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The three-dimensional Petersen-torus network 3PT is based on the Petersen graph and has recently been proposed as an interconnection network. 3PT is better than the well-known 3D torus and 3D honeycomb mesh in terms of diameter and network cost. In this paper, we propose one-to-all and all-to-all broadcasting algorithms for 3PT(

An interconnection network of parallel computing systems consists of a set of microprocessors, local memories, and communication links for data transmission between processors. The efficiency of interprocessor communication is critical for parallel computing systems, and the diameter of an interconnection network is an important parameter describing the efficiency of communication. Consequently, routing and the diameter of the network are major primitives with scope for improving the performance of interconnection networks.

An interconnection network can be modeled as an undirected graph

Because a delay will occur whenever a packet passes through a node, the efficiency of communication can be improved by minimizing the diameter, and by minimizing the delay in transferring a packet from a source node to a destination node under the worst-case scenario for the network. As a result, with a given fixed number of interconnection resources (i.e., nodes and edges of an interconnection network), being able to construct an interconnection network with a diameter as small as possible is a very significant factor in the design of an interconnection network [

The three-dimensional Petersen-torus interconnection network

The remainder of this paper is organized as follows. In Section

The three-dimensional Petersen-torus interconnection network

We have

In

We have the following.

The longitudinal edge is

The latitudinal edge is

The diagonal edge is

The reverse-diagonal edge is

The dimensional edge is

The wraparound edge is

In this section, we analyze the one-to-all broadcasting of

The one-to-all broadcasting time of the Petersen graph is 4 in the SLA model and 2 in the MLA model.

The following symbols are defined for broadcasting between modules.

Source module: Module received the message by Condition 1.

Destination module: Module for Step 5.

If

If

The conditions for the one-to-all broadcasting of

Perform broadcasting between modules via internal and dimensional edges as follows.

When

When

In

All nodes located inside

The modules that received a message through a

The conditions for the one-to-all broadcasting of

Perform broadcasting between modules via internal and dimensional edges as follows.

When

When

All nodes located inside

The modules that received a message through a

Algorithm

through an external edge.

[

Figure

Example of the one-to-all broadcasting among modules of

The one-to-all broadcasting time of the Petersen-torus network

It is proven by dividing the broadcasting time into three cases depending on the number of edge types used for broadcasting.

The maximum broadcasting time for Condition 1 is

We assume that the two edges used are

We assume that the two edges used are

The one-to-all broadcasting time of the Petersen-torus network

It is proven by dividing the broadcasting time into three cases depending on the number of edge types used for broadcasting.

The maximum broadcasting time for Condition 1 is

We assume that the two edges used are

We assume that the two edges used are

To analyze the all-to-all broadcasting time of

transmission is accomplished simultaneously.

transmission is accomplished simultaneously.

located inside cycle

Therefore, the all-to-all broadcasting time of the Petersen graph under SLA model is 6, which is the sum of 4 (i.e., the number of messages transmitted between nodes in the interior of each cycle at Step 1 in Algorithm

The all-to-all broadcasting time of the Petersen graph is 6 under SLA model and 3 under MLA model.

Algorithms

Petersen graph with the SLA model in Algorithm

Petersen graph with the SLA model in Algorithm

Petersen graph with the MLA model in Algorithm

Petersen graph with the MLA model in Algorithm

The all-to-all broadcasting time of

The all-to-all broadcasting time of

Figure

Example of Steps 2 and 3 of the all-to-all broadcasting in

The conditions for the all-to-all broadcasting of

Every module performs the following broadcasting via internal and latitudinal edges.

When

When

Every module performs the following broadcasting via internal and longitudinal edges.

When

When

Every module performs the following broadcasting via internal and dimensional edges.

When

When

The all-to-all broadcasting time of

The all-to-all broadcasting time of

Broadcasting is one of the major primitives with the scope for improving the performance of interconnection networks and is significantly influenced by broadcasting algorithms. In this paper, we proposed and analyzed algorithms for one-to-all and all-to-all broadcasting in

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper was supported by Sunchon National University Research Fund in 2011. The authors are grateful to the anonymous referees and the editor for their helpful suggestions.