Close to linear space routing schemes

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 from s , routes a message from s to t on a path whose length is O(k +m^1/k) . The total space used by our routing scheme is mn^O(1/√(log n)) , which is almost linear in the number of edges of the graph. We present also a routing scheme with n^O(1/√(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 kn^O(1/√(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 ( 2009 ), that was presented in the context of dynamic graph algorithms, with several new ideas and observations.