Close to linear space routing schemes

Let (Formula presented.) be an unweighted undirected graph with n vertices and m edges, and let (Formula presented.) 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 (Formula presented.) from s, routes a message from s to t on a path whose length is (Formula presented.). The total space used by our routing scheme is (Formula presented.), which is almost linear in the number of edges of the graph. We present also a routing scheme with (Formula presented.) header size, and the same stretch (up to constant factors). In this routing scheme, the routing table of every(Formula presented.) is at most (Formula presented.), 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.