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DEFINITION AND ANALYSIS OF ORTHOGONAL NETWORKS Dindar Öz MS Student Boğaziçi University Department Of Computer Engineering ABSTRACT In this paper, a class of interconnection network topologies is explained. Orthogonal binary vectors are main components of this connection model. Firstly, orthogonal graphs is defined by two integers ,m , n , and a set Q it has 2m nodes and 2m - n links for each mode defined in the set Q.The name “orthogonal” comes from the connection propeerty of two nodes, which is explained later. 1. INTRODUCTION Consider we have p processors and p*p memory modules. Each processors has an id and their acces rule is defined according to the that id. Think each memory module has an index of i,j ,where Mi,j denotes memory module of ith row and jth column. Then the acces rule of this system is defined as Pk accesses Mij if k= i for all j ( By rows ) or k= j for all i ( By columns Here ,it can be easily seen that each processors has access to exactly 2p-1 memory modules, and each memory modules ,except an the diagonal, are acessed by 2 processors. We define this acces rule by vector operations to make calculations and analysis easier. If we write the indices of modules and processors in binary mode then we get binary vectors. Then the access rule becomes; If we generalize the access rule by mode q then The rule simply states that, the processor Px has access to the memory module My if its index matches on n bits starting at the bit position q. 2. ORTHOGONAL GRAPHS Let Ym denotes the set of all binary vectors of length m. So the cardinality of Ym is 2^m. For example if m= 2 then Ym = { 00, 11, 01, 10 }. Let Q be the orderet set of nonnegative integers smaller than m. For m=2 , Q = { 0, 1} . We define a new oparation · as below; The · operator maps a vector from Ym to Ym by shifting the entering vector’s bits to left by a given integer q . Let z be the special vector of length m which has starting n bits as 1 and all other remaaining bits as 0 . The inner product of mode q is defined as; and if inner product is equal to 0 then two vectors are said to be orthogonal mode of q The previous result appears also here. Two vectors are orthogonal if and only if they match on n bits starting at bit position q. An orthogonal graph G (n,m,Q * ) where Q * Í Q , is an undirected graph with 2m nodes. Alink (edge) exists between two nodes y and y’ if and only if there exists a q Î Q * such that y ^ q y’ . N(q) is defined as the number of nearest neighbours under modeq Theorem 1 (Degree) . The degree of an orthogonal graph G(n,m, Q * ) is given by D = N( q1 ) + N( q2 ) + .... + N( q# Q * )+ - N ( q1 , q2 ) – N ( q1 , q3 ) - .... N( q# Q * - 1 , q# Q * ) - N( q1 , q2 ,... q# Q * ) where # Q * denotes rhe cardinality of Q * . Here the expression is acquired by summing up all neighbours in all modes in Q * . İf we calculate the number of neigbours for a single mode in Q * , it can be easily seen that N(q) = 2m - n where q Î Q * For special set of Q * , there may be no neighbour that is connected by more than one mode. This special set is called set of disjoint modes. In this case, since all N() functions which takes more than one modes as a parameter drops to zero. Then the degree expression gets simpler; D = ( 2m - n - 1 ) ´ # Q * , if Q * is disjoint set of modes. Theorem 2 ( Connectivity ) An orthogonal graph G (n,m, Q * ), where Q * = ( q0, q1,...ql ) , l £ m is of one component ( connected ) if and only if q0 · z n Ù q1 · z n Ù ......ql · z n = 0 . This equation simply means that the zeros in the mask vectors of set Q * must cover all the bitso of length m. If this coverage happens then it gets possible to go from any node two any other node in the graph by changing 0 bits in some mask vector in Q * , which implies connectivity. Theorem 3 (Diameter) Given a connected orthogonal graph G(n, m, Q * ) there exists a set Q |* Í Q * whose cardinality is k such that é m ù ê m ú ê mod(m - n + 1) ú m ê ú£ k £ 2 ê ú+ ê ú ê - nú m ê - n + 1ú ë ëm û ê m- n ú û In the best case every modes covers m-n different bits ( 0-bits covers, and there are m-n bits in a mask vector). For all bits to be covered , m / (m-n) modes is used. So the left side of the inequality is qlear. Consider the worst case. Every bit position canbe covered by at most two different modes in Q * uniquely. If the third mode existed , then one of those three modes would be obsolete, and therefore should not have been taken in diameter calculation. So 2* (m/ (m-n)) is the worst case. 3.OMEGA GRAPHS Omega graph is a subset of orthogonal graph with disjoint set of modes. It requires that we choose m and n such that for some integer w ³ 2 , m = w (m-n) or m = wm/(w-1). Q * is such that for all qi and qi + 1 in Q * qi + 1 - qi mod m = m - n and # Q * = w . These graphs called w - graphs wG (n , m ) . 4 G (3 , 4) , 2 G ( 2 , 4) are some examples. Note that m G (m-1 , m) is special graph called hypercube. ( See Figure 1) Figure 1. 4 G( 3, 4), Hypercube 4.CONCLUSION A special connection topology , orthogonal graph is introduced. Orthogonal multiprocessing systems ( OMP ) provides p processors accesses to p*p memory modules. The definition of orthogonal graphs is very general definition and based on binary vector operatios. Access rule in orthogonal systems is defined by vector orthogonality. By means of these rules , we can say that , there is no resource conflict in orthogonal systems since system allows only one processors to access to a certain memory modules in a certain time. Omega graphs , hyper cubes and MDA systems are all subset of our general orthogonal network set. 5.REFERENCES 1. “Definition and Analysis of a Class of Spanning Bus Orthogonal Multiprocessing Systems” Isaac D.Scherson Department of Electrical Engineering Princeton University Princeton New Jersey 08544

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posted: | 12/27/2011 |

language: | English |

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