Interval routing in reliability networks

Abstract

In this paper, we consider routing with compact tables in reliability networks. More precisely, we study interval routing on random graphs G ( B , p ) obtained from a base graph B by independently removing each edge 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 p = 0.5 , that is the percolation model at the critical phase. We show a lower bound of Ω ( log n / ( δ + 2 ) ) 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 ( n 0.827 ) , and thus that there exists a trivial shortest-path routing scheme using at most O ( n 0.827 ) 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 ( 16 - r ( δ + 2 ) 1 - r r - 4 ( log n ) 1 - 1 / r ) , for every additive stretch δ ⩾ 0 and for every integral dimension r ∈ [ 1 , log 2 n ] .