Fault‐tolerant processor interconnection networks

Abstract

As suitable topology for interconnection networks of multiprocessors, De Bruijn graphs have been proposed and a number of investigations have been conducted on their fault tolerance. A De Bruijn graph is a directed graph with maximum degree d (the maximum number of links that can be connected to one processor), diameter k (maximum number of repeaters between two processors) and number of nodes dk (number of processors). A Kautz graph is a directed graph with maximum degree d, diameter k and number of nodes dk + dk‐1. Both have the smallest diameter among the graphs with maximum degree d. This paper proves the following: (1) If d ‐2 nodes in a De Bruijn graph are removed (if d ‐2 processors are faulty), its diameter becomes d + 1, which is only 1 larger than the original diameter. (2) If d ‐3 nodes in a Kautz graph are removed, its diameter becomes k + 1 and if d ‐1 nodes are removed, the diameter is k + 2 or less, which is at most 2 larger than the original diameter. (3) The values of (1) and (2) are at most 1 larger than the lower bound under the limitation of degree d, (4) A fault‐tolerant routing algorithm can be realized easily. This algorithm is better than any existing algorithm because of its short routes.

<title>Massively parallel Kautz and de Bruijn topologies for the free-space optical interconnection of data arrays</title>

“…Remark 4: The following results (5) to (8) show that the product operation preserves most of the properties of ordinary de Bruijn networks (Section 2). However, for the sum of de Bruijn networks and for the sum and product of Kautz networks the situation is more complicated.…”

“…(8) is valid for any order of the product. The difference between Eqs (8). and(10)is caused by the self-loops in de Bruijn networks.…”

Wide Diameter of Cayley Digraphs of Finite Cyclic Groups

“…, x n , s) with s ∈ [0, t − 1]. Imase and others [7] proved that d t−1 (B(t, n)) = n + 1. Pradhan and Reddy [11] showed that the t-wide diameter of the undirected version of the de Bruijn graph UB(t, n) is at most 2n.…”

Diameter vulnerability of iterated line digraphs in terms of the girth

“…The Kautz digraph K(d, D) is the iterated line digraph L D−1 K d+1 , where K d+1 denotes the complete symmetric digraph on d + 1 vertices [17]. Diameter vulnerability of the Kautz digraphs has been studied by finding disjoint paths between any pair of vertices [7,16]. Now, we apply Theorem 3.7 and Theorem 3.9 to this family.…”