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Analysis and Design of a Half Hypercube Interconnection Network

Shahid Rajaee Teacher Training UniversityFaculty of Computer Engineering

PRESENTED BY:

Amir Masoud Sefidian

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Today’s Lecture Content

• Introduction

• Definition and Properties of the Half Hypercube

• Connectivity of Half Hypercube

• Routing Algorithm and Diameter of Half Hypercube

• Conclusion

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Today’s Lecture Content

• Introduction• Definition and Properties of the Half Hypercube

• Connectivity of Half Hypercube

• Routing Algorithm and Diameter of Half Hypercube

• Conclusion

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Introduction• Need to high-performance parallel processing is increasing because modern

application require real-time processing.• Parallel processing system can connect thousands of processors with via an

interconnection network.• The most common parameters for evaluating the performance of interconnection

networks: • Connectivity• Diameter (relevant to message transmission time) • Degree (relevant to hardware cost)

• Typical parameter to evaluate interconnection network performance:• Network cost = degree * diameter

• A parallel computer design increase hardware costs of an interconnection network, because of the increased number of processor.

• Lower degree can reduce hardware costs but Delay increase and throughput are degraded.

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Introduction• Hypercube is a typical interconnection network topology:

• Is node- and edge-symmetric with a simple routing algorithm.• Simple recursive structure. • Easily embedded in various types of existing interconnection networks (flexibility).

• Hypercube drawback:• Increasing network costs associated with the increased degree when the number of nodes increases.

• Proposed variations of the hypercube to improve this shortcoming :• Multiple Reduced Hypercube • Twisted cube • Folded hypercube • Connected hypercube network • Extended hypercube

• New variation of the hypercube proposed in 2013 by Jong-Seok Kim, Mi-Hye Kim and Hyeong-Ok Lee:

• n-dimensional Half Hypercube 𝑯𝑯𝒏

• reduce the degree by approximately half• has the same number of nodes as a hypercube.

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Today’s Lecture Content

• Introduction

• Definition and Properties of the Half Hypercube• Connectivity of Half Hypercube

• Routing Algorithm and Diameter of Half Hypercube

• Conclusion

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Definition and Properties of the Half Hypercube• Denote an n-dimensional half hypercube as 𝐻𝐻𝑛 and represent its node with n binary bits:• Node S in 𝐻𝐻𝑛 denoted: 𝑠𝑛𝑠𝑛−1𝑠𝑛−2…𝑠𝑖 …𝑠3𝑠2𝑠1 (𝑛 ≥ 3).• There are two types of edge:

• h-edge : connects node 𝑆(= 𝑠𝑛𝑠𝑛−1𝑠𝑛−2…𝑠𝑖 …𝑠3𝑠2𝑠1) to a node that has the complement in exactly one h position of the bit string of node S(1 ≤ ℎ ≤ 𝑛/2 )

• sw(swap)-edge: connects node 𝑆(= 𝑠𝑛𝑠𝑛−1𝑠𝑛−2…𝑠ℎ𝑠ℎ−1…𝑠3𝑠2𝑠1) to a node in which the 𝑛/2 leftmost bits of the bit string of S and the 𝑛/2 bits on the right-side of S starting from the 𝑛/2 bit are swapped(h= 𝑛/2 ).

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sw-edge Examplewhen n is even:

Connect to

(𝑠𝑛𝑠𝑛−1𝑠𝑛−2… 𝑠 𝑛/2 +1) and (𝑠 𝑛/2 𝑠 𝑛/2 −1𝑠 𝑛/2 −2… 𝑠3𝑠2𝑠1) of S are exchanged.

When n is odd:

Connect to

(𝑠𝑛𝑠𝑛−1𝑠𝑛−2… 𝑠 𝑛/2 +1) and (𝑠 𝑛/2 𝑠 𝑛/2 −1𝑠 𝑛/2 −2… 𝑠3𝑠2) of S are exchanged.

