Almost optimal permutation routing on hypercubes

• Authors: Berthold Vöcking
• Venue: STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
• Year: 2001

This paper deals with permutation routing on hypercube networks in the store-and-forward model. We introduce the first (on-line and off-line) algorithms routing any permutation on the d-dimensional hypercube in d+o(d) steps. The best previously known results were 2d+o(d) (oblivious on-line) and 2d-3 (off-line). In particular, we presenta randomized, oblivious on-line algorithm with routing time d + O(d/log d),a matching lower bound of d + &OHgr;(d/log d) for (randomized) oblivious on-line routing, anda deterministic, off-line algorithm with routing time d+O(\sqrt{d\log d}).Previous algorithms lose a factor of two mainly because packets are first sent to intermediate destinations in order to resolve congestion. As a consequence, the maximum path length becomes 2d - o(d). Our algorithms use intermediate destinations as well, but we introduce a simple, elegant trick ensuring that the routing paths are not stretched too much. In fact, we achieve small congestion using paths of length at most d.The main focus of our work, however, lies on the scheduling aspect. On one hand, we investigate well-known and practical scheduling policies for on-line routing, namely Farthest-to-Go and Nearest-to-Origin. On the other hand, we present a new off-line scheduling scheme that is based on frugal colorings of multigraphs. This scheme might be of interest for other sparse scheduling problems, too.