Scale-Free Compact Routing Schemes in Networks of Low Doubling Dimension

Abstract

We consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2 α balls of half radius. There are two variants of routing-scheme design: (i) labeled (name-dependent) routing, in which the designer is allowed to rename the nodes so that the names (labels) can contain additional routing information, for example, topological information; and (ii) name-independent routing, which works on top of the arbitrary original node names in the network, that is, the node names are independent of the routing scheme. In this article, given any constant ϵ ∈ (0, 1) and an n -node edge-weighted network of doubling dimension α ∈ O (loglog n ), we present —a (1 + ϵ)-stretch labeled compact routing scheme with ⌈log n ⌉-bit routing labels, O (log 2 n /loglog n )-bit packet headers, and ((1/ϵ) O (α) log 3 n )-bit routing information at each node; —a (9 + ϵ)-stretch name-independent compact routing scheme with O (log 2 n /loglog n )-bit packet headers, and ((1/ϵ) O (α) log 3 n )-bit routing information at each node. In addition, we prove a lower bound: any name-independent routing scheme with o ( n (ϵ/60) 2 ) bits of storage at each node has stretch no less than 9 − ϵ for any ϵ ∈ (0, 8). Therefore, our name-independent routing scheme achieves asymptotically optimal stretch with polylogarithmic storage at each node and packet headers. Note that both schemes are scale-free in the sense that their space requirements do not depend on the normalized diameter Δ of the network. We also present a simpler nonscale-free (9 + ϵ)-stretch name-independent compact routing scheme with improved space requirements if Δ is polynomial in n .