A Compact Routing Scheme and Approximate Distance Oracle for Power-Law Graphs - Chen, Wei; Sommer, Christian; Teng, Shang-Hua; Wang, Yajun (ACM Transactions on Algorithms 9(1): 4 (2012). Extended abstract published in Distributed Computing (DISC), 2009., 2012)
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 a shortest path from source to destination. We adapt the compact routing scheme by Thorup and Zwick 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, Chung, and Lu. Our result is the first theoretical bound coupled to the parameter of the power-law graph model for a compact routing scheme. In particular, we prove that, for stretch 3, instead of routing tables with ˜O(n1/2) bits as in the general scheme by Thorup and Zwick, expected sizes of O(nγlog n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n1+γlog n), with γ=(τ-2)/(2τ-3)+ε, where τ in (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 and using addresses and message headers with O(log nlog 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 for stretch 3 and obtain a new upper bound of expected ˜O(n1+γ) for space and preprocessing for random power-law graphs.