Interval Routing in Reliability Networks

In this paper we consider routing with compact tables in reliability networks. More precisely, we study interval routing on random graphs B p obtained from a base graph B by independently destroying each edge of B with a failure probability 1 p. We focus on additive stretched routing for n-node random graphs for which the base B is a square mesh and with p 0 5, that is the percolation model at the critical phase. We show a lower bound of Ω logn δ 1 on the number of intervals required per edge for every additive stretch δ 0. On the other side, our experimental results show that the size of the largest biconnected components is Θ n0 827 , and thus that there exists a trivial shortest-path routing scheme using at most O n0 827 intervals per edge. The results are extended to random meshes of higher dimension. We show that asymptotically almost surely, the number of intervals per edge for a random r-dimensional mesh with n nodes is Ω δ 1 1 rr 4 logn 1 1 r , for every additive stretch δ 0 and for every integral dimension r 1 log2 n .