3-disjoint paths fault-tolerant min

A. Abraham et al. (Eds.): ACC 2011, Part I, CCIS 190, pp. 21–33, 2011. © Springer-Verlag Berlin Heidelberg 2011

3-Disjoint Paths Fault-Tolerant Multi-stage Interconnection Networks

Ravi Rastogi1, Rohit Verma1, Nitin2, and Durg Singh Chauhan3

1 Department of Computer Science & Engineering and Information Technology, Jaypee University of Information Technology, Waknaghat, Solan-173234,

Himachal Pradesh, India [email protected]

2 College of Information Science and Technology, The Peter Kiewit Institute, University of Nebraska at Omaha, Omaha-68182-0116,

Nebraksa, United States of America [email protected]

3 Uttarakhand Technical Univesity, Post Office Chandanwadi, Prem Nagar, Sudohwala, Dehradun-248007, Uttarakhand, India

[email protected]

Abstract. In this paper, we have compared the existing 3-Disjoint Paths Fault-tolerant Omega Multi-stage Interconnection Network (3-DON) with newly pro-posed 3-Disjoint Fault-tolerant Gamma Interconnection Network (3-DGMIN) using the concept of reachable sets and coloring scheme. A 3-Disjoint network can concurrently send packets from the source node to increase the arrival ratio or tolerate a maximum of 2 faults in the network by re-routing the packet through another path. We have used red blue, green and yellow color for the co-loring the nodes. The 3-DON is better than existing Omega Multi-stage Inter-connection Network (OMIN) for every performance parameter except the cost. Moreover, the new 3-DGMIN is also better than existing Gamma Multi-stage Interconnection Network (GIN) for every performance parameter. Further, the experimental results show that the 3-DGMIN outperforms 3-DON when com-pared for the throughput.

Keywords: Multi-stage Interconnection Network, Fault-tolerance, 3-Disjoint Paths, Omega Network, Gamma Network, Reachable Sets, Coloring Schemes.

1 Introduction and Motivation

Multi-stage Interconnection Networks (MINs) [1-10] are used to design a network in which there are several independent paths between two modules being connected which increases the available bandwidth. With the performance requirement of the switches exceeding several terabits/sec and teraflops/sec, it becomes imperative to make them dynamic and fault-tolerant. For high reliability and performance, several methods have been suggested that provide fault-tolerance to MINs [11-23]. The basic idea for fault-tolerance is to provide multiple paths for a source–destination pair, so that alternate paths can be used in case of a fault in a path. However, to guarantee

22 R. Rastogi et al.

1–fault tolerance, a network should have a pair of alternate paths for every source-destination pair, which are Disjoint in nature [24-32]. Now-a-days applications of MINs are widely used for on-Chip communication. In past number of techniques has been used to increase the reliability and fault-tolerance of MINs, a survey of the fault-tolerance attributes of these networks is found in [1-6]. The modest cost of unique paths MINs makes them attractive for large multiprocessors systems, but their lack of fault-tolerance, is a major drawback. To mitigate this problem, three hardware options are available:

1. Replicate the entire network, 2. Add extra stages, 3. And /or Add chaining links. 4. Rearranging of the connection patterns with the addition or deletion of hard-

ware links.

In addition to this, MINs can be designed to achieve fault tolerance and collision solving by providing a set of disjoint paths. Many researchers have done sufficient work on providing 1-fault tolerance to the MINs however; little attention has been paid to design the 3-Disjoint Paths Fault-tolerant MINs.

