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their own routing algorithms Hence, their produced schedules can only be accurate and ef cient if the target parallel system can be directed to use the computed routes However, that is rarely the case in a generic parallel system, especially at the application level (Culler and Singh [48]) Fortunately, using the target system s own routing algorithm does not imply that for every target system a different routing algorithm must be mplemented in scheduling Most parallel computers employ minimal routing, which means they choose the shortest possible path, in terms of number of edges, through the network for every communication An example is dimension ordered routing in a multidimensional mesh (Culler and Singh [48], Kumar et al [108]) A message is rst routed along one dimension, then along the second dimension, and so on until it reaches its destination Given the graph based representation of the network, nding a shortest path can be accomplished with a BFS (breadth rst search Algorithm 1) Thus, the BFS can be used as a generic routing algorithm in the topology graph, which serves, at least, as a good approximation in many cases Shortest Path in Topology Graph with BFS Although BFS is an algorithm for directed and undirected graphs, it can readily be applied to the topology graph The only graph concept used in BFS is that of adjacency (in the for loop, Algorithm 1) As this is already de ned for directed and undirected edges (Section 31), it only remains to de ne adjacency for hyperedges De nition 74 (Adjacency Hyperedge) Let V be a nite set of vertices and H a nite set of hyperedges A vertex u V, u H, H H is adjacent to all vertices v H-u and all vertices v H-u are adjacent to u The set H H:u H H-u of all vertices adjacent to u V is denoted by adj(u) Now in the topology graph, the total set of all vertices adjacent to a given vertex u is the union of the two adjacent sets induced by the directed edges and the hyperedges So with this de nition of vertex adjacency, the BFS can be applied to the topology graph without any modi cation and returns a shortest path in terms of number of edges Complexity As routing depends on the algorithm of the target parallel system, there is no general time complexity expression that is valid for all networks On that account, the routing complexity shall be denoted generically by O(routing) The algorithm for routing in regular networks is usually linear in the number of network vertices or even of constant time For example, in a fully connected network it is obviously O(1), as it is in a network with one central switch (Figure 77(b)) In a topology graph for a mesh network of any dimension (Section 221), it is at most linear in the number of processors O(P) Whenever it is possible to calculate the routes for a given system once and then to store them; for example, in a table, O(routing) is just the complexity of the length of the route For example, in a ring network the length of a route (ie, the number of links) is O(P); hence, O(routing) O(P). convert pdf to word using itextsharp c# Free libraries SDK to generate Word and PDF file | The ASP.NET Forums
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