The Complexity of Shortest Path and Dilation Bounded Interval Routing

Interval routing is a popular compact routing method for point-to-point networks which found industrial applications in novel transputer routing technology 13]. Recently much eeort is devoted to relate the eeciency (measured by the dilation or the stretch factor) to space requirements (measured by the compactness or the total number of memory bits) in a variety of compact We add new results in this direction for interval routing. For the shortest path interval routing we apply a technique from 4] to some intercon-nection networks (shuue exchange (SE), cube connected cycles (CCC), butterry (BF) and star (S)) and get improved lower bounds on compactness in the form (n 1=2?"), any " > 0, for SE, (p n=log n) for CCC and BF, and (n(log logn=logn) 5) for S, where n is the number of nodes in the corresponding network. Previous lower bounds for these networks were only constant 7]. For the dilation bounded interval routing we give a routing algorithm with the dila-tion d1:5De and the compactness O(p n logn) on n-node networks with the diameter D. It is the rst nontrivial upper bound on the dilation bounded interval routing on general networks. Moreover, we construct a network on which each interval routing with the dilation 1:5D ? 3 needs the compactness at least (p n). It is an asymptotical improvement over the previous lower bounds in 19] and it is also better than independently obtained lower bounds in 20].