Upper bounds on the connection probability for 2-D meshes and tori

Abstract

Mesh is an important and popular interconnection network topology for large parallel computer systems. A mesh can be divided into submeshes to obtain the upper bounds on the connection probability for the mesh. Combinatorial techniques are used to get closer upper bounds on the connection probability for 2-D meshes compared with the existing upper bounds we have known. Simulation results of meshes of various sizes show that our upper bounds are close to the exact connection probability. The combinatorial methods and tools used in this paper can be used to study the connection probabilities for other networks.

Introduction

Many different topologies are used to model the architectures of large parallel computers, the important and popular ones being meshes and tori. Basically a $d$-dimensional mesh has $k_0\times k_1\times \cdots \times k_{d-1}$ nodes, with $k_i$ nodes along dimension $i(k_i\ge 2)$. 2-dimensional and 3-dimensional meshes are studied by many researchers, abbreviated as 2-D and 3-D meshes, respectively. Tori can be considered as meshes with wraparound connections to achieve vertex and edge symmetry. Being simple and scalable, meshes and tori have not only become the basic topologies for many theoretical researches [19], but have also been adopted in many famous commercial multicomputer systems. For example, 2-D mesh computers include Intel Touchstone DELTA [27], Standford DASH [16], MIT Alewife [1], and Goodyear Aerospace MPP [5], etc. 3-D meshes include Blue gene Supercomputer [2], Tera Computer System [3], and MIT J-machine [21]. Alpha 21364 is a 2-D torus and Cray T3D is a 3-D torus [6].

A system is said to be fault tolerant if it can remain functional in the presence of faults [13]. A massively parallel computer system is complex, which increases the possibility of having some faulty components (e.g., processors/nodes or communication links/edges). Hence, fault tolerance of a system plays a crucial role in its actual performance. One of the functionality criteria determines that a system is functional if and only if there is a nonfaulty communication path between each pair of nonfaulty processors [23]. That is, if all nonfaulty nodes and links in the system form a connected graph–which can be used to simulate the entire system to mask the effects of faults–then the system is fault tolerant.

Some graph theoretic concepts are used to develop deterministic measures of the fault tolerance. Conventionally, network fault tolerance is defined as the maximum number of nodes that are tolerant to failure without inducing a possible disconnection in the network [22], a number which is smaller than the connectivity of the network by one. Accordingly, the network fault tolerance of a regular graph topology with degree $n$ is at most $n-1$. For a 2-D mesh, the network fault tolerance is only 1, because the failure of the two neighboring nodes of a nonfaulty corner node would disconnect the corner node from any other nonfaulty node (if any) of the system. As a matter of fact, this traditional measure of fault tolerance is quite conservative. Consequently, much effort has been devoted to introduce more realistic measures. Many measures, either deterministic or probabilistic, have been proposed over the years. Esfahanian [13] introduced the concept of forbidden faulty sets, in which components cannot be faulty at the same time. Esfahanian’s results showed that an $n$-cube can tolerate up to $2n-3$ node failures and remain connected provided that, for each node $v$ in the network, all the nodes which are directly connected to $v$ do not fail at the same time. Latifi et al. [15] proposed the concept of k-safeness, based on the idea of forbidden faulty sets, and showed that a $k$-safe $n$-cube can tolerate up to $2^k(n-k)-1$ faulty nodes. All of these results demonstrate that the traditional fault tolerance measure tends to underestimate the fault tolerance of large networks. In fact, although many deterministic fault-tolerant routing algorithms for meshes and tori have been proposed [14], [7], [28], [12], little research has focused on deterministic fault tolerance measures of meshes or tori.

Probabilistic measures consider a random graph model to characterize a system. These measures evaluate the probability of the system being functional by assuming that nodes or links can fail independently with known probabilities. Probabilistic measures of fault tolerance can often model the real world better than deterministic ones because the real world is too complex to be studied in a deterministic way. Several studies have been conducted on probabilistic measures. Najjar and Gaudiot [20] introduced network resilience, a probabilistic measure of network fault tolerance expressed as the probability of a disconnection. However, the problem of computing the connection/disconnection probability of a general network has proved to be NP-complete [11], [24]. In contrast, computing upper and lower bounds on the connection/disconnection probability are more efficient. For example, [26], [9], [8] studied the lower bounds on the reliability of hypercubes, while Angel et al. [4], Chen et al. [10], and Liang et al. [17] investigated upper bounds and lower bounds on the connection probability of meshes.

While the lower bound on the connection probability seems to be of more interest, the upper bound is also of great importance. First of all, to achieve a desired network connection probability, the individual node failure probability can absolutely not be greater than some upper bound. In fact, the gap between the known lower and upper bounds is very large for meshes with large sizes. Nevertheless, the upper bound can still provide a reference to the corresponding lower bound. Although simulations demonstrate a good estimation to the connection probability, the theoretical analysis demonstrates accurate results.

Chen et al. [10] gave upper bounds on the connection probability for 2-D meshes based on rhombuses. A rhombus of a 2-D mesh is defined as a connected subgraph consisting of five nodes — one node located at the center and each of the other four nodes located on one direction. However, there are two important factors causing the weakness of upper bounds determined in this fashion. First, Chen et al. [10] derived the upper bounds by distinguishing disjoint rhombuses. A rhombus is of too small a size to be a good tool in studying the connection probability for meshes. Second, because the degree of a node on the boundary of a grid-based mesh is smaller than 4, such a node has a greater probability to be an isolated node than those nodes of degree 4 in the middle of the mesh. This factor is not considered in [10].

Motivated by the aforementioned deficiencies, in this paper, we study the upper bounds on the connection probability for 2-D meshes and 2-D tori, and further obtain tighter upper bounds. To achieve better upper bounds, we deal with nodes on the boundary and in the middle of a mesh based on combinatorial tools such as The Principle of Inclusion and Exclusion, and divide the meshes into small submeshes instead of rhombuses. Simulation results show that the upper bounds we have derived are much closer to the possible exact connection probability than previously determined bounds.

The main contribution of this paper is giving combinatorial method tools to obtain tighter upper bound on the connection probability without too complicated computation, which is much more better than that in [10]. For most meshes of large sizes studied in [10], the improvements are large. The methods and tools are much more important than the results themselves. If a new network was designed, people often want to know the connection probability to tell if this network is a good one. Although such a newly designed network may be much complicated than meshes, the tools and methods used to study the bounds on the connection probabilities for meshes may be used to study such a network. The methods and tools may be used to study the reliabilities for more objects.

To simplify our discussion, here we only consider node faults rather than link faults.

The rest of this paper is organized as follows. Section 2 overviews related work. Section 3 introduces the notations and the methods used in this paper. In Section 4, we give the main results of this paper, including some novel combinatorial approaches and new upper bounds on the connection probability for 2-D meshes. The extensive simulation results are presented in Section 5. Section 6 concludes the paper.