A Necessary and Sufficient Condition for Deadlock-Free Wormhole Routing

An important open problem in wormhole routing has been to find a necessary and sufficient condition for deadlock-free adaptive routing. Recently, Duato has solved this problem for a restricted class of adaptive routing algorithms. In this paper, a necessary and sufficient condition is proposed that can be used for any adaptive or nonadaptive routing algorithm for wormhole routing, as long as only local information is required for routing. The underlying proof technique introduces a new type of dependency graph, the channel waiting graph, which omits most channel dependencies that cannot be used to create a deadlock configuration. The necessary and sufficient condition can be applied in a straightforward manner to most routing algorithms. This is illustrated by proving deadlock freedom for a partially adaptive nonminimal mesh routing algorithm that does not require virtual channels and a fully adaptive minimal hypercube routing algorithm with two virtual channels per physical channel. Both routing algorithms are more adaptive than any previously proposed routing algorithm with similar virtual channel requirements.