Deterministic routing with bounded buffers: turning offline into online protocols

In this paper we present a deterministic protocol for routing arbitrary permutations in arbitrary networks. The protocol is analyzed in terms of the size of the network and the routing number of the network. Given a network H of size n, the routing number of H is defined as the maximum over all permutations π on [n] of the minimal number of steps to route π offline in H. We can show that for any network H of size n with routing number R our protocol needs O(log_R n·R) time to route any permutation in H using only constant size edge buffers. This significantly improves all previously known results on deterministic routing. In particular our result yields optimal deterministic routing protocols for arbitrary networks with diameter Ω(n^ε) or bisection width O(n^1-ε), ε>0 constant. Furthermore we can extend our result to deterministic compact routing. This yields, e.g., a deterministic routing protocol with runtime O((log n)/(log log n) R) for arbitrary bounded degree networks if only O(log n) bits are available at each node for storing routing information. Our proofs use a new protocol for routing arbitrary r·s-relations in r-replicated s-ary Multibutterflies in optimal time O(log, n)