We have been inspired by the work presented by the authors in [31]. Therefore, in this paper, we study the fault-tolerance of multiprocessor systems with 3-Disjoint multistage interconnection networks. The characterization of 3-Disjoint paths with respect to reachable sets and coloring scheme is introduced and is used to discuss fault-tolerance of the network under a given set of fault conditions. A 3-Disjoint net-work can concurrently send packets from the source node to increase the arrival ratio or tolerate a maximum of two faults in the network by re-routing the packet through another path [31]. This paper presents

1. Design of reachable sets and coloring scheme to analyze the fault-tolerance capability of any network,

2. Design scheme that enables the GIN to be 3-Disjoint with minimal hardware cost involved,

3. Comparison of the proposed 3-DGMIN with other existing 3-DON proposed in [32] with respect to network parameters,

4. Simulation results of the designed networks to realize the proposed fault tole-rant capability.

As per our proposed method, design schemes and simulation results, a designer can analyze and develop cost-efficient 3-Disjoint paths networks. We have taken Omega and Gamma Multi-stage Interconnection Network as running example throughout this paper.

The rest of the paper is organized as follows, Section II, describes the theory re-garding the application of Reachable Sets and Coloring Schemes to the MIN and more specifically converting them into the 3-Disjoint Paths MIN. The techniques are well supported by the theorem and definitions. Section III, provides the techniques of using Reachable Sets and Coloring Scheme to convert the existing Omega Network into 3-Disjoint Paths Fault-tolerant Omega MIN and to convert the existing Gamma Network into 3-Disjoint Paths Fault-tolerant Gamma MIN. Further, it shows various

3-Disjoint Paths Fault-Tolerant Multi-stage Interconnection Networks 23

examples of parallel communication between different source and destinations sup-ported by pseudocode and followed by the conclusion and the references.

2 Concept of Coloring Schemes and Reachable Sets for Disjoint MINs

In this section, the application of Reachable Sets and Coloring Schemes to the MIN and to convert them into the 3-Disjoint Paths MIN have been discussed. The tech-niques are well supported by the theorem and definitions.

We can compute the reachable sets for some specific destination nodes as accord-ing to their routing behavior for the given network. According to definitions, we have defined the reachable set at different stages to include switches that can deliver pack-ets to particular destination nodes.

Definition 2.1: A reachable set ( , ) for , switch at the stage is defined as a set of switches at the 1 stage that can deliver packets to the , switch.

Definition 2.2: Reachable set , for switch at the i stage is defined as a set of switches at the stage that can deliver packets to , switch.

For a typical Interconnection Network, atleast one path between each source and destination pair nodes, the reachable set , contains all the source nodes where,

the number of stages in the network, the final stage, any destination node in the current network.

2.1 Three Theorems for Supporting the Application of Coloring Schemes and Reachable Sets to the Disjoint MINs

For a network to be 3-Disjoint, there must exits atleast, 3-Disjoint paths between each source-destination pair. We considered the 3-Disjoint paths to be labeled with three colors-Red, Green and Blue. We start coloring by calculating the first (reachable set for a destination node. We then backtrack to calculate the reachable set in order to reach the source nodes, where is the number of stages in a network.

Theorem 2.1.1: The final destination node(s) must obtain packets from at least three nodes from the prefinal stage, which are subsequently labeled as red, green and blue.

( , ), where i the final stage, j any destination node.

Must contain at least three nodes and these are colored as red, green and blue.

1. All the nodes in the reachable set of Red, Green and Blue nodes in the prefinal stage are colored as Red, Green and Blue respectively.

2. If a node delivers packets to other nodes of varying color, then such a node remains in the reachable set of both the colors.

3. A node can be labeled with one or more colors. 4. A multicolored node can be considered as node of one color only.


Proof: We assume that packets are delivered to the final node by two nodes in the prefinal stage. The network, fails if the two nodes are faulty and therefore, the net-work is not 2 fault-tolerant. If the packets are delivered to the final node by three

Fig. 1. Topology of 16 x 16 Omega Multi-stage Interconnection Network

Fig. 2. Coloring Scheme of 16 x 16 Omega Multi-stage Interconnection Network

Stage 0 1 2 3 4

Fig. 3. 16 x 16 3-Disjoint Paths Fault-tolerant Omega Multi-stage Interconnection Network

