Close to Linear Space Routing Schemes

Abstract

Let G = (V,E) be an unweighted undirected graph with n-vertices and m-edges, and let k > 2 be an integer. We present a routing scheme with a poly-logarithmic header size, that given a source s and a destination t at distance Δ from s, routes a message from s to t on a path whose length is O(kΔ + m ^1/k). The total space used by our routing scheme is $\tilde{O}(mn^{O(1/\sqrt{\log n})})$, which is almost linear in the number of edges of the graph. We present also a routing scheme with $\tilde{O}(n^{O(1/\sqrt{\log n})})$ header size, and the same stretch (up to constant factors). In this routing scheme, the routing table of every v ∈ V is at most $\tilde{O}(kn^{O(1/\sqrt{\log n})}deg(v))$, where deg(v) is the degree of v in G. Our results are obtained by combining a general technique of Bernstein [6], that was presented in the context of dynamic graph algorithms, with several new ideas and observations.