A trade-off between space and efficiency for routing tables

Two conflicting goals play a crucial role in the design of routing schemes for communication networks. A routing scheme should use paths that are as short as possible for routing messages in the network, while keeping the routing information stored in the processors' local memory as succinct as possible. The efficiency of a routing scheme is measured in terms of its stretch factor -the maximum ratio between the length of a route computed by the scheme and that of a shortest path connecting the same pair of vertices. Most previous work has concentrated on finding good routing schemes (with a small fixed stretch factor) for special classes of network topologies. In this paper the problem for general networks is studied, and the entire range of possible stretch factors is examined. The results exhibit a trade-off between the efficiency of a routing scheme and its space requirements. Almost tight upper and lower bounds for this trade-off are presented. Specifically, it is proved that any routing scheme for general n -vertex networks that achieves a stretch factor k ≥ 1 must use a total of Ω( n 1+1/(2 k +4) ) bits of routing information in the networks. This lower bound is complemented by a family K ( k ) of hierarchical routing schemes (for every k ≥ l) for unit-cost general networks, which guarantee a stretch factor of O ( k ), require storing a total of O ( k 3 n 1+(1/h) log n )- bits of routing information in the network, name the vertices with O (log 2 n )-bit names and use O (log n )-bit headers.