新的Cayley圖連結網路之研究

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 (Interconnection network, Complete-rotation graph, Hypercube) This project present and analyze a new interconnection network
com- plete-rotation graph. An optimal routing algorithm and an optimal broadcasting algorithm of the network are proposed.

A novel routing tree for the network is developed for facilitating fault-tolerance.

Based on the routing tree, a fault tolerant routing algorithm, node disjoint paths, and the fault diameter can also be found in this project. Moreover, we present a scheme for embedding the hypercube into the complete-rotation graph with dilation 3. Besides, we propose a sche- me for embedding a mesh, trivalent Cayley graph, and degree four Cayley graph into the complete-rotation graph with dilation 3 and expansion 1, and embedding cube-connected-cycle into the complete-rotation graph with dila- tion 2 and expansion 1. In addition, we can find that the degree, diameter, and fault diameter of the complete-rotation graph is smaller than those of the hy- percube when these two graphs have the same number of vertices.

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Interconnection networks have

been an important research area for

highly parallel computers which com- municate by message passing.

We will present an optimal routing algorithm [3], a broadcasting algorithm [4] and find its diameter. It will be shown that the complete-rotation graph has a smaller diameter than those of the trivalent Cayley graphs and the degree four Cayley graph for the same number of nodes, while sharing its desirable property such as maximal fault tolerance.

We know fault-tolerant routing [5] and parallel paths [12] are very important in fault-tolerance consideration of inter- connection networks. Based on the sim- ple routing algorithm, a new routing tree is presented for facilitating fault-tolerance. The routing tree pos- sesses some outstanding properties which can be used to develop fault-tolerant routing as well as find par- allel paths and fault diameter [12,13].

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We know that complete-rotation graph is vertex symmetric. Let ab c be the source node and bc be the desti- a nation node. We can map the destination node to the identity node abc by renam-

ing the symbols as c

a b c b c a b a

b → , → , → , → , → and a → . Under this mapping the source c node becomes b c a . Then the paths between the original source and destina- tion nodes become isomorphic to the paths between the node b c a and the identity node abc in the renamed graph.

Thus in our subsequent discussion about

a path from a source node to a destina- tion node, the destination node is always assumed to be the identity node I with- out loss of generality.

Lemma 1 The algorithm of Sim- ple_Route correctly computes a path from an arbitrary node a a

L to the a

identity node I = t t

L . t

Theorem 1 For an arbitrary node a a

L in CR a

, the Simple_Route algorithm generates a path of length

≤ + n 1.

Observation 1 Based on Simple_Route algorithm, if u CR ∈

, then

distance( , ) u I =

k 1 if a t and a a k 1 if a t and a t

k otherwise

c n c

n * n n

l l

− = ≠

+ = ≠



 

 

where k = complemented symbol's number of u + uncomplemented sub- string(s) symbol's number of u .

Routing in interconnection net- works is the vital factor which decides the efficiency and throughput of the in- terconnection networks. Minimal path routing is preferred because the number of hops traveled by the messages are important in the interconnection net- works.

Lemma 2 In CR

, for an arbitrary node a a

L a

route to identity node I t t =

L . The Simple_Route algo- t

rithm at most increases by 1 step than the Optimal_Route algorithm.

Lemma 3 The algorithm of Opti-

mal_Route correctly computes a shortest

path from an arbitrary node a a

L a

to the identity node I = t t

L . t

Theorem 2 For an arbitrary node a a

L in CR a

, the algorithm Opti- mal_Route generates a path of length

≤ + n 1.

Theorem 3 The diameter of the graph CR

is given by D G (

) = +1. n

Routing a message from a node x to a node y in any Cayley graph is the same as routing from node xy

to the identity permutation[9]. The identity node is at level zero of the routing tree, and there are n +1 levels ranging from 0 to ( n +1 . The links of every node in ) CR

are marked by g and f

, for 1 ≤ ≤ − k n 1 ,where communication takes place along links.

Lemma 4 The algorithm of Fault_Tolerant Static Route correctly computes a path from an arbitrary node

a a

L to the identity node I . a

Theorem 4 The algorithm of Fault_Tolerant Static Route can tolerate at least n −1 node faults and the rout- ing distance at most increases by 6 steps than optimal routing distance.

Lemma 5 The graph G r ( ) is a greedy spanning tree rooted from node r in

CR

.

Lemma 6 Let B r ( ) be the broadcast- ing spanning tree rooted from node r in CR

where B r s ( )' vertices =

G r s ( )' vertices and B r s ( )' edges = reversing the direction of all edges of

G r ( ) .

Theorem 5 The graphs CR

has n node-disjoint paths whose lengths

≤ + n 3.

Theorem 6 ∆

( CR

) = n + 3.

Theorem 7 The fault diameter of CR

is n + 3.

Theorem 8 A hypercube of dimension

 

n + log n can be embedded into an n -dimension complete-rotation graph using Binary Reflected Gray Code (BRGC) with dilation 3 and expansion

 

n × 2

/ 2

.

Theorem 9 A hypercube of dimension n+logn-1 can be embedded into an n+logn-1 dimension complete-rotation graph using Binary Reflected Gray Code (BRGC) with dilation 3 and expan- sion(n+logn-1) ×2

/2

where

n = 2 , k ≥ 2 .

Theorem 5.3 A mesh of n ×2 can be

embedded into an n -dimension com- plete-rotation using Binary Reflected Gray Code (BRGC) with dilation 3 and expansion 1.



The complete-rotation network has smaller diameter than those of the triva- lent Cayley graph and the degree four Cayley graph. In particular, when we construct the network using the same nodes as those of a hypercube, the com- plete-rotation has smaller degree and diameter than those of the hypercube.

Regardless of its small diameter,

many parallel algorithms can be exe-

cuted in complete-rotation with the same

time complexity as in hypercube, mesh,

cube-connected-cycles, trivalent Cayley

graph and degree four Cayley graph. In

addition, we can find that the degree,

diameter, and fault diameter of the com-

plete-rotation graph is smaller than those of the hypercube when these two graphs have the same number of vertices. Thus the complete-rotation graph is a good alternative for massively parallel system.

 

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