Journal of Applied Mathematics

Journal of Applied Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 935737 | 10 pages | https://doi.org/10.1155/2014/935737

Broadcasting Algorithms of Three-Dimensional Petersen-Torus Network

Academic Editor: Guangchen Wang
Received19 Nov 2013
Accepted04 Mar 2014
Published10 Apr 2014

Abstract

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() under SLA (single-link available) and MLA (multiple-link available) models.

1. Introduction

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 , where is the set of nodes and is the set of edges of graph . Each processor is an element of and two arbitrary processors and are connected by a communication link . In , each processor is represented as a node and a communication link between two processors is represented as an edge. The distance between and in is defined as the length of a shortest path connecting and , denoted as . The diameter of is defined as the maximal value of the distances between all pairs of nodes in , denoted as (i.e., ).

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 [1]. Broadcasting is also one of the major parameters determining the performance of interconnection networks and is significantly influenced by the efficiency of broadcasting algorithms [2]. Broadcasting is a basic data communication method for interconnection networks, corresponding to message transmission between nodes [3]. In general, messages are disseminated between nodes in two ways: one-to-all broadcasting, whereby messages are sent from a source node to all other nodes in the network, and all-to-all broadcasting, where messages are sent from all nodes to all other nodes in the network [27]. Broadcasting algorithms are commonly based on two communication models: single-port or all-port communication [6, 7]. In the single-port communication model, each node transmits messages using only one link incident on it at each stage of broadcasting, whereas in all-port communication, each node transmits messages using all links incident on it at each stage of broadcasting. The former is known as the single-link-available (SLA) model, and the latter is the multiple-link-available (MLA) model [8].

The three-dimensional Petersen-torus interconnection network    was proposed by Seo and is based on the Petersen graph with a fixed four-degree network [9]. The network costs are improved compared to 3D mesh variation networks that have an equivalent number of nodes as . In this paper, we propose and analyze algorithms for one-to-all and all-to-all broadcasting in under SLA and MLA models. The results are as follows. The one-to-all broadcasting time in under SLA model is when odd, when even. The one-to-all broadcasting time in under MLA model is when odd, when even. The all-to-all broadcasting time in under SLA model is .   The all-to-all broadcasting time in under SLA model is when , , and are odd and is when , , and are even.

The remainder of this paper is organized as follows. In Section 2, we describe the properties of In Section 3, we propose the one-to-all broadcasting in under SLA and MLA models and we show the all-to-all broadcasting in under SLA and MLA models. Section 5 summarizes and concludes the paper.

2. Preliminaries

The three-dimensional Petersen-torus interconnection network    is based on the Petersen graph [10], which is a regular node- and edge-symmetric graph with 10 nodes, a degree of 3, and a diameter of 2. is also a regular graph and has nodes, edges, and a fixed degree of 4. is defined as follows: , where is a set of nodes and is a set of edges. An edge that connects two arbitrary nodes and is denoted by . A node in is represented by Definition 1 [9].

Definition 1. We have

In , a Petersen graph is located at the intersection of the -, -, and -axes on a coordinate plane and is called a module. The address of a module is represented by and the address of a node in a module is represented by , where , , and are the -, -, and -axes of the module and is a node address in the module (i.e., in the Petersen graph). 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 [9], where the symbol represents the modulus operator. Figure 1 shows constructed using Definition 2. For neatness, in Figure 1, wraparound edges are omitted at all basic modules.

Definition 2. We have the following.(1)The longitudinal edge is .(2)The latitudinal edge is .(3)The diagonal edge is .(4)The reverse-diagonal edge is .(5)The dimensional edge is .(6)The wraparound edge is , , , ,     ,  .

3. One-to-All Broadcasting of 3PT

In this section, we analyze the one-to-all broadcasting of under SLA and MLA models. The broadcasting of must be calculated by dividing the network into two cases: and . However, we only analyze the case of , because the results of the two cases are very similar. In a previous work [8], the one-to-all broadcasting of the Petersen graph was analyzed as Lemma 3.

