Space-eeciency for Routing Schemes of Stretch Factor Three

We deal with routing algorithms on arbitrary n-node networks. A routing algorithm is a deterministic distributed algorithm which routes messages from any source to any destination. It includes not only the classical routing tables, but also the routing algorithm that generates paths with loops. Our goal is to design routing algorithms which minimize, for each router of the network, the amount of routing information that needs to be stored by the router in order to implement its own local routing algorithm. So as to simplify the implementation of a routing algorithm, names of the routers can be chosen in advance. We take also into account the eeciency of the routing, i.e., the length of the routing paths. The stretch factor is the maximum ratio, taken over all source-destination pairs, between the length of the paths computed by the routing algorithm and the distance between the source and the destination. We show that there exists an n-node network on which every routing algorithm of stretch factor s < 3 requires at least a total of (n 2) bits of routing information, whereas for stretch factor s = 3 the best known upper bound is O(n 3=2 log 2 n) bits in total 1]. We show a similar gap for the space complexity of routing schemes on the subclass of networks of diameter 2.