Petersen-Torus Networks for Multicomputer Systems

The new Petersen-torus networks

“…An edge that connects two arbitrary nodes A and B is denoted (A, B). A node in PT(m, n) is represented by Definition 1 [22].…”

“…The edges can be divided into internal and external edges, where an internal edge connects two arbitrary nodes in a module (i.e., an internal edge is that of the Petersen graph), and an external edge connects two nodes located in different modules. Definition 2 describes the external edges of PT(m, n) [22], where the symbol % represents the modulus operator. Definition 2 1) The longitudinal edges are ((x, y, 6), (x, (y + 1)%n, 9)).…”

“…Therefore, the maximum distance between the two nodes is 2. In the routing algorithms reported here, we only describe routing by external edges, because routing by internal edges is equivalent to that described in the previous study [22]. We denote |x s − x d | as d x (i.e., d x = |x s − x d |) and y s − y d as d y (i.e., d y = |y s − y d |).…”

“…Based on Definition 2, Seo et al showed that diam(PT (m, n)) is 3(Max( m 2 , n 2 )) + 2 [22], and reduced it to 2(Max( m 2 , n 2 )) + 4 using an optimal routing algorithm in a previous study [28]. The proof proceeds as follows: "In the intermediate module between successive latitudinal edges, the length of internal path is 2, and in the intermediate module between successive diagonal edges, it is 1" [28].…”

One-to-One Embedding between Hyper Petersen and Petersen-Torus Networks

“…In the two graphs, one point is a basic module, and in mapping into a PT basic module from an HP basic module, the address of a Petersen Graph is mapped into the exactly same node. For example, xb of HP node [1001,5] is 01 and yb is 10, therefore it is mapped into the PT node (1,2,5). (v) and yb(u)≠yb(v).…”