Lemma 3. 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.(i): The message transmission via an internal edge.(ii): The message transmission via an external edge.(iii): An internal node of a module received the message by .(iv)-edge: Reverse-diagonal edge for message transmission from to .(v)-edge: Reverse-diagonal edge for message transmission from to .(vi)-edge: Diagonal edge for message transmission from to .(vii)-edge: Diagonal edge for message transmission from to .(viii)Source module: Module received the message by Condition 1.(ix)Destination module: Module for Step 5.(x): Modules that receive a message through diagonal edges and forward the message to adjacent modules via longitudinal, latitudinal, or reverse-diagonal edges or through reverse-diagonal edges and forward the message to adjacent modules via longitudinal, latitudinal, or reverse-diagonal edges.(xi)If , then ; if , then where .(xii)If , then , if , then , and if , then where .

The conditions for the one-to-all broadcasting of under SLA model are as follows.

3.1. Conditions for the One-to-All Broadcasting of 3 PT under SLA Mode 1

Condition 1. Perform broadcasting between modules via internal and dimensional edges as follows. (1) or or 0).(2)(i)When odd, (ii)When even,
In , if or 4 or 6 or 9, then remove the message transmission by , and if or 5, then remove the message transmission by or 5).

Condition 2. All nodes located inside receive a message in the following order:(1); ,(2); ; ; ,(3); .

Condition 3. The modules that received a message through a -edge transmit the message in the following order, and the modules that received a message through a -, -, or -edge also transmit the message in a similar way to the following orders. However, when performing , if the modules to receive a message (i.e., the modules to which nodes , , , and belong) already had the message, the message is not sent to the modules:(1),(2);  ,(3);  ;  ,(4);  ;  ;,(5);  ;  ;  .
The conditions for the one-to-all broadcasting of under MLA model are as follows.

3.2. Conditions for Using the MLA Model

Condition 1. Perform broadcasting between modules via internal and dimensional edges as follows. (1)); ).(2)(i)When odd, (ii)When even, In , if or 4 or 6 or 9, then remove the message transmission by , and if or 5, then remove the message transmission by or 5).

Condition 2. All nodes located inside receive a message in the following order:(1); ; ; ,(2); ; ; .

Condition 3. The modules that received a message through a -edge transmit the message in the following order, and the modules that received a message through a -, -, or -edge also transmit the message in a similar way to the following orders. However, when performing Condition 3, if the modules to receive a message already had the message, the message is not sent to the modules:(1);  ;  ,(2);  ;  ;  ;  ;  ;  ,(3);  ;  ;  .

Algorithm 1 shows one-to-all broadcasting algorithm (OABA) of under SLA and MLA models.

Step  1. Perform Condition and .
Step  2. A message is transmitted from a source module with the message to eight modules adjacent to the source module
through an external edge.
Step  3. The modules that received a message via diagonal or reverse-diagonal edges perform Condition .
Step  4. Repeat Step  3 until the message is transmitted to all modules in .
Step  5. The node with the message transmit the message to other nodes in the same module using the broadcasting scheme in
[8] when its all eight adjacent modules (except for the modules connected via dimensional edges) already had the message.

Figure 2 shows an example of the one-to-all broadcasting among modules of in under SLA model. Numbers represent the arrival time of a message to the corresponding module, and numbers in parentheses represent the arrival time of a message to all nodes in the module. Arrows represent the processes of message transmission.

Theorem 4. The one-to-all broadcasting time of the Petersen-torus network is when odd and when even under SLA model.

