| Binary tree structures have been very useful in solving divide-and-conquer type of problems. Embedding binary trees into another network|the host network|helps in designing solutions for the host network using the known solutions on binary trees. Embedding arbitrary binary trees into networks, in particular into the hypercube, has been addressed in the literature. The latter was achieved with load 1 and constant dilation. The n-star graph is a recently introduced interconnection network for massively parallel systems. It enjoys symmetry and fault tolerance properties that make it a viable alternative to the hypercube. In this paper, we address the problem of embedding arbitrary binary trees into the n-star graph. This work is the rst to present such an embedding. The tree has be(n ? 1)!c or fewer vertices. The embedding leads to load 1 and constant dilation for all values of n. It therefore enables the star graph to e ciently simulate an arbitrary binary tree with only a constant factor of communication delay. Keywords| Star graph, interconnection network, massively parallel, binary tree, embedding, load, dilation, tree partitioning.

In this paper we present a routing algorithm that uses the depth rst search approach combined with a backtracking technique to route messages on the star graph in the presence of faulty links. The algorithm is distributed and requires no global knowledge of faults. The only knowledge required at a node is the state of its incident links. The routed message carries information about the followed path and the visited nodes. The algorithm routes messages along the optimal, i.e., the shortest path if no faults are encountered or if the faults are such that an optimal path still exists.In the absence of an optimal path, the algorithm always nds a path between two nodes within a bounded number of hops if the two nodes are connected. Otherwise, it returns the message to the originating node. We provide a performance analysis for the case where an optimal path does not exist. We prove that for a maximum of n ? 2 faults on a graph with N = n! nodes, at most 2i + 2 steps are added to the path, where i is O( p n).Finally, we use the routing algorithm to present an e cient broadcast algorithm on the star graph in the presence of faults.

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