Stage 0 1 2 3 4

Fig. 4. Coloring Scheme of 16 x 16 3-Disjoint Paths Fault-tolerant Omega Multi-stage Inter-connection Network with 0111 as the Destina-tion Node


Fig. 5. Topology of 8 x 8 Gamma Multi-stage Interconnection Network

Fig. 6. Coloring Scheme of 8 x 8 Gamma Multi-stage Interconnection Network

Fig. 7. 8 x 8 3-Disjoint Paths Fault-tolerant Gamma Multi-stage Interconnection Net-work

Fig. 8. Coloring Scheme of 8 x 8 3-Disjoint Paths Fault-tolerant Gamma Multi-stage Inter-connection Network with 0111 as the Destina-tion Node

nodes in the prefinal stage and the two nodes are simultaneously faulty, then the network does not fail (assuming that there are no other faults in the network) as there is a third path available.

Theorem 2.1.2: Each source node must deliver packets to atleast three nodes of dif-ferent color. In other words, all the source nodes must be labeled with all the three colors.

Proof: We assume that the source node delivers packets to two nodes in the first stage. The entire network fails, if the two nodes are simultaneously faulty and


therefore, the network is not 2 fault-tolerant. If the source node delivers packets to three nodes in the first stage and the two nodes are simultaneously faulty, then the network does not fail (assuming that there are no other faults in the network) as a third path is available.

Theorem 2.1.3: For a network to be 3-Disjoint their must exist at least one node of each color (including multicolored nodes) at each stage in the network.

Proof: The 3-Disjoint paths are labeled as Red, Green and Blue. Each of the three paths delivers packets from source to destination and pass through all the intermediate stages in the network. If there exists at least one path between each source-destination node pairs then all the source nodes have all the colors i.e. Red, Green and Blue.

3 Comparison of Disjoint Paths Fault-Tolerant Multi-stage Interconnection Networks on the Basis of Architecture, Coloring Schemes and Reachable Sets

3.1 Disjoint Paths 16 x 16 Omega Fault-Tolerant Multi-stage Interconnection Network

Analysis of Existing Omega Multi-stage Interconnection Network. An OMIN (see figure 1) is described by the perfect k-shuffle permutation for 01. Connection pattern is selected to be . MINs interconnect N input/output ports using k x k switches,

switch stages, each with / switches and /

)) total number of switches [1-5]. As the MINs size increases the cost also increases and the reduction in MINs, switch cost comes at the price of performance. The Network has the property of being blocking and the contention is more likely to occur on network links moreover the paths from different sources to different destina-tions share one or more links.

Analysis by Coloring Scheme and Reachable Sets. As shown in the figure 2,

1. Every destination node obtains packets from two nodes in the previous stage, 2. Every source node delivers packets to only one node at stage 0. The packet

follows either the red or the green path to reach the destination node.

As shown in the figure 2, the reachable set for the destination node are as follows:

1. 3,3 = 1(Red), 5(Green), 2. 3,3 = 0(Red), 4(Red), 2(Green), 6(Green), 3. 3,3 = 0(Red), 1(Green), 2(Red), 3(Green), 4(Red), 5(Green), 6(Red),

7(Green).

Description of 16 x 16 3-Disjoint Paths Fault-tolerant Omega Multi-stage Inter-connection Network. Out of four hardware options as listed these options as listed in Secion 1, we have choose “to add an extra stages to the network” in order to improve to convert the omega network into fault-tolerant network called as 3-DON. A 3-DON (see figure 3) of size 2 is similar to Omega Network, except the source nodes 2 and 2 1 are combined into one 2 x 4 switch and with an extra stage. The 2 x 4 switches at the 2 stage deliver packets to


1. 2, 1, 1 2 (where is not equal to 0 or 1), 2. , 1 2 (where is equal to 0), 3. , 1 2 (where is equal to 1).