Proof. It is proven by dividing the broadcasting time into three cases depending on the number of edge types used for broadcasting.
Case  1. Consider the case when broadcasting is performed via internal edges, dimensional edges, and only one external edge, such as -, -, -, or -edges.
The maximum broadcasting time for Condition 1 is when odd and when even. The broadcasting time for Condition 2 is 3. The maximum broadcasting time via external edges except dimensional edges, such as -, -, -, or -edges, is . The maximum total value from performing in each module of except for the source and destination modules is . The internal broadcasting time for the destination module is 4 under SLA model by Lemma 3. Therefore, the broadcasting time for this case is when odd and when even.
Case  2. Consider the case when broadcasting is performed via internal edges, dimensional edges, and the combination of two types of external edges, for example, -edge and latitudinal edges, or -edge and -edges.
We assume that the two edges used are -edge and longitudinal edges. The maximum broadcasting time for Condition 1 is when odd and when even. The broadcasting time for Condition 2 is 3. The maximum broadcasting time via -edge and longitudinal edges is . The internal broadcasting time of module is 4 because the interior of , which receives a message via a -edge and transmits the message to adjacent modules through a longitudinal edge, must meet Condition 3. The maximum total value from performing in each module of except for the source and destination modules is . The internal broadcasting time for the destination module is 4 under SLA model by Lemma 3. Therefore, the broadcasting time for this case is when odd and when even.
Case  3. Consider the case when broadcasting is performed via internal edges, dimensional edges, and the combination of three types of external edges, for example, -edge, -edge, and latitudinal edges.
We assume that the two edges used are -edge, -edge, and latitudinal edges. The maximum broadcasting time for Condition 1 is when odd and when even. The broadcasting time for Condition 2 is 3. The maximum broadcasting time via -edge, -edge, and longitudinal edges is . The internal broadcasting time of module is 4 because the interior of , which receives a message via a -edge and transmits the message to adjacent modules through a -edge, must meet Condition 3. The internal broadcasting time of module is 4 because the interior of , which receives a message via a -edge and transmits the message to adjacent modules through a longitudinal edge, must meet Condition 3. The maximum total value from performing in each module of except for the source and destination modules is . The internal broadcasting time for the destination module is 4 under SLA model by Lemma 3. Therefore, the broadcasting time for this case is when odd and when even.

Theorem 5. The one-to-all broadcasting time of the Petersen-torus network is when   odd and when   even under MLA model.

Proof. It is proven by dividing the broadcasting time into three cases depending on the number of edge types used for broadcasting.
Case  1. Consider the case when broadcasting is performed via internal edges, dimensional edges, and only one external edge, such as -, -, -, or -edges.
The maximum broadcasting time for Condition 1 is when odd and when even. The broadcasting time for Condition 2 is 2. The maximum broadcasting time via external edges except dimensional edges, such as -, -, -, or -edges is . The maximum total value from performing in each module of except for the source and destination modules is . The internal broadcasting time for the destination module is 2 under MLA model by Lemma 3. Therefore, the broadcasting time for this case is when odd and when even.
Case  2. Consider the case when broadcasting is performed via internal edges, dimensional edges, and the combination of two types of external edges, for example, -edge and latitudinal edges, or -edge and -edges.
We assume that the two edges used are -edge and longitudinal edges. The maximum broadcasting time for Condition 1 is when odd and when even. The broadcasting time for Condition 2 is 2. The maximum broadcasting time via -edge and longitudinal edges is . The internal broadcasting time of module is 2 because the interior of , which receives a message via a -edge and transmits the message to adjacent modules through a longitudinal edge, must meet Condition 3. The maximum total value from performing in each module of except for the source and destination modules is . The internal broadcasting time for the destination module is 2 under SLA model by Lemma 3. Therefore, the broadcasting time for this case is when odd and when even.
Case  3. Consider the case when broadcasting is performed via internal edges, dimensional edges, and the combination of three types of external edges, for example, -edge, -edge, and latitudinal edges.
We assume that the two edges used are -edge, -edge, and latitudinal edges. The maximum broadcasting time for Condition 1 is when odd and when even. The broadcasting time for Condition 2 is 2. The maximum broadcasting time via -edge, -edge, and longitudinal edges is . The internal broadcasting time of module is 2 because the interior of , which receives a message via a -edge and transmits the message to adjacent modules through a -edge, must meet Condition 3. The internal broadcasting time of module is 2 because the interior of , which receives a message via a -edge and transmits the message to adjacent modules through a longitudinal edge, must meet Condition 3. The maximum total value from performing in each module of except for the source and destination modules is . The internal broadcasting time for the destination module is 2 under SLA model by Lemma 3. Therefore, the broadcasting time for this case is when odd and when even.

4. All-to-All Broadcasting of 3PT

To analyze the all-to-all broadcasting time of , we first examine the all-to-all broadcasting time of the Petersen graph because the basic modules consisting of are the Petersen graph. Algorithms 2 and 3 describe the all-to-all broadcasting algorithms for the Petersen graph based on the SLA model and the MLA model, respectively. Let a cycle consisting of the nodes of the Petersen graph be let and a cycle consisting of the nodes of the Petersen graph be .

Step  1. A message is transmitted between nodes in each cycle. Each message is transmitted as follows and each message
transmission is accomplished simultaneously.
; ; ;
; ; ;
; ; ;
.
Step  2. All nodes located inside cycle transmit the message to all nodes located inside cycle .
Step  3. All nodes located inside cycle transmit the message to all nodes located inside cycle .