𝑆(= 𝑠𝑛𝑠𝑛−1𝑠𝑛−2… 𝑠 𝑛/2 +1 𝑠 𝑛/2 𝑠 𝑛/2 −1𝑠 𝑛/2 −2… 𝑠3𝑠2𝑠1)

𝑠 𝑛/2 𝑠 𝑛/2 −1𝑠 𝑛/2 −2… 𝑠3𝑠2𝑠1 𝑠𝑛𝑠𝑛−1𝑠𝑛−2… 𝑠 𝑛/2 +1

𝑆(= 𝑠𝑛𝑠𝑛−1𝑠𝑛−2… 𝑠 𝑛/2 +1 𝑠 𝑛/2 𝑠 𝑛/2 −1𝑠 𝑛/2 −2… 𝑠3𝑠2𝑠1)

𝑠 𝑛/2 𝑠 𝑛/2 −1𝑠 𝑛/2 −2… 𝑠3𝑠2 𝑠𝑛𝑠𝑛−1𝑠𝑛−2… 𝑠 𝑛/2 +1 𝑠1

if two parts of ⌊𝑛/2⌋ bits to exchange in the bit string of node S are the same, the node S connects to one’s complement of the binary number of node S.

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Example:

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Example:

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Example (Wrong Figure!!):

5-dimensional half hypercube (HH5)

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Degree of 𝐻𝐻𝑛:𝑛

2+ 1

number of h-edges (1 ≤ ℎ ≤𝑛

2)

sw-edge

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Lemma: An 𝐻𝐻𝑛 graph is expanded with recursive structures.

Proof An 𝐻𝐻𝑛 graph is constructed with the nodes of two (n-1)-dimensional 𝐻𝐻𝑛−1

graphs by adding one sw-edge or h-edge.The address of each node in an 𝐻𝐻𝑗 graph is represented as j bit strings with binary

numbers {0, 1}(binary number of length j).

𝐻𝐻𝑗0 j +1 position of the address of a node 0

𝐻𝐻𝑗1 j +1 position of the address of a node is 1

Case 1 When expanded from an odd-dimension (j) to an even-dimension (k=j+1)

Construct an 𝐻𝐻𝑘 graph by connecting a node of 𝐻𝐻𝑗0 and a node of 𝐻𝐻𝑗

1 in 𝐻𝐻𝑗.

The sw-edges in 𝐻𝐻𝑗 will be replaced with new sw-edges in the expanded 𝐻𝐻𝑘 graph.

A node of 𝐻𝐻𝑗0 the address bit of which is binary 1 at the n/2 position, will be

connected to a node of 𝐻𝐻𝑗1 through an sw-edge in 𝐻𝐻𝑘. When the bit is binary 0,

the node will be connected to a node of 𝐻𝐻𝑗0.

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Case 2: When expanded from an even-dimension (k) to an add-dimension (l=k+1):

Construct an 𝐻𝐻𝑙 graph by connecting a node of 𝐻𝐻𝑘0 and a node of 𝐻𝐻𝑘

1 in 𝐻𝐻𝑘.The sw-edges in 𝐻𝐻𝑘 will be replaced with new sw-edges in the expanded 𝐻𝐻𝑙

graph.

A node of 𝐻𝐻𝑘0 the address bit of which is binary 1 at the 𝑛/2 position, will be

connected to a node of 𝐻𝐻𝑘1 through an sw-edge in 𝐻𝐻𝑙. When the bit is binary 0,

the node will be connected to a node of 𝐻𝐻𝑘0.

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Lemma : There exist 2 𝑛/2 𝑛/2 -dimensional hypercube structures in an 𝐻𝐻𝑛.

Proof: node 𝑆(= 𝑠𝑛𝑠𝑛−1𝑠𝑛−2…𝑠ℎ …𝑠3𝑠2𝑠1)

• The h-edge of 𝐻𝐻𝑛 is equal to the edge of a hypercube.• The structure of a partial graph constructed from 𝐻𝐻𝑛 via the h-edge is the same as

the structure of a 𝑛/2 -dimensional hypercube.• Assume that a partial graph that has the same structure as an 𝑛/2 -dimensional

hypercube is 𝐻𝐻𝑛𝑛/2

in 𝐻𝐻𝑛 and refer it as a cluster. • The address of each node in a cluster is 𝑠ℎ …𝑠3𝑠2𝑠1.