Similarly, the destination nodes 2 and 2 1 are also combined into a 2 x 4 switch. These 2 x 4 switches recieve packets from

1. 2, 1, 1 2 (where is not equal to 0 or 1), 2. , 1, 2 (where is equal to 0), 3. , 1, 2 (where is equal to 1).

Analysis by Coloring Scheme and Reachable Sets. As shown in figure 4,

1. Every destination node obtains packets from four nodes in the previous stage, 2. Every source node delivers packets to three node of different color including

the source node 0 and 7, therefore, there exists only 3-Disjoint paths between the source 0 (0000) and destination 4 (0100). Hence, the network is perfectly 3-Disjoint,

3. There is a node of every color at each stage of the network; hence, there exists 3-Disjoint paths from each of the source nodes.

As shown in figure 4, the reachable set for the destination node are as follows:

1. 3,4 = 1(Red), 2(Green), 4(Blue), 5(Yellow), 2. 3,4 = 0(Red), 1(Green), 2(Blue, Yellow), 4(Red), 5(Green),

6(Blue, Yellow), 3. 3,4 = 0(Red, Green), 1(Blue, Yellow), 2(Red, Green), 3(Blue, Yellow), 4(Red, Green), 5(Blue, Yellow), 6(Red, Green), 7(Blue, Yellow).

3.2 Disjoint Paths 8 x 8 Gamma Fault-Tolerant Multi-stage Interconnection Network

Analysis of Existing Gamma Multi-stage Interconnection Network. A GIN (see figure 5) of size 2 has 1 stages labeled from 0 to and each stage involves switches. Switches of sizes 1 x 3 and 3 x 1 are coupled with the first and the last stage respectively. Each switch at intermediate stages is a 3 x 3 crossbar. Each switch at stage has three output link connections to switches at stage 1 according to the plus-

minus-2 function. The switch at stage has three output links to switches 2 , and 2 at each consecutive stage [1-5], [28-31].

Analysis by Coloring Scheme and Reachable Sets. As shown in the figure 6,

1. Every destination node obtains packets from two nodes (in the previous stage) only,

2. Every source node delivers packets to three nodes. These three nodes are not of different colors,

3. There exist only two disjoint paths from source nodes 000, 001, 010, 011 to the destination node. There exists only one disjoint path from source nodes 100, 101, 110, 111 to the destination node 100,

4. Therefore, the network is neither 2-Disjoint nor 3-Disjoint.


As shown in the figure 6, the reachable set for the destination node are as follows:

1. 3,4 = 0(Red), 4(Green), 2. 3,4 = 0(Red), 2(Red, Green), 4(Green), 6(Green), 3. 3,4 = 0(Red, Green), 1(Red, Green), 2(Green), 3(Green).

Fig. 9. Comparison of 3-Disjoint Paths 16 x 16 Omega and 8 x 8 Gamma Fault-tolerant Multi-stage Interconnection Networks based on the parameters suggested on the y-axis against the throughput values given on the x-axis.

Description of Proposed 8 x 8 3-Disjoint Paths Fault-tolerant Gamma Multi-stage Interconnection Network. Out of four hardware options as listed in Secion 1, we have used “to rearrange the connection patterns with the addition or deletion of the hardware links” in order to convert the existing gamma network into fault-tolerant network called as 3-DGMIN. A 3-DGMIN (see figure 7) of size is similar to gamma network, except the source nodes and are combined into one 2 x 4 switch. These 2 x 4 switches deliver packets to

1. 2, 1, 1 2 (where is not equal to 0 or 1), 2. , 1, 2 3 (where is equal to 0), 3. , 1, 2 3 (where is equal to 1).

Similarly, the destination nodes 2 and 2 1 are also combined into a 2 x 4 switch. These 2 X 4 switches recieve packets from

1. 2, 1, 1 2 (where is not equal to 0 or 1), 2. , 1, 2 3 (where is equal to 0), 3. , 1, 2 3 (where is equal to 1).