Step  1. A message is transmitted between nodes in each cycle. Each message is transmitted as follows and each message
transmission is accomplished simultaneously.
; ; ; ; ; ;
; ; ; ; ; ;
; ; ; ; ; ;
; .
Step  2. All nodes located inside cycle transmit their message to all nodes located inside cycle ; at the same time, all nodes
located inside cycle transmit their message to all nodes 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 2) plus 2 (i.e., the number of messages transmitted between two cycles at Step 2 and Step 3 in Algorithm 2). The all-to-all broadcasting time of the Petersen graph under MLA model is 3 from the sum of 2 (i.e., the number of messages transmitted between nodes located inside each cycle at Step 1 in Algorithm 3) and 1 (i.e., the number of messages transmitted between two cycles at Step 2 Algorithm 3). Consequently, the following lemma is proven.

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

Algorithms 4 and 5 describe the all-to-all broadcasting algorithms for based on the SLA model and the MLA model, respectively.

Step  1. Perform broadcasting among nodes located inside each module using the all-to-all broadcasting algorithm for the
Petersen graph with the SLA model in Algorithm 2.
Step  2. Every module performs the following broadcasting via internal and latitudinal edges.
.
Step  3. Every module performs the following broadcasting via internal and longitudinal edges.
.
Step  4. Every module performs the following broadcasting via internal and dimentional edges.
.
Step  5. Perform broadcasting among nodes located inside each module using the all-to-all broadcasting algorithm for the
Petersen graph with the SLA model in Algorithm 2.

Step  1. Perform broadcasting among nodes located inside each module using the all-to-all broadcasting algorithm for the
Petersen graph with the MLA model in Algorithm 3.
Step  2. Perform Condition .
Step  3. Perform Condition .
Step  4. Perform Condition .
Step  5. Perform broadcasting among nodes located inside each module using the all-to-all broadcasting algorithm for the
Petersen graph with the MLA model in Algorithm 3.

The all-to-all broadcasting time of under SLA model is as follows. Step 1 takes 6 from Lemma 6. The broadcasting time taken at Step 2 is from the number of broadcasts by internal edges plus those by longitudinal edges . The broadcasting time taken at Step 3 is from the number of broadcasts by internal edges plus those by latitudinal edges . The broadcasting time taken at Step 4 is from the number of broadcasts by internal edges plus those by dimensional edges . The number of broadcasts at Step 5 is 6 from Lemma 6. Therefore, the all-to-all broadcasting time of under SLA model is . Consequently, the following theorem is proven.

Theorem 7. The all-to-all broadcasting time of    under SLA model is  .

Figure 3 shows an example of Steps 2 and 3 of the all-to-all broadcasting in under SLA model. Here, we have not drawn the external edges of because of the complexity of the graph. The arrows represent the processes of message transfer.

The conditions for the all-to-all broadcasting of under MLA model are as follows.

4.1. Conditions for Using the MLA Model

Condition 1. Every module performs the following broadcasting via internal and latitudinal edges.(1)(i)When odd, (ii)When even,

Condition 2. Every module performs the following broadcasting via internal and longitudinal edges. (2)(i)When odd, (ii)When even,

Condition 3. Every module performs the following broadcasting via internal and dimensional edges.(3)(i)When odd, (3)(ii)When even,

The all-to-all broadcasting time of under MLA model is as follows. Step 1 takes 3 from Lemma 6. The broadcasting time taken at Step 2 is when is odd from the number of broadcasts by internal edges plus the number of broadcasts by longitudinal edges and is when is even from the number of broadcasts by internal edges plus the number of broadcasts by longitudinal edges . The broadcasting time taken at Step 3 is when is odd from the number of broadcasts by internal edges plus the number of broadcasts by latitudinal edges and is when is even from the number of broadcasts by internal edges plus the number of broadcasts by longitudinal edges . The broadcasting time taken at Step 4 is when is odd from the number of broadcasts by internal edges plus the number of broadcasts by dimensional edges and is when is even from the number of broadcasts by internal edges plus the number of broadcasts by longitudinal edges . The number of broadcasts at Step 5 is 3 from Lemma 6. Therefore, the all-to-all broadcasting time of under MLA model is