Optimal-stretch name-independent compact routing in doubling metrics

We consider the problem of name-independent routing in doubling metrics. A doubling metric is a metric space whose doubling dimension is a constant, where the doubling dimension of a metric space is the least value α such that any ball of radius r can be covered by at most 2^α balls of radius r/2. Given any δ > 0 and a weighted undirected network G whose shortest path metric d is a doubling metric with doubling dimension α, we present a name-independent routing scheme for G with (9+δ)-stretch, (2+ 1/δ)^O(α)(log Δ)²(log n)-bit routing information at each node, and packet headers of size O(log n), where Δ is the ratio of the largest to the smallest shortest path distance in G. In addition, we prove that for any ε ∈. (0, 8), there is a doubling metric network G with n nodes, doubling dimension α ≤ 6 - log ε, and Δ = O(2^1/εn) such that any name-independent routing scheme on G with routing information at each node of size o(n ^(ε60)2)-bits has stretch larger than 9 - ε. Therefore assuming that Δ is bounded by a polynomial on n, our algorithm basically achieves optimal stretch for name-independent routing in doubling metrics with packet header size and routing information at each node both bounded by a polylogarithmic function of n.