tori are introduced.

Introduction

Torus-based interconnection networks are deployed in several high performance parallel computers. Examples are Ametak 2010 [31], the MIT J-Machine [20] (3D mesh), the Mosaic [30], [29], and the Cray T3D/T3E [21], [28] (3D torus).

In these multicomputer systems, there is a limited number of resources that each processor needs to access. These resources could be I/O nodes, control nodes, redundant fault-tolerant components, routing tables, or software packages. They need to be distributed in the system so that all processors can access them in a comparable manner to enhance the system's overall performance. For example, when designing an I/O system for a multicomputer, its interconnection network is divided into clusters each of which has one or more I/O nodes. On one hand, the I/O system designer tries to control the latency of the cluster local I/O traffic by limiting the number of nodes in a cluster and making each of them within a maximum distance d or less from some local I/O node in the same cluster. On the other hand, the designer would like to control the latency of the non-local I/O traffic, i.e. traffic among different clusters, by uniformly distributing the I/O resources over the clusters so that each cluster receives comparable amount of non-local I/O requests. This reduces the risk of having parts of the network congested while others are barely utilized. In general, this problem is called “the resource placement problem”. It has intensively been studied in hypercubes [13], [14], [17], [27], [32] and tori [1], [2], [3], [8], [9], [10], [18], [25], [26].

The applications of the resource placement strategies in toroidal networks are not restricted to the design of an I/O system for a multicomputer. Torus-based interconnection networks are used as router/switch fabrics [15], for processor memory interfacing [28], and for I/O interfacing [23]. Strategies for resource placements in these cases have several applications in terms of placing queuing buffers, port interfaces, or redundant fault-tolerant components.

In this paper, the perfect distance-d resource placement problem in 3D tori is investigated. In a perfect distance-d placement the resources are placed in the system so that any non-resource node is at distance d or less from exactly one resource node. In [8], [10], the authors using Lee distance error correcting codes have shown that a perfect distance-d placement is possible in a 2D torus when both dimensions are a multiple of (2d²+2d+1). Quasi-perfect placements for 2D tori have been introduced in [1], [4]. A placement is quasi-perfect distance-d if every node is either at distance d or less from exactly one resource node, or at distance (d+1) from one or more resource nodes. In [5], [6] the effect of a resource placement strategy on the average network latency in 2D tori is simulated. It is shown that the perfect/quasi-perfect placements result in superior performance over other placements techniques.

The major contributions of this paper are: (1) proving the non-existence of any unknown linear perfect distance-1 placements in 3D tori, (2) introducing perfect distance-d placements in irregular tori with respect to d as to be defined in Section 2, and (3) introducing quasi-perfect placements for 3D tori.

The rest of this paper is organized as follows:

Section 2 gives the definitions, the required mathematical background, and the previous known results. Section 3 shows the results on the non-existence of any unknown linear perfect distance-1 placements. Section 4 presents perfect distance-d placements in irregular tori with respect to d. Section 5 introduces quasi-perfect placements for 3D tori. Finally, conclusions are given in Section 6.

Section snippets

Preliminaries and background

The Lee distance is a metric used in the field of error correcting codes. It has been shown in [12] that the Lee distance is a natural metric useful in defining the interconnection characteristic of the toroidal networks. Many topological properties of a toroidal network can be derived from this useful metric [12].

Mixed radix notation: In a mixed radix notation, any integer I, 0⩽I<K=k_n−1k_n−2⋯k₀, is represented as an n-dimensional vector X=x_n−1x_n−2⋯x₀ over K=k_n−1k_n−2⋯k₀, where x_i∈{0,1,2,…,k_i−1}.

On existence of perfect distance-1 placements

The known perfect distance-1 placements are:

1.
The perfect distance-1 placements for (7i×7j×7k) regular tori with respect to d=1, where $i, j,$ and k are positive integers (the basic tiling block is shown in Fig. 1).
2.
The perfect distance-1 placements for (2×3i×6j) irregular tori with respect to d=1, where i and j are positive integers (the basic tiling block is shown in Fig. 2).
3.
The perfect distance-1 placement for a (2×2×2) irregular torus with respect to d=1 (shown in Fig. 3).

It is clear that these

Perfect distance-d placements in irregular tori

Golomb and Welch have proved that a full packing sphere of radius 2 cannot tile a closed 3D toroidal space. Furthermore, they conjectured that a perfect code does not exist in a 3D toroidal space if t>1 [16]. However, codes (or placements) exist for any t (or d ) in irregular 3D toroidal spaces with respect to t (or d ) as to be shown in the following theorem.

Theorem 4.1

Let T be a 2×2×(8d−4) irregular torus with respect to d, d⩾2. Then, the placement P is perfect distance-d, where $P={(0,0,0), (0,4d−2,0),$

Quasi-perfect placements

A placement is quasi-perfect distance-d in a torus T if the following two conditions hold:

•
Let a and b be any two resources in T, and S_a and S_b be the sets of nodes at a distance of d or less from a and b, respectively, then S_a∩S_b=φ.
•
Any node in T is either at distance l⩽d from exactly one resource node, or at distance (d+1) from some resource nodes.

These two conditions imply that a quasi-perfect distance-d placement would be optimal in terms of minimizing the maximum distance between any

Conclusion

The problem of distance-d resource placement in 3D tori is surveyed and investigated. Previous studies show the non-existence of perfect distance-d placements for 3D regular tori with respect to d, d⩾2. In this paper, the non-existence of any unknown linear perfect distance-1 placements in 3D tori is proved. A class of perfect distance-d placements for 3D irregular tori is presented. Furthermore, quasi-perfect placements for 3D tori are introduced. These placements could be a proper alternative

Acknowledgements

We thank the referees for their constructive comments. B. Bose's work is supported in part by the NSF under Grant CCR-0105204.

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On resource placements in 3D tori

Abstract

Introduction

Section snippets

Preliminaries and background

On existence of perfect distance-1 placements

Perfect distance-d placements in irregular tori

Quasi-perfect placements

Conclusion

Acknowledgements

Inform. Control

Resource placement in torus-based networks

IEEE Trans. Comput.

Algebraic Coding Theory

Lee distance and topological properties of k-ary n-cubes

IEEE Trans. Comput.

Efficient resource placement in hypercubes using multiple-adjacency codes

IEEE Trans. Comput.