• 𝐻𝐻𝑛 consist 2 𝑛/2 of clusters: • because the number of bit strings that can be configured by the bit string

𝑠𝑛𝑠𝑛−1𝑠𝑛−2…𝑠ℎ is 2 𝑛/2 .

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Today’s Lecture Content

• Introduction

• Definition and Properties of the Half Hypercube

•Connectivity of Half Hypercube• Routing Algorithm and Diameter of Half Hypercube

• Conclusion

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Connectivity of 𝐻𝐻𝑛

• Node (edge) connectivity is the minimum number of nodes (edges) that must be removed to disconnect an interconnection network to two or more parts without duplicate nodes.

• if degree = node connectivity said that the interconnection network has maximum fault-tolerance.

• Characteristic notation of interconnection network G:

• Node connectivity: 𝑘(𝐺)

• Edge connectivity: 𝜆(𝐺)

• Degree of interconnection network: d(𝐺)

It has been proven that 𝑘(𝐺) ≤ 𝜆 𝐺 ≤ d(𝐺)

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Theorem 1 The connectivity of 𝐻𝐻𝑛

• 𝑘(𝐻𝐻𝑛) = 𝜆 𝐻𝐻𝑛 = d(𝐻𝐻𝑛)• Proof:• Let’s prove that 𝐻𝐻𝑛 remains connected even when n nodes are deleted from 𝐻𝐻𝑛.• Characteristic notation of interconnection network G:

• Node connectivity: 𝑘(𝐺)

• Edge connectivity: 𝜆(𝐺)

• Degree of interconnection network: d(𝐺)

It has been proven that 𝑘(𝐺) ≤ 𝜆 𝐺 ≤ d(𝐺)

From Lemma 2: 𝐻𝐻𝑛 is composed of clusters, all clusters in 𝐻𝐻𝑛 are hypercubes and two arbitrary nodes are connected via an sw-edge.

Assuming that X is a partial graph of 𝐻𝐻𝑛 where |X| = n, it will be proven that 𝑘 𝐻𝐻𝑛 ≥𝑛

2+ 1 by

demonstrating that 𝐻𝐻𝑛 remains connected even after the removal of X:Dividing two cases in accordance with the location of X. We denote the 𝐻𝐻𝑛 in which X is deleted as 𝐻𝐻𝑛 − 𝑋 and a node of HHn as S.

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Case 1: When X is located in one cluster of 𝐻𝐻𝑛

• Degree of each node in a cluster is 𝑛/2 .

If 𝑛/2 nodes adjacent to an arbitrary node S of the cluster are the same as the nodes to be deleted from X, 𝐻𝐻𝑛 is divided into two components:

• an interconnection network 𝐻𝐻𝑛 − 𝑋. • a node S.

• all nodes in a cluster are linked to other clusters in 𝐻𝐻𝑛 via sw-edges.

• 2 𝑛/2 -1 clusters in which X is not located are also connected to other clusters via sw-edges.

• Therefore, 𝐻𝐻𝑛 − 𝑋 is always connected when X is included in only a cluster of 𝐻𝐻𝑛.

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Case 1: When X is located in one cluster of 𝐻𝐻𝑛

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Case 2: When X is located across two or more clusters of 𝐻𝐻𝑛

As the nodes of X to be deleted are included across two or more clusters of 𝐻𝐻𝑛 ,the number of nodes to be deleted from a cluster is at most 𝑛/2 − 1 .

Even if 𝑛/2 − 1 nodes adjacent to an arbitrary node S of a cluster are deleted, thenodes of the cluster in which node S is included remain connected because thedegree of a node in a cluster is 𝑛/2 .Although the other node to be removed is located in a different cluster, and not in the cluster that includes node S, it is still clear that 𝐻𝐻𝑛 remains connected.

I 𝑘 𝐻𝐻𝑛 ≥ 𝑛 ≥𝑛

2+ 1 because 𝐻𝐻𝑛 always remains connected after removing X from any clusters in

𝐻𝐻𝑛 and

(𝐼𝐼) 𝑘(𝐻𝐻𝑛) ≤𝑛

2+ 1 because the degree of 𝐻𝐻𝑛 is

𝑛

2+ 1 .

