Distributed computing on transitive networks: The torus
We identify the class of transitive networks as being of particular interest for distributed computation with known topology. This class includes the ring and the complete network as well as the 2-dimensional grid network with boundary connections, i.e. the torus. We consider the bit and message complexity of computing non-constant functions on an asynchronous torus where processors are identical except that they have one input bit. We show that every function can be computed with bit complexity O(n√n) on an √n × √n torus and show that this bound is optimal for many functions including AND, OR and XOR. We also give non-constant functions with a bit complexity linear in the number of processors on the torus. These bounds are better than those possible for the ring network.
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