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<p><it>Abstract</it>—We consider a distributed broadcasting algorithm for injured hypercubes using incomplete spanning binomial trees. An injured hypercube is a connected hypercube with faulty nodes. The incomplete spanning binomial tree proposed in this paper is a useful structure for implementing broadcasting in injured hypercubes. It is defined as a subtree of a regular spanning binomial tree that connects all the nonfaulty nodes. We show that in an injured <it>n</it>-dimensional hypercube with <it>m</it> faulty nodes, there are at least 2<super><it>n</it></super>− 2<super><it>m</it></super> source nodes (called <it>l</it>-nodes), each of which can generate an incomplete spanning binomial tree. A method is proposed to locate a large subset of the <it>l</it>-node set using the concept of safety level. The safety level of each node in an <it>n</it>-dimensional hypercube can be easily calculated through <it>n</it>− 1 rounds of information exchange among neighboring nodes. An optimal broadcast initiated from a safe node is proposed. When a nonfaulty source node is unsafe and there are at most <it>n</it>− 1 faulty nodes in an injured <it>n</it>-dimensional hypercube, the proposed broadcasting scheme requires at most <it>n</it>+ 1 steps.</p>
Binomial trees, broadcasting, fault tolerance, hypercubes.
"Safety Levels-An Efficient Mechanism for Achieving Reliable Broadcasting in Hypercubes", IEEE Transactions on Computers, vol. 44, no. , pp. 702-706, May 1995, doi:10.1109/TC.1995.10003
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