One to One

One to All

All to One

All to All

Two to One

Two to Two

One to Half

Network

Upper Half To

Lower Half of Netwo

rk

Alternate

Half to Half(0,2,4)to(1,3,5)

Upper Half to Upper Half

Half to All

All To Half

Gamma 97.6 75.7 88.3 48.74 95.8 90.55 88.4 76 82.56 61.6 59.3 63.5

Omega 94.75 51.45 61.5 52.85 93.1 85.13 61.56 48.59 45.1 48.5 44.6 48.4

30

50

70

90

110

Thr

ough

put


Analysis by Coloring Scheme and Reachable Sets. As shown in figure 8, 1. Every destination node obtains packets from three nodes in the previous stage, 2. There exist three disjoint paths from source nodes 000, 001, 010, 011 to the

destination node and from source nodes 100, 101, 110, 111 to the destination node 100,

3. There is a node of every color at each stage of the network; hence, there exists 3-Disjoint paths from each of the source nodes.

As shown in figure 8, the reachable set for the destination node are as follows:

1. 3,4 = 2(Red), 3(Green), 5(Blue), 6(Yellow), 2. 3,4 = 0(Red), 1(Green), 2(Red), 3(Green, Blue), 4(Red, Yellow),

5(Green), 6(Yellow), 3. 3,4 = 0(Red, Green, Blue), 1(Red, Green, Yellow), 2(Red, Green,

Yellow), 3(Red, Green, Yellow).

Pseudocode. In this section, we present the Pseudocode for sending the data from source to destination

Input: Source node(s), Destination node(d), 2 1 Output: List of all available paths between the source and destination node pair

stage_s(s/2,d,0,str); 1. stage_s(int s,int d,int n,String str) int a[]={-1,-1,-1,-1}; if(s==0) a[0]=s; a[1]=s+1; a[2]=s+2; a[3]=s+3; else if(s==N) a[0]=s-3; a[1]=s-2; a[2]=s-1; a[3]=s; else if((s>-1)&&(s<=N)) a[0]=2*s-2; a[1]=2*s-1; a[2]=2*s+2; a[3]=2*s+3; for(int i=0;i<4;i++) stage12(a[i],d,n+1,str+"-"+a[i]); 2. stage12(int s,int d,int n,String str) int a[]={-1,-1,-1}; a[0]=s-2; a[1]=s; a[2]=s+2; for(int i=0;i<3;i++) if((a[i]>-1)&&(a[i]<=N)) stage_f(a[i],d,n+1,str+"-"+a[i]); 3. stage_f(int s,int d,int n,String str) int a[]={-1,-1,-1,-1}; if(s==0) a[0]=s; a[1]=s+1; a[2]=s+2;


else if(s==N) a[0]=s-2; a[1]=s-1; a[2]=s; else if((s>-1)&&(s<=N)) a[0]=s-2; a[1]=s-1; a[2]=s+1; a[3]=s+2; for(int i=0;i<4;i++) if(a[i]==d) print(str+"-"+d);

3.3 Comparison of 3-Disjoint Paths 16 x 16 Omega and 8 x 8 Gamma Fault-Tolerant Multi-stage Interconnection Networks

Testbed, Experimental Setup and Simulation Outputs. We have designed both the networks using the Fast Interconnections tool and the architectural design of the soft-ware is already published in [33-34]. We have build this tool using Java Technology (i.e. JDK 1.6) and this version is running on top of the IBM System x, running with Novell's SUSE Linux Enterprise Server 11. We have used advanced java features to build our system. The most important part of the tool is designing of the components, which are used to design disjoint paths MINs. We have design them in paint and stored them in component library. We have provided the access of this component within the tool using ComponentChooser class.

