The performance of adaptive routers on worst case permutations
Chaotic routing [4, 13, 14] is a randomized, nonminimal adaptive routing algorithm for multicomputers. An adaptive routing algorithm is one in which the path a packet takes from its source to its destination may depend on other packets it encounters. Such algorithms potentially avoid network bottlenecks by routing packets around “hot spots.” Minimal adaptive routing algorithms have the additional advantage that the path each packet takes is a shortest one.
Chinn, Leighton, and Tompa  provide a lower bound for permutation routing problems on the n×n mesh for a large class of deterministic minimal adaptive algorithms. Specifically, they prove that for any such routing algorithm, there exists a permutation that requires Ω(n2/k2) steps to route all the packets in the permutation, where k is the number of packets a node can contain.
We present experimental results showing the performance of the Chaos router on permutations for which a deterministic minimal adaptive version of the Chaos router performs poorly. The results show that on these worst case permutations, the time the Chaos router takes to deliver all packets in the permutation closely fits a polynomial in n whose degree is 3/2. From these experiments, we conjecture that no practical router for the n×n mesh can route arbitrary permutations in time proportional to n, even though the mesh topology has the bandwidth to do so.
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