A compact routing scheme and approximate distance oracle for power-law graphs

Compact routing addresses the tradeoff between table sizes and stretch, which is the worst-case ratio between the length of the path a packet is routed through by the scheme and the length of an actual shortest path from source to destination. We adapt the compact routing scheme by Thorup and Zwick [2001] to optimize it for power-law graphs. We analyze our adapted routing scheme based on the theory of unweighted random power-law graphs with fixed expected degree sequence by Aiello et al. [2000]. Our result is the first analytical bound coupled to the parameter of the power-law graph model for a compact routing scheme.

Let n denote the number of nodes in the network. We provide a labeled routing scheme that, after a stretch--5 handshaking step (similar to DNS lookup in TCP/IP), routes messages along stretch--3 paths. We prove that, instead of routing tables with Õ(n^1/2) bits (Õ suppresses factors logarithmic in n) as in the general scheme by Thorup and Zwick, expected sizes of O(n^γ log n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n^1+γ log n), with γ = τ-22/τ-3 + ε, where τ∈(2,3) is the power-law exponent and ε 0 (which implies ε < γ < 1/3 + ε). Both bounds also hold with probability at least 1-1/n (independent of ε). The routing scheme is a labeled scheme, requiring a stretch--5 handshaking step. The scheme uses addresses and message headers with O(log n log log n) bits, with probability at least 1-o(1). We further demonstrate the effectiveness of our scheme by simulations on real-world graphs as well as synthetic power-law graphs.

With the same techniques as for the compact routing scheme, we also adapt the approximate distance oracle by Thorup and Zwick [2001, 2005] for stretch-3 and we obtain a new upper bound of expected Õ(n^1+γ) for space and preprocessing for random power-law graphs. Our distance oracle is the first one optimized for power-law graphs. Furthermore, we provide a linear-space data structure that can answer 5--approximate distance queries in time at most Õ(n^1/4+ε) (similar to γ, the exponent actually depends on τ and lies between ε and 1/4 + ε).