Direct Routing

Direct routing is a special case of bufferless routing in which packets are not allowed to conflict with each other. The task is to compute the injection times of the packets so that they don’t conflict. A well known lower bound on the routing time of any algorithm is Ω(C + D), where the congestion C is the maximum number of paths that use any edge, and the dilation D is the maximum length of any path. We study the extent to which direct routing algorithms can achieve this lower bound. We present a simple greedy direct routing algorithm for arbitrary routing problems that has routing time O(C · D). We show that this routing time is worst case optimal. In particular, we construct a “hard” routing problem on the mesh, for which any direct routing algorithm has routing time Ω(C ·D), while C +D = Θ( √ C ·D). We then consider many interesting routing problems on commonly used network topologies. We show that variants of the simple greedy algorithm achieve optimal routing time. The routing problems and corresponding routing times (rt) are: i. Trees: arbitrary routing problems on arbitrary tree topologies (rt = O(C +D)). ii. Mesh with n nodes: permutations (rt = O( √ n)); arbitrary one-bend paths (rt = O(C +D)). iii. Butterfly with n inputs: random destinations and permutations (rt = O(lg n) w.h.p.). iv. Hypercube with n nodes: random destinations and permutations (rt = O(lg n) w.h.p.). Computer Science Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA. Email: [email protected] Computer Science Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA. Email: [email protected] Department of Computer Science, University of Cyprus, P. O. Box 20537, Nicosia CY-1678, Cyprus. Email: [email protected] Department of Computer Engineering and Informatics, University of Patras, Rion, 265 00 Patras, Greece, & Computer Technology Institute, P. O. Box 1122, 261 10 Patras, Greece. Email: [email protected]