3-DON. In this section, we are showing the output of the simulation program designed for the 3-Disjoint Omega Multi-stage Interconnection Networks.

Node 0 to Node 4. The set of all available paths between node0 and node4 are:- 0-0-0-0-0-2-4.....................................................................100% 0-0-0-0-1-2-4.....................................................................97% 0-0-0-1-3-2-4.....................................................................96% 0-0-1-2-4-2-4.....................................................................97% 0-0-2-4-0-2-4.....................................................................95% 0-0-2-4-1-2-4.....................................................................92% 0-0-2-5-3-2-4.....................................................................91% ====================================================== Number Of paths=7 Average Value=95.42857

Node 0 to Node 4 and Node 5. The set of all available paths between node0 and node4 are:-

0-0-0-0-0-2-4.....................................................................100% 0-0-0-0-1-2-4.....................................................................97% 0-0-0-1-3-2-4.....................................................................96% 0-0-1-2-4-2-4.....................................................................97% 0-0-2-4-0-2-4.....................................................................95% 0-0-2-4-1-2-4.....................................................................92% 0-0-2-5-3-2-4.....................................................................91% ====================================================== Number Of paths=7


The set of all available paths between node0 and node5 are :- 0-0-0-0-0-2-5.....................................................................86% 0-0-0-0-1-2-5.....................................................................83% 0-0-0-1-3-2-5.....................................................................83% 0-0-1-2-4-2-5.....................................................................87% 0-0-2-4-0-2-5.....................................................................81% 0-0-2-4-1-2-5.....................................................................78% 0-0-2-5-3-2-5.....................................................................78% ====================================================== Number Of paths=7 Average Value=88.85714

3-DGMIN. In this section, we are showing the output of the simulation program de-signed for the 3-Disjoint Gamma Multi-stage Interconnection Networks.

Node 0 to Node 4. The set of all available paths between node0 and node4 are:- 0-0-0-2-4.....................................................................100% 0-0-1-3-4.....................................................................99% 0-0-2-2-4.....................................................................97% 0-0-3-3-4.....................................................................96% 0-0-3-5-4.....................................................................95% ===================================================== Number of Paths=5 Average Value=97.4

Node 0 to Node 4 and Node 5. The set of all available paths between node0 and node4 are:-

0-0-0-2-4.....................................................................100% 0-0-1-3-4.....................................................................99% 0-0-2-2-4.....................................................................97% 0-0-3-3-4.....................................................................96% 0-0-3-5-4.....................................................................95% =================================================== Number of Paths=5 The set of all available paths between node0 and node5 are:- 0-0-1-3-5.....................................................................92% 0-0-2-4-5.....................................................................93% 0-0-3-3-5.....................................................................88% =================================================== Number of Paths=3 Average Value=95.0

From the subsection 3.3 (see figure 9) and the values of throughput generated by the software [33-34] shows that, it is depicted that the 8 x 8 3-DGMIN outperforms the 16 x 16 3-DON for most of the fault-tolerant communication patterns setup between source and the destination nodes and hence the 3-DGMIN is better than 3-DON.

4 Conclusion

This paper, presents a novel method to study the fault tolerance capability of MINs under multiple faults. The characterization of 3-Disjoint paths with respect to reachable sets and coloring scheme is introduce to discuss the fault-tolerance of the network under


a given set of fault conditions. Further, we have designed a 16 x 16 3-Disjoint Paths Fault-tolerant Omega Multi-stage Interconnection Network. This Disjoint network is modified version of existing Omega Multi-stage Interconnection Network and provides 3-Disjoint paths to tolerate two switches or link faults between any source-destination pair (s). Furthermore, we have designed a 8 x 8 3-Disjoint Paths Fault-tolerant Gamma Multi-stage Interconnection Network. This Disjoint network is modified version of existing Gamma Multi-stage Interconnection Network and provides 3-Disjoint paths to tolerate two switches or link faults between any source-destination pair (s).

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