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 \(\varDelta \) from s, routes a message from s to t on a path whose length is \(O(k\varDelta +m^{1/k})\). The total space used by our routing scheme is \(mn^{O(1/\sqrt{\log n})}\), which is almost linear in the number of edges of the graph. We present also a routing scheme with \(n^{O(1/\sqrt{\log n})}\) header size, and the same stretch (up to constant factors). In this routing scheme, the routing table of every \(v\in V\) is at most \(kn^{O(1/\sqrt{\log n})}deg(v)\), where deg(v) is the degree of v in G. Our results are obtained by combining a general technique of Bernstein (2009), that was presented in the context of dynamic graph algorithms, with several new ideas and observations.