I , (𝐼𝐼) ⇒ connectivity of 𝐻𝐻𝑛 , 𝑘(𝐻𝐻𝑛) =𝑛

2+ 1 .

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Today’s Lecture Content

• Introduction

• Definition and Properties of the Half Hypercube

• Connectivity of Half Hypercube

• Routing Algorithm and Diameter of Half Hypercube

• Conclusion

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Routing Algorithm and Diameter of 𝐻𝐻𝑛

• This section analyzes a simple routing algorithm and diameter of 𝐻𝐻𝑛.

• Assume :• Initial node is 𝑆 = 𝑠𝑛𝑠𝑛−1𝑠𝑛−2…𝑠𝑛/2…𝑠3𝑠2𝑠1• Destination node is T = t𝑛 t𝑛−1t𝑛−2…t𝑛/2… t3t2t1

• Simple routing algorithm can be considered in two cases depending on whether n is an even or an odd number.

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Case 1 When n is an even number:

Denote S with two 𝑛/2 bit strings A(= 𝑠𝑛𝑠𝑛−1𝑠𝑛−2… 𝑠𝑛/2) and B(= 𝑠𝑛2−1…𝑠3𝑠2𝑠1) => S=AB

Denote T with two 𝑛/2 bit strings C(= t𝑛t𝑛−1t𝑛−2… t𝑛/2) and D(= t𝑛2−1…t3t2t1) => T=CD

S=(0010) => A=00,B=10T=(1101) => C=11,D=01

Corollary 1: When n is even, length of the shortest path is

2 ∗ 𝑛/2 + 1 = 𝑛 + 1by the above algorithm.

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Case 2 When n is an odd number:

initial node is 𝑆 = 𝑠𝑛𝑠𝑛−1𝑠𝑛−2… 𝑠 𝑛/2 +1 𝑠 𝑛/2 𝑠 𝑛/2 −1… 𝑠3𝑠2𝑠1destination node is T = t𝑛t𝑛−1t𝑛−2…𝑡 𝑛/2 +1 𝑡 𝑛/2 𝑡 𝑛/2 −1… 𝑡3𝑡2𝑡1

Denote S with two 𝑛/2 bit strings: A(= 𝑠𝑛𝑠𝑛−1𝑠𝑛−2… 𝑠 𝑛/2 +1) and B(𝑠 𝑛/2 𝑠 𝑛/2 −1… 𝑠3𝑠2)

=> S = AB𝑠1Denote T with two 𝑛/2 bit strings: C(= 𝑡𝑛𝑡𝑛−1𝑡𝑛−2… 𝑡 𝑛/2 +1) and D(𝑡 𝑛/2 𝑡 𝑛/2 −1… 𝑡3𝑡2)

=> T = CD𝑡1

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Simple routing algorithm when n is odd:

S=(00000) => A=00,B=00T=(11111) => C=11,D=11

Corollary 2: When n is odd:

Length of the shortest path is 2 ∗𝑛

2+ 2

by the above algorithm.

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Routing Algorithm and Diameter of 𝐻𝐻𝑛

Through the proposed simple routing algorithms, we can see an upper bound forthe diameter of 𝐻𝐻𝑛, thus proving Theorem 2:

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Conclusion:• A new variation of the hypercube (half hypercube interconnection network) that

• reduced the degree by approximately half• has the same number of nodes as a hypercube.

• Evaluate the effectiveness of the proposed half hypercube.• Analyzed its connectivity and diameter properties. • Analyzed a simple routing algorithm of 𝐻𝐻𝑛 and presented an upper bound for the

diameter of 𝐻𝐻𝑛.• These results demonstrate that the proposed half hypercube is an appropriate

interconnection network for implementation in a large-scale system.• More works is available on:

• “Broadcasting and Embedding Algorithms for a Half Hypercube Interconnection Network”

by: Mi-Hye Kim, Jong-Seok Kim and Hyeong-Ok Lee

Reference : http://link.springer.com/chapter/10.1007/978-94-007-6738-6_65

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